Local and Analytic Cyclic Homology (EMS Tracts in Mathematics)

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EMS Tracts in Mathematics 3

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EMS Tracts in Mathematics Editorial Board: Carlos E. Kenig (The University of Chicago, USA) Andrew Ranicki (The University of Edinburgh, Great Britain) Michael Röckner (Universität Bielefeld, Germany, and Purdue University, USA) Vladimir Turaev (Institut de Recherche Mathématique Avancée, Université Louis Pasteur et CNRS, Strasbourg, France) Alexander Varchenko (The University of North Carolina at Chapel Hill, USA) This series includes advanced texts and monographs covering all fields in pure and applied mathematics. Tracts will give a reliable introduction and reference to special fields of current research. The books in the series will in most cases be authored monographs, although edited volumes may be published if appropriate. They are addressed to graduate students seeking access to research topics as well as to the experts in the field working at the frontier of research. 1 Panagiota Daskalopoulos and Carlos E. Kenig, Degenerate Diffusions 2 Karl H. Hofmann and Sidney A. Morris, The Lie Theory of Connected Pro-Lie Groups

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Ralf Meyer

Local and Analytic Cyclic Homology

M

M

S E M E S

S E M E S

European Mathematical Society

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Author: Ralf Meyer Mathematisches Institut Georg-August-Universität Göttingen Bunsenstraße 3–5 37073 Göttingen Germany E-Mail: [email protected]

2000 Mathematics Subject Classification: 19-02, 46-02; 46L80, 46A17, 46H30, 19D55, 19K35. Key words: Cyclic homology, bornology, Banach algebra, non-commutative geometry, functional calculus, K-theory.

ISBN 978-3-03719-039-5 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2007 European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum FLI C4 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: info @ems-ph.org Homepage: www.ems-ph.org Typeset using the authors’ TEX files: I. Zimmermann, Freiburg Printed in Germany 987654321

Preface

This monograph deals with two closely related variants of periodic cyclic homology, called analytic and local cyclic homology. I introduced analytic cyclic homology in my doctoral thesis ([65]) in 1999, based on the entire cyclic cohomology due to Alain Connes. There I observed the relevance of bornological vector spaces and studied the formal properties of analytic cyclic homology, including a more conceptual proof of the excision theorem. I was inspired by the work of Michael Puschnigg on local cyclic homology and Joachim Cuntz’s construction of bivariant K-theories ([17]), which had emphasised the role of abstract homological properties. Large parts of this monograph are based on my thesis. But as quite some time has passed since then, I have rewritten it almost entirely. The main change in content is the inclusion of bivariant local cyclic homology. When formulated suitably, this theory is quite close to the analytic theory. The main difference is that complete bornological vector spaces are replaced by inductive systems of Banach spaces. Although the definitions are similar, the local theory has much better formal properties. Much of the additional material is needed to define the local theory and establish its nice properties. Although we also deal with periodic cyclic homology here, this mainly serves to point out similarities and differences between the periodic and the analytic and local cyclic theories. If you are unfamiliar with periodic cyclic homology, you may want to consult [20] first in order to get a rough idea of its basic properties.

Contents

Preface

v

Introduction

1

1

2

Bornological vector spaces and inductive systems 1.1 Basic definitions . . . . . . . . . . . . . . . . 1.2 Some functional analysis . . . . . . . . . . . 1.3 Constructions with bornological vector spaces 1.4 Categories of inductive systems . . . . . . . . 1.5 Dissecting bornological vector spaces . . . . 1.6 Metrisability and the approximation property

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9 13 22 29 53 59 67

Relations between entire, analytic, and local cyclic homology 2.1 Several definitions . . . . . . . . . . . . . . . . . . . . . 2.2 Comparison of analytic and local cyclic homology . . . . 2.3 The local homotopy category of chain complexes . . . . 2.4 Some counterexamples with compact Lie groups . . . .

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3 The spectral radius of bounded subsets and its applications 3.1 The spectral radius . . . . . . . . . . . . . . . . . . . 3.2 Locally multiplicative bornological algebras . . . . . . 3.3 Analytically nilpotent bornological algebras . . . . . . 3.4 Isoradial homomorphisms . . . . . . . . . . . . . . . . 3.5 Examples of isoradial homomorphisms . . . . . . . . . 3.6 Isoradial hulls of subalgebras . . . . . . . . . . . . . . 3.7 Passage to inductive systems . . . . . . . . . . . . . . 4

Periodic cyclic homology via pro-nilpotent extensions 4.1 Pro-algebras . . . . . . . . . . . . . . . . . . . . 4.2 Homotopy invariance of periodic cyclic homology 4.3 Excision in periodic cyclic homology . . . . . . . 4.4 Exterior products . . . . . . . . . . . . . . . . .

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5 Analytic cyclic homology and analytically nilpotent extensions 5.1 Lanilcurs and the analytic tensor algebra . . . . . . . . . . 5.2 Analytic cyclic homology via analytic tensor algebras . . . 5.3 Basic properties . . . . . . . . . . . . . . . . . . . . . . . 5.4 Excision in analytic and local cyclic homology . . . . . .

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viii 6

Contents

Local homotopy invariance and isoradial subalgebras 6.1 Local homotopy equivalences . . . . . . . . . . . 6.2 Approximate local homotopy equivalences . . . . 6.3 Application to isoradial homomorphisms . . . . . 6.4 Local and approximate local homotopy category .

7 The Chern–Connes character 7.1 The bivariant Chern–Connes character . . . . 7.2 The Universal Coefficient Theorem . . . . . . 7.3 The character for idempotents and invertibles 7.4 Finitely summable Fredholm modules . . . . 7.5 The character for general Fredholm modules .

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259 260 266 270 273 279

Appendix. Algebraic preliminaries A.1 Chain complexes over additive categories . . . A.2 Basic constructions with algebras and modules . A.3 Non-commutative differential forms . . . . . . A.4 Fedosov type products . . . . . . . . . . . . . A.5 Homological algebra for modules . . . . . . . . A.6 Hochschild homology and cyclic homology . . A.7 Biprojective algebras . . . . . . . . . . . . . .

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291 291 298 308 314 321 328 338

Bibliography

343

Notation and Symbols

349

Index

355

Introduction

The two basic kinds of homology theories for algebras that are considered in non-commutative geometry are K-theory and cyclic .co/homology. Periodic cyclic homology  HP .A/ extends de Rham cohomology HdR .M / for manifolds in the sense that   HPn C 1 .M / Š

M

m HdR .M /

m2nC2Z

for any smooth compact manifold M ; here C 1 .M / denotes the Fréchet algebra of smooth functions on M . The dual theory HP .A/ is called periodic cyclic cohomology. Both are special cases of bivariant periodic cyclic homology HP .A; B/. We are going to study two closely related (in fact, almost identical) variants of periodic cyclic homology called analytic cyclic homology HA and local cyclic homology HL . Both have dual cohomology theories and extend to bivariant theories. Several of the tools we develop to study these theories are useful for other purposes as well. Therefore, even if you are not at all interested in cyclic homology, you may find that Chapters 1 and 3 and parts of Chapter 2 contain interesting ideas in functional analysis; the construction in Chapter 6 can also be applied in the context of bivariant K-theory. The Appendix contains a brief survey of some preliminaries on homological algebra in symmetric monoidal categories and constructions with differential forms. Each chapter has its own introduction which summarises its contents. In this introduction to the whole book, we first discuss the shortcomings of periodic cyclic homology that justify the existence of analytic and local cyclic homology, and we discuss their relationship to the entire cyclic cohomology of Alain Connes. Then we discuss some of the preliminaries from functional analysis and algebra that we need to understand analytic and local cyclic homology and mention a few particularly important ideas, including excision and invariance for isoradial subalgebras.

What is wrong with periodic cyclic homology? We get HP .A/ by a limiting process from the cyclic homology HC .A/, which is in turn computable from Hochschild homology HH .A/ by a spectral sequence. Hence periodic cyclic homology only gives reasonable results for algebras whose Hochschild homology is sufficiently rich. Since Hochschild homology is closely related to differential forms, this requires a certain amount of differentiability. Therefore, periodic cyclic homology yields poor results for C  -algebras; similarly, de Rham cohomology only makes sense for smooth manifolds. When we study a C  -algebra A, we use HP .A1 / for an appropriate dense subalgebra A1 , which plays the role of the subalgebra C 1 .M / of smooth functions in the algebra C .M / of continuous functions.

2

Introduction

This works well enough in many concrete cases. But HP .A1 / may depend on the choice of the subalgebra A1 . For instance, consider a connected Lie group G, say, the circle group T 1 . Let C  .G/ be the group C  -algebra of G. Let C 1 .G/  C  .G/ be the dense subalgebra of smooth functions. Let R.G/  C 1 .G/ be the dense subalgebra that is spanned by the matrix coefficients of irreducible representations. Both R.G/ and C 1 .G/ behave like 0-dimensional spaces, so that their periodic cyclic homology agrees with the 0th Hochschild homology, HH0 . The latter turns out to be different for R.G/ and C 1 .G/ (see §2.4). Although C 1 .G/ is a more obvious choice of smooth subalgebra, R.G/ gives better results in the sense that the Chern–Connes character K .A/ ˝Z C Š K .A1 / ˝Z C ! HP .A1 / is an isomorphism for A1 D R.G/, but not for A1 D C 1 .G/. Another drawback of periodic cyclic cohomology is its built-in finite dimensionality: it only admits Chern–Connes characters for finitely summable Fredholm modules.

Entire to analytic cyclic homology and cohomology Entire cyclic cohomology extends periodic cyclic cohomology by, roughly speaking, allowing infinite-dimensional cochains with sufficiently slow growth. The main application is to construct Chern–Connes characters for #-summable Fredholm modules (the summability condition restricts the growth of the singular values of certain compact operators). The JLO cocycle defined in [53] (and named after its creators Jaffe, Lesniewski, and Osterwalder) provides an explicit formula for such a character. The JLO cocycle and related formulas are useful for index computations such as the proof of the local index formula of Alain Connes and Henri Moscovici ([10]); thus entire cyclic cohomology is interesting even for algebras like C 1 .M / for which we know that it coincides with periodic cyclic cohomology. Entire cyclic cohomology goes beyond Hochschild cohomology, but it does not yet address the first shortcoming of periodic cyclic cohomology mentioned above: a result of Masoud Khalkhali ([62]) yields HE .A/ D HP .A/ for nuclear C  -algebras, so that we seem to gain nothing. Nevertheless, some apparently small technical changes yield a theory with considerably better properties. First we pass from entire cyclic cohomology to the dual homology theory HE (see also [34]). To define this theory properly, we first need a category of algebras on which to define it. The growth condition for entire cyclic cochains only depends on the collection of (von Neumann) bounded subsets of A, which we call its von Neumann bornology. A bornology is a suitable family of subsets, called bounded subsets, and a bornological algebra is an algebra over R or C with a bornology such that the multiplication is bounded. This category of algebras is the natural domain for entire cyclic homology and cohomology. The chain complex that defines entire cyclic homology carries a bornology by construction, but it carries no obvious topology in general.

Introduction

3

We may choose another bornology on a topological algebra than the von Neumann bornology. In fact, our default choice is the bornology of precompact subsets because this yields considerably better results. To avoid confusion with the existing entire theories, we call our theories analytic cyclic homology and cohomology and denote them by HA and HA . Both are defined for complete bornological algebras. When we apply them to topological algebras, it is understood that we equip them with the precompact bornology; in contrast, entire cyclic homology and cohomology use the von Neumann bornology. Another reason to change notation is that the apparent importance of entire functions for entire cyclic cohomology is an artefact created by looking only at a cohomology theory. The relevant function algebra is the algebra C..t // of analytic power series, that is, power series with a non-zero radius of convergence. The space of entire functions only appears because it is the dual space of C..t //. Analytic power series in several noncommuting variables are a crucial ingredient in our conceptual approach to analytic cyclic homology. Unlike the periodic and entire cyclic theories, analytic cyclic homology yields good results for C  -algebras (with the precompact bornology). It is split-exact, C  -stable, invariant under continuous homotopies, and additive for C  -completions of infinite direct sums. Standard results related to the Universal Coefficient Theorem for Kasparov theory yield a Chern–Connes character K .A/ ˝ C ! HA .A/ and show that it is an isomorphism if A belongs to the bootstrap category (see §7.2). In particular, for locally compact topological spaces X we have   HA C0 .X; C/ Š K .X / ˝ C Š Hc .X I C/: We also construct a Chern–Connes character from the Kasparov K-homology K .A/ to analytic cyclic cohomology HA .A/; notice that we require no summability condition. This should suffice to make you curious about analytic cyclic homology. But a lot of preparation is needed until we can prove these results. The good news is that many of these preliminary results are useful for other purposes as well; therefore, large parts of this book may be useful to readers with no interest in cyclic homology. The main difficulty in learning about analytic cyclic homology is that it requires a good deal of both functional analysis and homological algebra; even more, the results that we need from these areas are omitted in most textbooks for beginners.

Background from functional analysis: bornologies Bornological vector spaces have not received much attention by functional analysts, although there are several situations where they work better than topological vector spaces. As a general rule, this happens whenever we combine (homological) algebra and functional analysis. I have used bornological vector spaces for problems in representation theory and homological algebra in [68]–[72]. We spend some time reviewing the basic theory of bornological vector spaces in Chapter 1.

4

Introduction

Besides basic issues of functional analysis, we particularly emphasise categorical constructions. Much of this is needed in order to discuss the relationship between complete bornological vector spaces and inductive systems of Banach spaces. This relationship is crucial for the more exciting results about analytic cyclic homology mentioned above. Since inductive systems are rather abstract objects, the only elegant way to deal with them is via general methods from category theory. We use some of these techniques already in the context of bornological vector spaces – where they are not yet so crucial – to exhibit the similarities between both setups. It is not surprising that tensor products are a crucial ingredient for our cyclic theories and therefore duely studied in Chapter 1. In addition, approximation properties play an important technical role in several proofs. In connection with tensor products, we emphasise symmetric monoidal categories. It has already been noticed by Guillermo Cortiñas and Christian Valqui that this is the right context for studying cyclic homology theories. When we study Hochschild homology or periodic cyclic homology for, say, Banach or complete locally convex topological algebras, then we use the completed projective tensor product and restrict attention to continuous maps. Here we study ! algebras in the categories Cborn of complete bornological vector spaces and Ban of inductive systems of Banach spaces. In each case, the definitions are essentially the same but employ different tensor products. Careful readers should wonder which properties of the cyclic theories extend to these more exotic kinds of algebras. The notion of a symmetric monoidal category formalises the basic associativity, commutativity, and monoidal properties of tensor products that we need to define algebras and modules and do homological algebra with them. All the basic features of Hochschild, cyclic, and periodic cyclic homology extend to algebras in any Q-linear symmetric monoidal category. Another crucial concept from functional analysis is the notion of . joint/ spectral radius for bounded subsets in bornological algebras. The spectral radius %.S / of a bounded subset S in a bornological algebra is defined exactly as for a single element: it is the infimum of the set of scalars r > 0 for which .r

1

1

S/

WD

1 [

.r 1 S /n

nD1

is bounded. This notion contains more information than the spectral radius of single elements because the elements of S need not commute. Roughly speaking, spectral radius estimates ensure that certain power series in several non-commuting variables converge. The joint spectral radius is an intrinsically bornological concept because it deals with subsets. Even for commutative algebras, where the joint spectral radius usually contains no more information than the spectral radius for single elements, the bornological framework is more suitable to study the functional calculus. This was noticed already by Lucien Waelbroeck ([107]). We use the spectral radius to define two classes of bornological algebras with good functional calculus. A bornological algebra is called locally multiplicative if

Introduction

5

%.S/ < 1 for all bounded subsets S, and analytically nilpotent if %.S / D 0 for all bounded subsets S . Both classes of algebras play an important role for analytic cyclic homology. The spectral radius generates a useful notion of smooth subalgebra. Let A and B be locally multiplicative bornological algebras. A bounded homomorphism  f W A ! B with dense range (in a suitable sense) is called isoradial if % f .S /I B D %.S I A/ for all bounded subsets S  A. In most applications, f is injective, so that we may view A as a dense subalgebra of B. The isoradiality condition means that A  B is closed under functional calculus in several non-commuting variables. We check that this condition behaves nicely with respect to various constructions like extensions and tensor products, and we study several important examples. It is much harder to get a good notion of smooth subalgebra in the context of topological algebras; the most useful definition is due to Bruce Blackadar and Joachim Cuntz (see [3]).

Some relevant algebraic notions ! We study algebras and modules in the somewhat exotic categories Cborn and Ban. As we have already explained, we use the framework of symmetric monoidal categories to deal with this. We recall a few basic facts about algebras and modules in this generality in Chapter 1 and in the Appendix. We also need various familiar results about Hochschild homology and periodic cyclic homology; we briefly discuss them in the Appendix, mostly in the general framework of symmetric monoidal categories. ! The categories Cborn and Ban are not Abelian, so that we need homological algebra over non-Abelian categories. A side effect of this is that the passage from chain complexes to homology forgets too much information. For chain complexes of vector spaces, the homology functor is an equivalence of categories from the homotopy category of chain complexes to the category of vector spaces; thus we lose no information at all. For chain complexes in more general Abelian categories, things already get more complicated; but at least we can detect exactness of chain complexes using homology. Once we are in non-Abelian categories, even this fails. The lesson is that we should not take homology and instead consider functors with values in suitable homotopy categories or derived categories of chain complexes. This is crucial in [69], [71], [72] where we need certain chain complexes to be contractible and not just exact. Similarly, we usually treat analytic cyclic homology as a functor HA to the category of chain complexes of bornological vector spaces. This point of view is already implicit in the definition of bivariant cyclic homology theories because the elements of the bivariant group HA0 .A; B/ are exactly the morphisms HA.A/ ! HA.B/ in a suitable homotopy category of chain complexes. The most sophisticated homological algebra that we need enters in the definition of bivariant local cyclic homology; our definition is essentially equivalent to the original one by Michael Puschnigg in [86], [88]; but we use a more efficient category of algebras. This both simplifies and generalises the theory.

6

Introduction

The dual theory HA is not as well-behaved as we would like; for instance, we   cannot compute HA .A/ for A WD C0 .0; 1 , although we know that HA.A/ is locally contractible, that is, on each bounded subset of HA.A/ we can define a contracting homotopy. But these maps do not fit together to a global map, and the failure of the Hahn–Banach Theorem for bornological vector spaces prevents us from computing the cohomology. To repair this defect, we replace ordinary cohomology by an appropriate derived functor (called local cohomology). This yields local cyclic cohomology and bivariant local cyclic cohomology. These constructions become more transparent in the context of inductive systems of chain complexes. The basic idea is that we replace an inductive system of chain complex by its homotopy direct limit. There is a variant HL .A/ of HA .A/ as well, which repairs the lack of exactness of the completion functor for bornological vector spaces. But this makes no difference in practice: we show in Chapter 2 that HL .A/ Š HA .A/ if A is a Fréchet algebra (with the precompact bornology) or if A has a suitable approximation property. Even though the two theories agree for all practical purposes, we must first distinguish them to prove such a statement. The assertions about the analytic cyclic homology of C  -algebras are based on this isomorphism and therefore implicitly use the local cyclic theory.

The Cuntz–Quillen approach to cyclic theories The chain complexes HA.A/ and HL.A/ are defined in Chapter 2 as completions of the cyclic bicomplex because this approach to cyclic homology is probably known to most readers. But later we switch to another chain homotopy equivalent complex that is more adequate for the analytic and local cyclic theories. The cyclic bicomplex relates periodic cyclic homology to Hochschild homology and thus provides a useful scheme to compute periodic cyclic homology and cohomology. But since the whole point of the analytic and local cyclic theories is to go beyond the limitations of Hochschild homology, the cyclic bicomplex offers no clues how to understand these theories. The approach of Joachim Cuntz and Daniel Quillen ([25], [26]) is more useful because it allows to treat periodic, analytic, and local cyclic homology in a very similar fashion. We use it both to prove properties shared by all three theories and to establish the special features of the local cyclic theory. The Cuntz–Quillen approach uses two ingredients: a completed tensor algebra and the X-complex. The X-complex is a very small quotient of the cyclic bicomplex, which is therefore easy to study by hand; we recall its definition in §A.6.3. The most important ingredient is the completed tensor algebra. The usual tensor algebra is defined by a universal property: algebra homomorphisms TA ! B correspond to linear maps A ! B; it is not so interesting because it completely forgets the algebra structure. The completed tensor algebras that we study have a similar universal property for linear maps that are almost multiplicative in a suitable sense.

Introduction

7

To make this precise, we need the curvature of a linear map f W A ! B, which is the bilinear map !f W A  A ! B;

.a1 ; a2 / 7! f .a1  a2 /  f .a1 /  f .a2 /:

The map f is an algebra homomorphism if and only if !f D 0. We say that f has analytically nilpotent curvature if the spectral radius of !f .S; S /  B vanishes for each bounded subset S  A. We briefly call such maps lanilcurs. The analytic tensor algebra T A is defined so that algebra homomorphisms T A ! B correspond to lanilcurs A ! B. This universal property is our main tool. We will see that HA.A/ is chain homotopy equivalent to X.T A/. The pro-tensor algebra that appears in the Cuntz–Quillen approach to periodic cyclic homology can be characterised by a similar universal property, which we state in Chapter 4. Various formal properties like homotopy invariance for smooth homotopies, invariance under nilpotent extensions, stability for algebras of nuclear operators, and additivity can be proved easily in this framework. Furthermore, it allows us to compute some simple examples. The deepest common property of the periodic, analytic, and local cyclic theories is excision. One way to formulate this is that an extension of algebras I  E  Q with a bounded linear section induces an exact triangle HA.I / ! HA.E/ ! HA.Q/ ! HA.I /Œ1

in the homotopy category of chain complexes; this induces various long exact sequences for homology. Excision is crucial for many computations. It was established by Joachim Cuntz and Daniel Quillen ([27]) for the periodic cyclic theory and by the author and Michael Puschnigg for the analytic and local cyclic theories. The elegant proof we present here appeared previously in [66].

Invariance for smooth subalgebras A remarkable property of local cyclic homology is that the obvious map i W C 1 .M / ! C.M / for a smooth compact manifold M is an HL-equivalence. More generally, the same statement holds for any isoradial homomorphism that has approximate bounded linear sections in a suitable sense. We want to indicate why this is true. Along the way, we see why it is important to use precompact bornologies and invert local chain homotopy equivalences. The following sketch is made more precise in Chapter 6. Any smoothing operator yields a bounded linear map s W C .M / ! C 1 .M /. There is a sequence .sn / of smoothing operators such that i ı sn and sn ı i converge towards the identity maps on C .M / and C 1 .M /; this convergence is not uniform in norm, but it is uniform on precompact subsets. This is the point where it is crucial to use the

8

Introduction

precompact bornology on C.M /. Since i ı sn converges towards  the identity map,  the curvature !iısn D i ı !sn converges towards !id D 0. Thus % i ı !sn .K; K/ ! 0; now we use that i is isoradial to conclude that % !sn .K; K/ ! 0. That is, the maps .sn / approximately have analytically nilpotent curvature for n ! 1. The same argument that shows that lanilcurs induce maps on T A also shows that maps with sufficiently small curvature such as sn induce maps on bounded subsets of T A. More precisely, for each bounded submultiplicative disk S  T A there is n 2 N such that sn induces an algebra homomorphism on the Banach subalgebra of T A generated by S . The composite maps i ı sn and sn ı i are homotopic to the identity map via an affine homotopy. Again, these homotopies have approximately analytically nilpotent curvature. As a result, the map i induces a local homotopy equivalence T C 1 .M / ! T C.M /. Since the X-complex for quasi-free algebras is homotopy    invariant (for smooth homotopies, say), the induced chain map X T C 1 .M / ! X T C .M / is a local chain homotopy equivalence. The definition of the bivariant local cyclic theory ensures that such chain maps become invertible. This finishes the proof that i is an HL-equivalence.

What is missing? Finally, there are a few important aspects of the infinite cyclic cohomology theories that we study that are left out in this book due to limitations of time, space, and energy. First, there is the JLO cocycle ([53]) and its bivariant generalisation by Denis Perrot ([79], [80]). Instead, I present another character construction that does not require any summability condition. It would be worthwhile to compare these constructions. Secondly, there is the work of Michael Puschnigg on the local cyclic homology of group algebras ([85], [87]). It is difficult to compute local cyclic homology by hand. Besides some easy cases, all the computations we shall do in this book use formal properties of the theory to reduce the problem to a K-theory computation. It is an important point that there are examples where local cyclic homology can be computed directly and is more accessible than K-theory. Thirdly, Christian Voigt has extended periodic and analytic cyclic cohomology theories to the equivariant setting for algebras with actions of groups or even quantum groups in [103]–[105]. This theory is also based on suitable completed tensor algebras, so that many of our arguments carry over to it.

Chapter 1

Bornological vector spaces and inductive systems

Both analytic and local cyclic homology require a certain amount of functional analysis. For this purpose, we need a suitable category of vector spaces with additional structure. For analytic cyclic homology, we use the category Born1=2 of bornological vector spaces and its full subcategories Cborn of complete and Born of separated bornological vector spaces. For local cyclic homology, these are replaced by the closely related ! ! ! categories Norm1=2, Norm, and Ban of inductive systems of semi-normed spaces, normed spaces, and Banach spaces. We are going to introduce these categories and explain how they are related and how to do functional analysis in them. Bornological vector spaces were never fashionable and have been neglected for some time. Nevertheless, I have found that they provide an ideal setting for bringing together (homological) algebra and functional analysis. Therefore, this chapter may also be interesting for readers who are not primarily interested in cyclic homology. We develop the theory of bornological vector spaces from scratch because it is hardly known even among functional analysts. Much of this theory goes back to Henri HogbeNlend and Lucien Waelbroeck ([46], [47], [49], [50], [106]–[108]). Given our interests, we emphasise algebraic and categorical properties of bornological vector spaces. We also include several results that are not immediately necessary for cyclic homology theories, but which are important for other homological computations. Since we only use convex bornologies, we drop the adjective “convex” from our notation and tacitly require all bornologies to be convex vector bornologies. The categories of inductive systems mentioned above have even better algebraic properties, at the expense of being less concrete. Therefore, we must use general notions of category theory to deal with them. In order to understand these abstract notions, we usually compare what happens for inductive systems and bornological vector spaces. There is a functorial way to write any bornological vector space as a direct limit of an inductive system of semi-normed spaces. The resulting inductive systems are reduced, that is, all the structure maps are injective. This construction identifies the category ! Cborn of bornological vector spaces with the category Norm1=2red of reduced inductive systems of semi-normed spaces. If we restrict attention to separated or complete ! bornological vector spaces, we get equivalences of categories Born Š Normred and ! Cborn Š Banred . Thus the categories of inductive systems extend the more familiar categories of bornological vector spaces. In practice, we usually only need bornological vector spaces, but some general constructions like completions and completed tensor products work better in the larger category of inductive systems. The completion V c of a separated bornological vector space V is defined by the universal property that bounded linear maps V ! W with complete range “extend”

10

1 Bornological vector spaces and inductive systems

to V c . Unlike for topological vector spaces, the canonical map V ! V c need not be injective, we may even have V c D 0 for V ¤ 0. In contrast, the completion functor ! ! Norm ! Ban is a purely local construction and does not have such undesirable properties. Inductive systems are more powerful than bornological vector spaces because there are more of them and most constructions with bornological vector spaces extend in a straightforward fashion to the ambient categories of inductive systems. Therefore, readers with a sufficiently strong background in category theory to appreciate inductive systems should concentrate on them and think of bornological vector spaces merely as particularly nice inductive systems. Other readers that feel uncomfortable about inductive systems should stick to bornological vector spaces first: most examples of inductive systems that arise in practice are reduced and can therefore be viewed as bornological vector spaces. We introduce basic definitions concerning bornologies in §1.1. In §1.1.1, we motivate the theory of bornological vector spaces by the well-known relationship between continuous semi-norms and absorbing, absolutely convex subsets of topological vector spaces. The notion of a semi-norm leads to locally convex topological vector spaces, which are vector spaces with a family of semi-norms (subject to some simple axioms). The notion of an absolutely convex subset leads to .convex/ bornological vector spaces, which are vector spaces with a family of absolutely convex disks (subject to some simple axioms). After defining bornologies and bornological vector spaces in §1.1.2, we discuss several important examples in §1.1.4: the von Neumann bornology and the precompact bornology on a topological vector space, and the fine bornology on a vector space without additional structure. All these bornologies are still closely related to topologies on the underlying vector spaces. Later, we will meet bornologies which are unrelated to any reasonable topology. Our default bornology on a topological vector space is the precompact bornology, not the more commonly used von Neumann bornology. The precompact bornology usually behaves much better because many estimates in functional analysis automatically hold uniformly on precompact subsets. In order to avoid confusion, we call a subset of a topological vector space that is absorbed by all neighbourhoods of the origin “von Neumann bounded” instead of just “bounded.” In §1.1.5, we define boundedness for linear and multi-linear maps and equip spaces of bounded multi-linear maps with canonical bornologies; this yields an internal Hom functor. We argue that the notion of boundedness for multi-linear maps is closer to separate continuity than to continuity: a separately continuous multi-linear map between complete topological vector spaces is automatically (jointly) bounded with respect to the von Neumann bornologies. This provides many examples of bornological algebras with only separately continuous multiplication. In §1.2, we introduce the most basic notions of analysis. Since a separated bornological vector space is just a union of normed subspaces, all these notions are straightforward generalisations of classical ones for normed spaces. In §1.2.1, we introduce

1 Bornological vector spaces and inductive systems

11

convergent sequences, Cauchy sequences, and continuous functions, smooth functions, and real-analytic functions. We only consider functions from compact topological spaces into bornological vector spaces; we have no need for smooth maps between two bornological vector spaces here. We also introduce functions of bounded variation Œ0; 1 ! V for a bornological vector space V and define Stieltjes integrals of the form R1 0 f0 ˝ df1 . Our proof of homotopy invariance of periodic, analytic, and local cyclic homology works for continuous homotopies of bounded variation; but if you only care about homotopy invariance for sufficiently differentiable homotopies, then you may skip §1.3.7. We define closed subspaces and closed hulls in §1.2.2, using convergence of sequences. This yields a canonical topology on a bornological vector space, which may be rather pathological in general because the addition need not be jointly continuous. However, if we start with the precompact or von Neumann bornology on a metrisable locally convex topological vector space or a Silva space, then we get back the original topology. In §1.2.3, we study the bornological notions of precompact, relatively compact, and compact subsets. Since the precompact subsets form another bornology, we get a functor Cpt W Born ! Born, which generalises the construction of precompact bornologies on Fréchet spaces. This functor is a projection onto those bornological vector spaces in which all bounded subsets are precompact; such spaces are called Schwartz spaces in [51]. Some crucial results later on only work for spaces with a precompact bornology because they require certain approximations to work uniformly on bounded subsets. On precompact subsets, uniformity often comes for free. As a result, several notions of density become equivalent. We investigate this in §1.2.4 and §1.6.4; these rather technical results are used later in connection with isoradial algebra homomorphisms. If we are given a topological vector space and equip it with the precompact or von Neumann bornology, then we now have two ways of doing analysis: the bornological and the topological way. Remarkably, both approaches yield equivalent definitions for all the notions of analysis mentioned above if our topological vector space is metrisable. This permanence principle is essential for many applications. A slightly more detailed account can be found in [67], which also treats non-convex topologies and bornologies. In §1.3, we introduce several important constructions with bornological vector spaces. This includes the standard categorical constructions like subspaces, quotient spaces, kernels, cokernels, products, coproducts, direct limits (also called colimits), and inverse limits (also called just limits). We also recall the categorical notion of an extension and specify what it amounts to in categories of bornological vector spaces or bornological algebras. In §1.3.2, we discuss the separated quotient and completion functors, which force bornological vector spaces to become separated or complete, respectively. The most crucial construction for us is the projective tensor product ˝ and its y which are introduced in §1.3.6. They are defined by a universal completed version ˝, property. The completed tensor product is the completion of the uncompleted one and

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therefore is also affected by the problems with completions mentioned above. A basic result is that the completed projective bornological and topological tensor products are equivalent for Fréchet spaces. This essentially goes back to Alexander Grothendieck ([36]); we merely have to strengthen some of his results slightly. It is crucial for cyclic homology that we have well-defined tensor powers A˝n for all n 2 N. This requires the tensor product to be associative and commutative; in addition, it is useful to have a unit object for ˝. But associativity and commutativity and the unit property only work up to certain natural isomorphisms; these must satisfy suitable compatibility conditions in order for the theory to work. These technical issues have been studied by category theorists and are summarised in the notion of a symmetric monoidal category ([64], [91]). We examine these categorical notions in §1.3.8. In the Appendix we develop the basic machinery of homological algebra and cyclic homology in this framework, following Guillermo Cortiñas and Christian Valqui ([13]). Cyclic homology has been constructed in various categories of algebras with additional structure. In each case, it is asserted that everything works exactly as for algebras without additional structure. The formal approach clarifies such claims. The tensor product for bornological vector spaces has a right adjoint: the internal Hom functor, denoted Hom. If such an adjoint functor exists, a symmetric monoidal category is called closed. This is not the case in categories of topological vector spaces: there are dozens of topologies on spaces of linear operators, but none of them is truly canonical. Internal Hom functors also appear frequently in algebraic situations and are quite useful. Many formal properties of such functors follow from general categorical considerations; this saves us a certain amount of routine checking. This is particularly useful for categories of inductive systems, whose internal Hom functor is much less concrete and therefore hard to deal with directly. In §1.4, we study inductive systems of semi-normed, normed, and Banach spaces. These are closely related to bornological vector spaces, but they are less concrete because they have no elements or subsets. We can still define what a map from a set or a normed space into such an object is, but there are usually few “injective” maps. Nevertheless, these inductive systems share many properties with bornological vector spaces. All the constructions in §1.3 extend to the appropriate categories of inductive systems. In particular, these categories are still closed symmetric monoidal. The relationship between bornological vector spaces and inductive systems is investigated in §1.5. Each complete bornological vector space can be written in a canonical way as a direct union of Banach subspaces. This gives rise to a functor ! diss W Cborn ! Ban ! called dissection functor. There exist similar dissection functors Born ! Norm and ! Born1=2 ! Norm1=2. The direct limit construction provides a functor in the opposite direction. These functors and some of their properties are already known for quite some time: they are already used by Henri Hogbe-Nlend in [46]. Another article where they are studied is [82].

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We show in §1.5.1 that the dissection functor is fully faithful and right adjoint to the direct limit functor, and that its range is equivalent to the category of reduced inductive systems (§1.5.3). We also show that it is exact (§1.5.4) and explain the relationship between separated quotients and completions for bornological vector spaces and inductive systems (§1.5.5). The subcomplete bornological vector spaces are those for which the completion functor behaves nicely. In §1.6 we derive sufficient conditions for subcompleteness that involve metrisability and the approximation property. Metrisability captures a special property of precompact and von Neumann bornologies on Fréchet spaces. In addition, we study a stronger notion of dense range for bounded linear maps, which will be needed in Chapter 6.

1.1 Basic definitions Both bornological and topological vector spaces generalise semi-normed spaces. After explaining this, we define some basic notions concerning bornological vector spaces. The following theory works for vector spaces over R or C. Certain aspects should even work over non-Archimedean locally compact fields like Qp ; but we have not explored this case thoroughly. To simplify notation, we only write down the theory for C-vector spaces.

1.1.1 Semi-norms versus disks Let V be a vector space. A semi-norm  W V ! RC on V and its closed unit ball B  V determine each other uniquely: B D fx 2 V j .x/  1g;

.x/ D inf fr > 0 j x 2 r B g:

A subset B  V is the closed unit ball of a semi-norm if and only if it has the following properties: it is convex: tx C .1  t /y 2 B for all x; y 2 B, t 2 Œ0; 1; absolutely convex:   B  B for all scalars  with jj  1; internally closed: x 2 B provided r  x 2 B for all r 2 Œ0; 1/; absorbing: RC  B D V . The third condition (internally closed) distinguishes the open and closed unit ball. The unique semi-norm B with closed unit ball B is called the gauge semi-norm of B. Definition 1.1. A subset of a vector space that satisfies the first three properties (convex, absolutely convex, internally closed) is called a disk in V .

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Let V1 and V2 be semi-normed spaces with unit balls B1 and B2 , respectively. A linear map f W V1 ! V2 is continuous if and only if it is bounded, that is, f .B1 /  c B2 for some c 2 R0 . Even more, we have two equivalent descriptions of the operator norm of f using the closed unit balls and the semi-norms, respectively:  ˚  sup 2 f .x/ j 1 .x/  1g D kf k D inffr > 0 j f .B1 /  r B2 g: There are analogous statements for multi-linear maps. Summing up, semi-norms on V and absorbing disks in V are equivalent data for the purposes of functional analysis in semi-normed spaces. If B  V is a disk – not necessarily absorbing – then RC  B is a vector subspace of V and B is an absorbing disk in RC  B. Notation 1.2. Let B  V be a disk. We let VB  V be the vector space RC B equipped with the gauge semi-norm of B and call this the semi-normed subspace generated by B. Definition 1.3. A disk B  V is called norming if VB is a normed space; equivalently, we have v D 0 whenever C  v  B. A norming disk B  V is called complete if VB is complete; recall that complete normed spaces are also called Banach spaces.

1.1.2 Bounded subsets, bornologies and bornological vector spaces Many spaces that occur in functional analysis cannot be described adequately by a single semi-norm or a single disk. The obvious generalisation is to consider vector spaces with lots of semi-norms or lots of disks. The first approach leads to locally convex topological vector spaces, the other to convex bornological vector spaces. Definition 1.4. A .convex/ bornological vector space is a pair .V; S/, where V is a vector space and S is a set of subsets of V , subject to the following conditions: 1.4.1. if S1  S2 and S2 2 S, then S1 2 S; 1.4.2. if S1 ; S2 2 S, then S1 [ S2 2 S; 1.4.3. fxg 2 S for all x 2 V ; 1.4.4. if S 2 S and c 2 RC , then c  S 2 S; 1.4.5. any S 2 S is contained in a disk T with T 2 S. The set S is called the bornology of .V; S/; its elements are the bounded subsets of V . The first three conditions define the (uninteresting) notion of a bornology on a set. Since we do not consider non-convex bornologies here (they are defined in [50], [67]), we drop the word convex from our notation and tacitly require all bornological vector spaces to be convex.

1.1 Basic definitions

15

Definition 1.5. We usually denote a bornological vector space just by V and let Sc .V /  Sd .V /  S.V / be the sets of complete bounded disks, bounded disks, and bounded subsets of V . These sets carry two canonical partial orders: we write A  B if B contains A and A  B if B absorbs A, that is, A  c B for some c 2 R>0 . Definition 1.6. A bornological vector space V is called separated if all B 2 Sd .V / are norming, and complete if each bounded subset is contained in a complete bounded disk. Complete bornological vector spaces are automatically separated. We are mainly interested in complete bornological vector spaces, but incomplete and non-separated spaces occur as intermediate steps in several constructions. Definition 1.7. A directed set is a partially ordered set .S; / such that for any x; y 2 S there is z 2 S with z  x and z  y; equivalently, any finite subset of S has an upper bound. A subset T of a directed set .S; / is cofinal if for any x 2 S there is y 2 T with x  y. We often view a directed sets as a category with exactly one arrow x ! y if x  y and no arrow x ! y otherwise. Lemma 1.8. The sets Sc .V /, Sd .V /, and S.V / are directed with respect to both partial orders  and . The subset Sd .V / is always cofinal in S.V /; the subset Sc .V / is cofinal in S.V / if and only if V is complete. Proof. The families of subsets Sc .V /, Sd .V /, and S.V / are closed under the operation .B1 ; B2 / 7! B1 C B2 WD fx1 C x2 j x1 2 B1 ; x2 2 B2 g: This holds for S.V / because B1 C B2 is contained in the convex hull of 2 B1 [ 2 B2 . It is clear that B1 C B2 is again a disk if B1 and B2 are. It also inherits completeness because VB1 CB2 is a quotient of the Banach space VB1 ˚ VB2 , and quotients of Banach spaces are again complete. The cofinality assertions are trivial.

1.1.3 Disked hulls and complete disked hulls The sets Sd .V / and Sc .V / are closed under intersections. Therefore, if B 2 S.V / is arbitrary, then there are minimal subsets in Sd .V / and Sc .V / containing B. Definition 1.9. The minimal bounded disk containing B 2 S.V / is called the disked hull of B and denoted B } . The minimal complete bounded disk containing B 2 S.V / is called the complete disked hull of B and denoted B ~ .

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Disked hulls of bounded subsets always exist and are again bounded. If V is complete, then complete disked hulls of bounded subsets exist and are again bounded. If V is incomplete, then there are bounded subsets that are not contained in any complete disk, so that they have no complete disked hull. P PN More explicitly, B } is the set of all finite sums N nD0 n bn with nD0 jn j  1 and bn 2 B, n 2 C for 0  n  N . We will describe B ~ similarly (Lemma 1.43) once we have defined convergence for infinite series. Notation 1.10. We sometimes write A } B or x 2} B instead of A  B } or x 2 B } and A ~ B or x 2~ B instead of A  B ~ or x 2 B ~ . These relations are transitive, that is, x 2} S and S } T implies x 2} T , and so on.

1.1.4 Examples of bornologies Definition 1.11. Let V be just a vector space. The fine bornology on V is the smallest possible bornology on V ; a subset B  V belongs to this bornology if and only if B } F for some finite subset F  V , if and only if there is a finite-dimensional subspace W  V such that B  W and B is bounded in W in the usual sense. We write Fin.V / for V with the fine bornology. The fine bornology is always complete. It is a good choice if V has a countable basis. If V is bigger, this suggests that some non-trivial functional analysis may be happening. In many examples, we do not construct the bornology itself but only a generating set of bounded subsets. That is, we have got a vector space V and a collection S of subsets, subject to no requirements. Definition 1.12. The .convex/ bornology generated by S is the smallest bornology hhSii on V that contains S. This bornology always exists. It may fail to be separated and will usually not be complete. This can be helped in a second step by taking the separated quotient or the completion of .V; hhSii/. Example 1.13. Any bornology is generated in this sense by Sd .V /. A bornology is complete if and only if it is generated by Sc .V /. The fine bornology on a vector space is generated by the family of all finite subsets or by the empty set of subsets. Definition 1.14. Let V be a locally convex topological vector space. A subset B  V is von Neumann bounded if .B/  RC is bounded for each continuous semi-norm  W V ! RC ; equivalently, B is absorbed by each neighbourhood of the origin in V . The von Neumann bounded subsets form a (convex) bornology on V . We denote V together with this bornology by vN.V /.

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The bornological vector space vN.V / is separated if V is Hausdorff, and complete if V is complete (see [50]). This sufficient condition for completeness is far from necessary. A topological vector space is called quasi-complete if vN.V / is complete. The case of (semi-)normed spaces is particularly important. Let V be a seminormed space with unit ball D. Then B  V is bounded if and only if B is absorbed by D, that is, B  c D for some scalar c. Definition 1.15. Let V be a locally convex topological vector space. A subset B  V is precompact if for any neighbourhood of the origin U there is a finite set F  V such that B  F C U . If V is complete, then B  V is precompact if and only if its closure is compact. In general, B  V is precompact if and only if its closure in the completion of V is compact. The precompact subsets form a bornology on V , which we call precompact bornology. We write Cpt.V / for V equipped with this bornology. The bornological vector space Cpt.V / is separated if V is Hausdorff, and complete if V is (quasi-)complete (compact disks are automatically complete). Definition 1.16. A Fréchet space is a complete, metrisable, locally convex topological vector space. That is, the topology must be definable by an increasing sequence of semi-norms, and any Cauchy sequence must converge. Notation 1.17. A sequence converging to 0 is called a null-sequence. The following alternative description of the precompact bornology on a Fréchet space is often useful: Theorem 1.18. Let V be a Fréchet space. A subset B  V is precompact if and only if B ~ fxn j n 2 Ng for a null-sequence .xn / in V . If A  V is dense, we can also require xn 2 A for all n 2 N. Proof. This is proved already in [36]. We briefly sketch the argument. Let .Un / be a decreasing basis for the neighbourhoods of 0 in V . Using the precompactness of B, we recursively construct precompact subsets Bn  V and finite sets Fn  .Bn C 2n Un / \ A such that B0 D B, Bn  Fn C BnC1 , and BnC1  2n Un for all n 2 N. RecursivelyPpicking approximations in Fn , we can write any element of B as an all n 2 N. It is clear that S such infinite infinite series 1 nD0 xn with xn 2 Fn for S 1 1 n ~ n sums belong to the complete disked hull of 2 F , that is, B  n nD0 nD0 2 Fn . S1 n Enumerating the elements of nD0 2 Fn , we get a sequence .xn / with fxn j n 2 S Ng D 2n Fn , so that B ~ fxn j n 2 Ng (see Lemma 1.43). We have lim xn D 0 because 2n Fn  2n Bn C Un  Un1 C Un . Warning 1.19. What we call a von Neumann bounded subset is usually just called bounded. This works as long as topological vector spaces are always equipped with the von Neumann bornology. But our default bornology is the precompact one; this is crucial for local cyclic homology. In my thesis [65], I resolved the ambiguity by calling bounded subsets “small” and von Neumann bounded subsets “bounded”. I have decided since to reclaim the word “bounded” for the theory of bornological vector spaces.

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1.1.5 Bounded maps Definition 1.20. Let V1 and V2 be bornological vector spaces. A linear map l W V1 ! V2 is bounded if it maps bounded sets again to bounded sets, that is, l.B/ 2 S.V2 / for all B 2 S.V1 /. A set L of linear maps V1 ! V2 is equibounded or uniformly bounded if L.B/ WD fl.x/ j l 2 L; x 2 Bg 2 S.V2 /

for all B 2 S.V1 /:

A .bornological/ isomorphism is a bounded linear map l W V1 ! V2 with a bounded (two-sided) inverse. We often drop the adjective “bornological” here. Now we give names to several important categories: Born1=2 Born Cborn Norm1=2 Norm Ban Vect

bornological vector spaces separated bornological vector spaces complete bornological vector spaces semi-normed spaces normed spaces Banach spaces vector spaces

The morphisms in the first six categories are the bounded linear maps, the morphisms in Vect are the linear maps. These categories are related by fully faithful functors Vect

 Fin

/ Cborn O









Ban

/ Born O



/ Norm

/ Born1=2 O 



/ Norm1=2.

Here we identify Norm1=2 with a full subcategory of Born1=2. Notice that Norm D Norm1=2 \ Born;

Ban D Norm1=2 \ Cborn:

We will check below that the functor Fin is fully faithful, that is,   Hom Fin.V1 /; Fin.V2 / Š Hom.V1 ; V2 / for all vector spaces V1 ; V2 . Here Hom denotes the spaces of morphisms in Cborn and Vect, respectively. Notation 1.21. Let C be a category. We write A 22 C to denote that A is an object of C. The notation f 2 C means that f is a morphism in C. If A; B 22 C, then we often write Hom.A; B/ for the space of morphisms A ! B. We sometimes write HomC .A; B/ or C.A; B/ if we want to emphasise C. Definition 1.22. Let V1 and V2 be bornological vector spaces. We let Hom.V1 ; V2 / be the vector space of all bounded linear maps V1 ! V2 . The equibounded subsets form a bornology on Hom.V1 ; V2 /. Let Hom.V1 ; V2 / be the resulting bornological vector space.

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There is no similarly canonical topology on the space Hom.V1 ; V2 / of continuous linear mappings between two topological vector spaces V1 and V2 . The most canonical structure on Hom.V1 ; V2 / is, in fact, the bornology of equicontinuous sets of linear maps. The obvious functor Forget W Born1=2 ! Vect that forgets the bornology recovers Hom from Hom: Forget Hom.V; W / Š Hom.V; W /: We have V Š Hom.C; V / and Forget.V / Š Hom.C; V / for all V 22 Born1=2. Lemma 1.23. The space Hom.V1 ; V2 / is separated or complete if V2 is. If V1 ; V2 are .semi/-normed spaces or Banach spaces, so is Hom.V1 ; V2 /. Proof. The second assertion is trivial. The first one is proved in [50]. We also need multi-linear maps to study algebras and modules. Definition 1.24. Let V1 ; : : : ; VnC1 be bornological vector spaces. An n-linear map l W V1      Vn ! VnC1 is bounded if l.B1      Bn / 2 S.VnC1 / whenever Bj 2 S.Vj / for j D 1; : : : ; n. A set L of such maps is equibounded or uniformly bounded if L.B1   Bn / 2 S.VnC1 / whenever Bj 2 S.Vj / for j D 1; : : : ; n. Definition 1.25. We let Hom.n/ .V1   Vn I VnC1 / be the vector space of all bounded n-linear maps V1      Vn ! VnC1 . The equibounded subsets form a bornology on Hom.n/ .V1      Vn I VnC1 /. We write Hom.n/ .V1      Vn I VnC1 / for the resulting bornological vector space. Of course, we have Hom.1/ .V1 I V2 / D Hom.V1 ; V2 /. Definition 1.26. A (complete or separated/ bornological algebra is a (complete or separated) bornological vector space A together with a bounded, associative bilinear map m W A  A ! A. Many algebras that we consider have no unit element. Example 1.27. Let V1 ; : : : ; Vn be vector spaces and let VnC1 be an arbitrary bornological vector space. Then any n-linear map Fin.V1 /      Fin.Vn / ! VnC1 is bounded, that is   Hom.n/ .Fin.V1 /      Fin.Vn /I VnC1 / Š Hom.n/ V1      Vn I Forget.VnC1 / : For n D 1, this says that the functor Fin W Vect ! Born1=2 is the left adjoint of the forgetful functor Forget W Born1=2 ! Vect. Since Forget ı Fin.V / D V , we also get that the functor Fin W Vect ! Born1=2 is fully faithful. For n D 2, the above adjointness relation implies that the fine bornology functor maps algebras to complete bornological algebras.

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Example 1.28. Let V1 be a vector space, let S be a collection of subsets of V1 , and let hhSii be the bornology generated by S as in Definition 1.12. Let V2 be any bornological vector space. Then a linear map l W .V1 ; hhSii/ ! V2 is bounded if and only if l.B/ 2 S.V2 / for all B 2 S.

1.1.6 Boundedness versus continuity A continuous linear map f W V1 ! V2 between topological vector spaces maps von Neumann bounded subsets and precompact subsets again to such subsets. This means that the von Neumann bornology and the precompact bornology give rise to functors vN; Cpt W f locally convex topological vector spacesg ! Born1=2: Of course, vN.f / D 0 or Cpt.f / D 0 if and only if f D 0, that is, these two functors are faithful. But they are not fully faithful: there may be bounded linear maps that are not continuous. Let V1 and V2 be Fréchet spaces with the bounded or the precompact bornology (see Definitions 1.14 and 1.15). Then a linear map V1 ! V2 is continuous if and only if it is bounded as a map Cpt.V1 / ! Cpt.V2 / or vN.V1 / ! vN.V2 /. In other words, the functors vN and Cpt from the category of Fréchet spaces to the category of complete bornological vector spaces are fully faithful. This even carries over to multi-linear maps: Theorem 1.29. Let V1 ; : : : ; Vn be Fréchet spaces, let VnC1 be any locally convex topological vector space, and let f W V1      Vn ! VnC1 be an n-linear map. Then the following assertions are equivalent: (1) f is jointly continuous; (2) f is separately continuous; (3) if .vi;k /k2N are null-sequences in Vi for i D 1; : : : ; n, then the sequence  f .v1;k ; : : : ; vn;k / k2N is a von Neumann bounded sequence in VnC1 ; (4) f is bounded as a map Cpt.V1 /      Cpt.Vn / ! Cpt.VnC1 /; (5) f is bounded as a map vN.V1 /      vN.Vn / ! vN.VnC1 /; (6) f is bounded as a map Cpt.V1 /      Cpt.Vn / ! vN.VnC1 /. Hence a Fréchet algebra, endowed with either the von Neumann or the precompact bornology, is a complete bornological algebra. Proof. The equivalence of the first three properties is well known. Each of the last three properties implies the third one by Theorem 1.18.

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Whereas separately continuous multi-linear maps occur rather frequently, there is hardly any need to consider separately bounded multi-linear maps (the only relevant example I know is described in Warning 7.53). Lemma 1.30. Let V1 ; : : : ; Vn be complete bornological vector spaces and let W be a locally convex topological vector space. Let f W V1      Vn ! vN.W / be an n-linear map that is separately bounded in the sense that it is a bounded linear map on Vj if the other entries are fixed. Then f is already . jointly/ bounded. Proof. Let Bj  Vj be bounded complete disks for j D 1; : : : ; n. The restriction .V1 /B1      .Vn /Bn ! W is a separately continuous n-linear map defined on Banach spaces. By Theorem 1.29, separate continuity implies joint continuity and hence joint boundedness in this situation. Thus f is bounded. Definition 1.31. A locally convex topological vector space V is called an LF-space if there is an increasing sequence .Vn /n2N of subspaces S of V such that each Vn is a Fréchet space with respect to the subspace topology, Vn D V , and V carries the finest locally convex topology making the inclusions Vn ! V continuous for all n 2 N. This means that a linear map V ! V 0 into a locally convex space V 0 is continuous if and only if its restriction to Vn is continuous for all n 2 N. S Theorem 1.32 (see [101]). Let V D Vn be an LF-space. A subset of V is von Neumann bounded if and only if it is contained in and von Neumann bounded in Vn for some n 2 N. A subset of V is precompact if and only if it is contained in and precompact in Vn for some n 2 N. Theorem 1.33. Let V1 ; : : : ; VnC1 be LF-spaces and endow all of them with the von Neumann or the precompact bornology. An n-linear map l W V1      Vn ! VnC1 is bounded if and only if it is separately continuous. Thus bounded bilinear maps need not be jointly continuous in this case. Proof. By the universal property of the LF-topology, an n-linear map l W V1   Vn ! VnC1 is separately continuous if and only if for all Fréchet subspaces Vj0  Vj , the restriction of l to V10      Vn0 is separately continuous. Similarly, l is bounded if and only if its restriction to V10      Vn0 is bounded for all Fréchet subspaces by Theorem 1.32. Once we are in the world of Fréchet spaces, boundedness, separate continuity, and joint continuity are all equivalent by Theorem 1.29. Hence l is bounded if and only if it is separately continuous. Example 1.34. Interesting separately continuous bilinear maps are the convolution maps on spaces of compactly supported smooth functions on Lie groups. For instance, the space Cc1 .R/ of smooth compactly supported functions on the real line is an LFspace. It becomes an algebra for the convolution Z .f  g/.t / D f .t  s/  g.s/ ds: R

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This multiplication is evidently separately continuous. We claim that it is not jointly continuous. To prove this, consider the continuous semi-norm X .f / WD jf .n/ .n/j for all f 2 Cc1 .R/: n2N

If the convolution were jointly continuous, then there would be continuous semi-norms 1 ; 2 on Cc1 .R/ with .f  g/  1 .f /  2 .g/. Restrict 1 to functions with support in Œ1; 1. The restricted semi-norm 1 .f / can only take into account finitely many derivatives of f . For sufficiently big n, we have no control on the nth derivative of f near 0. Since .f g/.n/ D f .n/ g, the modulus of .f g/.n/ .n/ can become arbitrary while 1 .f /  2 .g/ remains bounded. Therefore, the convolution cannot be jointly continuous.

1.2 Some functional analysis 1.2.1 Convergent sequences and continuous functions We define various notions of functional analysis for bornological vector spaces by reduction to semi-normed subspaces: Definition 1.35. Let V be a bornological vector space. Recall that Sd .V / denotes the set of bounded disks in V . • A sequence .xn /n2N in V is .bornologically/ convergent towards x1 2 V if it converges to x1 in VB for some B 2 Sd .V /; that is, there is a null-sequence of scalars ."n /n2N such that xn  x1 2 "n B for all n 2 N. • A sequence .xn /n2N in V is a .bornological/ Cauchy sequence if it is a Cauchy sequence in VB for some B 2 Sd .V /.   P • A function f W X ! V for a set X is absolutely summable if x2X B f .x/ < 1 for some B 2 Sd .V /. In particular, this defines absolutely summable series in V for X D N. We denote the space of absolutely summable maps X ! V by `1 .X; V /. • A function f W X ! V from a compact space X to V is continuous if it is continuous as a function X ! VB for some B 2 Sd .V /. We write C .X; V / for the space of continuous functions X ! V . • A set S of continuous functions X ! V is uniformly bounded if it is a uniformly bounded set of functions X ! VB for some B 2 Sd .V /. That is, S.X / WD ff .x/ j x 2 X; f 2 Sg 2 S.V /:

1.2 Some functional analysis

23

• A set S of continuous functions X ! V is uniformly continuous if it is uniformly bounded and uniformly continuous as a set of functions X ! VB for some B 2 Sd .V /. • Let M be a compact C k -manifold, where k 2 N [ f1; !g; C 1 -manifold means smooth manifold, and C ! -manifold means real-analytic manifold. A function f W M ! V is C k if it belongs to C k .M; VB / for some B 2 Sd .V /. We write C k .M; V / for the space of C k -functions M ! V . We will only use continuous functions with values in complete bornological vector spaces. Both the uniformly bounded and the uniformly continuous subsets form bornologies on C.X; V /. These bornologies are complete or separated if V is so. If V D C, then C .X; C/ D C.X / is a Banach space; the uniformly bounded bornology is its von Neumann bornology, the uniformly continuous bornology is its precompact bornology by the Arzelà–Ascoli Theorem. Since we prefer the precompact bornology, we always equip C.X; V / with the uniformly continuous bornology. We also equip C k .M; V / for k 2 N [ f1; !g with a canonical bornology. If k  1, then a subset of C k .M; V / is considered bounded if X1 ı    ı Xm .S / is a uniformly continuous set of functions M ! V for any m  k and vector fields X1 ; : : : ; Xm on M . Equivalently, we may replace X1 ı    ı Xm by a differential operator of order k. For the space of real-analytic functions, we also require the radius of convergence of the Taylor series of a function to be bounded away from zero. Thus all elements of a bounded subset of C ! .M; V / may be extended holomorphically to some neighbourhood of M in the ambient complex manifold. Let V be a locally convex topological vector space. Then we can use the topology to define C k -functions X ! V and get locally convex topological vector spaces C .X; V / and C k .X; V / for k 2 N [ f1; !g. It is clear that a function X ! V that is C k with respect to the precompact or von Neumann bornology is C k with respect to the topology as well. The converse may be false in general, that is, there may be continuous functions X ! V that do not factor continuously through VB for any von Neumann bounded disk B. This problem does not occur for Fréchet spaces: Theorem 1.36 ([67]). Let V be a Fréchet space. A sequence in V is bornologically convergent or Cauchy in Cpt.V / if and only if it is convergent or Cauchy in the usual sense, if and only if it is bornologically convergent or Cauchy in vN.V /. A series in V is absolutely summable in Cpt.V / if and only if it is absolutely summable in V if and only if it is absolutely summable in vN.V /. Let X be a compact space. A function X ! V is continuous as a map to Cpt.V / if and only if it is continuous as a map to V if and only if it is continuous as a map to vN.V /. A subset of C .X; V / is precompact if and only if it is uniformly continuous as a set of functions X ! Cpt.V /. In other words, we have a natural bornological isomorphism     C X; Cpt.V / Š Cpt C.X; V / :

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A subset of C.X; V / isvon Neumann bounded if and only if it is a uniformly bounded subset of C X; vN.V / . Similar assertions hold for C k -functions M ! V for any k 2 N [f1; !g and any compact C k -manifold M . In particular, we have a natural bornological isomorphism     C k M; Cpt.V / Š Cpt C k .M; V / : Proof. Most of these assertions are verified in [67]; since the proofs are rather technical, we do not reproduce them here. The equivalent descriptions of convergent sequences and Cauchy sequences are contained in [67, Corollary 3.8]. The assertion about absolutely summable sequences amounts P to the well-known statement that we can rewrite an absolutely summable series n2N xn in a Fréchet space in the form xn D n  xn0 with .n / 2 `1 .N/ and a null-sequence .xn0 / in V . The assertions about spaces of continuous functions are contained in [67, Theorem 3.7]. We remark that [67] considers maps defined on metric spaces instead of compact spaces. Since any continuous map into a metrisable space factors through a metric space, we can apply the results of [67] even for compact spaces that are not metrisable (we are not going to use such spaces in the following, anyway). Allowing more general metric spaces is useful for certain purposes, but we have no need for this generalisation here. The assertions about C k -functions for k  1 are contained in [67, Corollary 3.9]. Thus it only remains to verify the assertions for k D !, that is, for real-analytic functions. For this we embed the given real-analytic manifold M in some complex manifold MC T. Let .Un / be a decreasing sequence of compact neighbourhoods of M in MC with Un D M . First, we must describe the locally convex topological vector space C ! .M; V /. Let A.Un ; V / be the space of bounded continuous functions Un ! V that are holomorphic on the interior of Un ; this is a closed subspace of C .Un ; V /. Any realanalytic function extends holomorphically to a function in A.Un ; V / for some n and, conversely, functions in A.Un ; V / restrict to real-analytic functions on M . Thus C ! .M; V / is the union of an increasing sequence of subspaces A.Un ; V /. By definition, the topology on C ! .M; V / is the finest locally convex topology for which the maps A.Un ; V / ! C ! .M; V / are all continuous. It turns out that a subset of C ! .M; V / is precompact if and only if it is precompact in A.Un ; V / for some n 2 N; we do not prove this fact here. Since A.Un ; V / is a closed subspace of C .Un ; V /, any precompact subset of A.Un ; V / is already precompact in A.Un ; VB / for some precompact disk B  V . Hence any precompact subset of ! C ! .M; V / is already precompact  in C !.M;  VB / for some precompact disk B  V . ! This shows that Cpt C .M; V / D C M; Cpt.V / . Literally the same argument works for von  Neumann  subsets instead of precompact subsets, so that we  bounded also get vN C ! .M; V / D C ! M; vN.V / . If X is a locally compact topological space, then we form its one-point compactification XC D X [ f1g and define C0 .X; V / WD ff 2 C .XC ; V / j f .1/ D 0g:

1.2 Some functional analysis

25

If U  X is an open subset, then we may identify C0 .U; V / with the space of continuous functions X ! V that vanish outside U . The space Cc .X; V / of compactly supported continuous functions X ! V is the union of the spaces C0 .U; V / for relatively compact open subsets U  X; a subset of Cc .X; V / is bounded if and only if it is contained in and bounded in C0 .U; V / for some relatively compact open subset U  X . The space Cck .M; V / for k 2 N [ f1g is defined similarly.

1.2.2 Closed subsets and the bornological topology Definition 1.37. A subset A of a bornological vector space V is called .bornologically/ closed if x 2 A whenever x D lim xn for a sequence .xn / in A. These subsets satisfy the axioms for a topology, which we call the bornological topology on V . Thus we get a notion of .bornological/ closure of a subset. Example 1.38. Theorem 1.36 shows that the bornological topology agrees with the given one on a Fréchet space because closed subspaces in both senses are detected by the same kind of convergent sequence. Example 1.39. Let V be a vector space. Then a sequence .xn / in Fin.V / converges if and only if all xn are contained in and converge in a fixed finite-dimensional vector subspace. Thus a subset A  Fin.V / is bornologically closed if and only if A \ W is closed for all finite-dimensional vector subspaces W . If V has a countable basis, then the bornological topology on V is locally convex and hence a vector space topology. More generally, it is shown in [33], [50], [99] that this happens for any Silva space (see Definition 1.158). Helge Glöckner kindly pointed out to me the following counterexample: Example 1.40. S Let V be an LF-space with the precompact or von Neumann bornology. Thus V D Vn for an increasing sequence of Fréchet subspaces Vn . A sequence in V converges if and only if it converges in Vn for some n 2 N in the usual sense. Therefore, a subset A of V is closed if and only if V \ Vn is closed in the Fréchet topology of Vn for all n 2 N. Thus the bornological topology on V is the inductive limit topology. It is shown by Aiichi Yamasaki ([111]) that V with this topology is no topological group, that is, the addition is not jointly continuous. Therefore, it does not turn V into a topological vector space. A bounded linear map f W V1 ! V2 respects convergence of sequences. Therefore, x  f .A/ for all A  V1 . it is continuous for the bornological topology and satisfies f .A/ Lemma 1.41. Let V be a bornological vector space. If A  V is convex or a vector x subspace, then so is A. Proof. Let A1 be the set of limit points of Cauchy sequences in A. Then A1 is convex or a vector subspace if A is. Similarly, these two properties are inherited by increasing x by transfinite induction, repeating these two steps. Hence A x unions. We can construct A also inherits the two properties.

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Warning 1.42. Complete disks need not be closed. Hence the closed disked hull of a subset S may be bigger than S ~ . Bornological closures of bounded subsets remain bounded in many cases such as for von Neumann, precompact, or fine bornologies. But this need not be true in general. Lemma 1.43. Let V be a complete bornological vector space and let B 2 S.V /. Its complete disked hull is ˇ  ˚P P ˇ B~ D n2N n bn n 2 C and bn 2 B for all n 2 N, and n2N jn j  1 : Proof. Let B 0 be the set of all such sums. It is clear that B 0  B ~ . For the converse inclusion, it suffices to check that B 0 is a complete disk in V . It is clear that B 0 is a disk. Completeness may be checked by hand. The slickest proof uses that quotients of Banach spaces are again Banach spaces. Let `1 .B/ be the set of all absolutely summable functions B ! C; this is a Banach space. The summation operator defines a bounded linear map `1 .B/ ! V , which maps the closed unit ball of `1 .B/ onto B 0 . Hence B 0 is complete.

1.2.3 Precompact bornologies Recall that we prefer to equip topological vector spaces with the precompact bornology instead of the von Neumann bornology. Now we formalise some properties of the resulting bornological vector spaces. Definition 1.44. Let V be a bornological vector space. A subset of V is called bornologically precompact, bornologically relatively compact, or bornologically compact, respectively, if it is precompact, relatively compact, or compact in VB for some B 2 Sd .V /. Precompact subsets are relatively compact, these are compact, and compact subsets are bounded and bornologically closed. A subset of a bornological vector space is relatively compact if and only if it is contained in a compact subset. If V is complete, then any precompact subset of V is relatively compact. The closed disked hull of a relatively compact subset is compact and agrees with its complete disked hull. If f W V1 ! V2 is a bounded linear map, then f maps precompact, relatively compact, and compact subsets again to such subsets of V2 . Theorem 1.45 ([67, Theorem 5.2]). Let V be a metrisable locally convex topological vector space. A subset of V is precompact for the topology if and only if it is bornologically precompact in Cpt.V / if and only if it is bornologically precompact in vN.V /. Similar statements hold for compact and relatively compact subsets. Definition 1.46. A bornology is called relatively compact or precompact if all bounded subsets are relatively compact or precompact, respectively.

1.2 Some functional analysis

27

The precompact bornology on a Fréchet space is relatively compact by Theorem 1.45. Precompact bornologies are also studied by Henri Hogbe-Nlend and Vincenzo B. Moscatelli in [51], where they are called Schwartz bornologies. A relatively compact bornology is necessarily complete: if a subset is compact, then its disked hull is again compact and hence complete. For complete bornological vector spaces, there is no difference between precompact and relatively compact subsets. Thus the difference between precompact and relatively compact bornologies is precisely that the former are required to be complete. Definition 1.47. The precompact subsets of V form a bornology. We write Cpt.V / for V equipped with this new bornology. This defines a functor Cpt W Born1=2 ! Born1=2. Proposition 1.48. The bornology on Cpt.V / is always precompact, and relatively compact if V is complete. Proof. Let S  V be bornologically precompact. Thus S is precompact in VB for some B 2 Sd .V /. By Theorem 1.45, S is bornologically precompact in Cpt.VB / and hence in Cpt.V /. If V is complete, then so is Cpt.V /, so that relative compactness and precompactness are equivalent. Let V1 and V2 be bornological vector spaces. Since precompact subsets are bounded, the identical map Cpt.V2 / ! V2 is bounded. If V1 has a precompact bornology, then a bounded linear map f W V1 ! V2 maps bounded subsets to precompact subsets because all bounded subsets of V1 are precompact. Hence   Hom V1 ; Cpt.V2 / Š Hom.V1 ; V2 /: This means that the functor Cpt is right adjoint to the embedding of the full subcategory of spaces with precompact bornology.

1.2.4 Notions of density We first define the weakest reasonable notion of density: Definition 1.49. A subset A of a separated bornological vector space V is called dense if its bornological closure is all of V . The bornological closure of a subset A can be constructed by transfinite induction, adding limit points of sequences or taking increasing unions in each step. Since the number of steps needed to arrive at the closure can be an arbitrary cardinal number, density is not enough if we want explicit approximations. We call a subset sequentially dense if any point of V is the limit of a sequence in A. We prefer the following refined notion:

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Definition 1.50 ([67, Definition 4.5]). A subset A of a separated bornological vector space V is called locally dense if for any S 2 Sd .V / there is T 2 Sd .V / with S  T such that the closure of A \ T in VT contains S . Local density implies sequential density because the subspaces VT are semi-normed, and sequential density implies density. Sometimes all three notions are equivalent: Theorem 1.51 ([67, Theorem 4.10]). Let V be a Fréchet space and let X  V be a subset. Then the following assertions are equivalent: • X is locally dense in Cpt.V /; • X is dense in Cpt.V / .in the sense of Definition 1.49/; • X is dense in V in the usual sense. Evidently, a subset of a Banach space is locally dense for the von Neumann bornology if and only if it is dense in the usual sense. The corresponding assertion for a Fréchet space requires an extra condition (see [67]). We are particularly interested in bounded linear maps f W W ! V with dense range. In this situation, we want to approximate elements of V by elements of the form f .x/ in a controlled fashion, as explained in the following definition. Definition 1.52. Let W and V be bornological vector spaces. A bounded linear map f W W ! V has uniformly dense range if, for any B 2 S.V /, there is a sequence of bounded (non-linear) maps n W B ! W such that f ı n .x/ converges uniformly to x for x 2 B. This means that there are T 2 Sd .V / and a null-sequence of scalars ."n / such that f ı n .x/  x 2 "n T for all x 2 B, n 2 N. The boundedness of the maps n W B ! W means that n .B/ is bounded for each n 2 N. In most cases S of interest, these maps are not uniformly bounded with respect to n 2 N, that is, n2N n .B/ is unbounded. Proposition 1.53. If f W W ! V is a bounded linear map with uniformly dense range, then f .W /  V is locally dense. The converse holds if V has a precompact bornology. Proof. It is clear that f .W / is locally dense if f has uniformly dense range. For the converse, we assume that V has a precompact bornology. Thus any B 2 S.V / is precompact in VD for some D 2 Sd .V /. Hence we can find a sequenceSof finite subsets Fn  B such that B  Fn C 2n D for all n 2 N. Let A D 1 nD1 Fn , this is a bounded countable subset of B. For each x 2 B, choose n1 .x/ 2 Fn with x  n1 .x/ 2 2n D. Since f .W / is locally dense, there is E 2 Sd .V / such that the2 closure  of f .W / \ VE in VE contains  A. For each x 2 A, we choose a sequence n .x/ n2N in W with limn!1 f n2 .x/ D x in VE . Passing to a subsequence, we   can achieve f n2 .x/  x 2 2n E for all n 2 N, x 2 A.

1.3 Constructions with bornological vector spaces

29

Now we let n WD n2 ı n1 W B ! A ! W . These maps are certainly bounded because n .B/ is finite for each n 2 N. By construction, f ı n .x/  x D f ı n2 ı n1 .x/  n1 .x/ C n1 .x/  x 2 2n E C 2n D for all x 2 B. This means that lim f ı n .x/ D x uniformly for x 2 B. Example 1.54. Let V be a Fréchet space equipped with the precompact bornology. Then the first half of Proposition 1.53 applies, that is, a bounded linear map f W W ! V has uniformly dense range if and only if f .W / is locally dense; this is in turn equivalent to density in the usual, topological sense by Theorem 1.51. In contrast, if we equip V with the von Neumann bornology, then the notion of uniformly dense range for f W W ! V depends on the bornology on W . For instance, consider the Banach space `1 .N/ and the dense subspace CŒt  of polynomials; this subspace is locally dense by Theorem 1.51. If we equip CŒt  with the fine bornology, then the map i W CŒt  ! `1 .N/ does not have uniformly dense range because a bounded map into CŒt is necessarily compact, whereas the identity map on `1 .N/ is not compact; if we equip CŒt with the subspace bornology from `1 .N/ instead, then i has uniformly dense range. Therefore, the property of having uniformly dense range depends on the bornology on W in this case. We have defined uniformly dense range by requiring the existence of approximate local sections n W B ! W because this is what is needed in our applications. The following technical criterion is easier to verify: Lemma 1.55. A bounded linear map f W W ! V has uniformly dense range if and only if for each B 2 Sd .V / there are an increasing sequence .Sn / in Sd S .W / and D 2 Sd .V / such that B  D, f .Sn /  D for all n 2 N, and the closure of f .Sn / in VD contains B. Proof. If we have approximate local sections n W B ! W , then we may put Sn WD subsets Sn as above, thenSeach x 2 B can be n .B/. Conversely, if we have  bounded  written as the VD -limit of f n .x/ for some sequence n .x/ 2 Sm . Re-indexing the sequence n , we can achieve that n .x/ 2 Sn for all n 2 N, so that all maps n are bounded. Warning 1.56. The class of maps with uniformly dense range does not have good permanence properties. For instance, it is unclear whether the composite of two maps with uniformly dense range has uniformly dense range again.

1.3 Constructions with bornological vector spaces 1.3.1 Subspaces and quotients Let V be a bornological vector space and let W  V be a subspace.

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Definition 1.57. The subspace bornology on W is defined by S.W / WD fB j B 2 S.V /; B  W g D fB \ W j B 2 S.V /g: We always equip subspaces with this bornology unless otherwise specified. This bornology is characterised by the property that a linear map f W X ! W is bounded if and only if it is bounded as a map to V . An analogous assertion holds for uniformly bounded sets of linear maps. That is, Hom.X; W / carries the subspace bornology from Hom.X; V /. Example 1.58. Let V be a locally convex topological vector space and let W  V be a subspace, endowed with the subspace topology. The subspace bornologies on W induced by vN.V / and Cpt.V / are equal to vN.W / and Cpt.W /, respectively. Definition 1.59. The quotient bornology on V =W is defined by S.V =W / WD f.B/ j B 2 S.V /g; where  W V ! V =W is the quotient map. We always equip quotients with this bornology. This bornology is characterised by the property that a linear map f W V =W ! X is bounded if and only if f ı  W V ! X is bounded. An analogous assertion holds for uniformly bounded sets of linear maps. That is, Hom.V =W; X / carries the subspace bornology from Hom.V; X /. Example 1.60. The quotient and subspace bornologies induced by a fine bornology are again fine. Lemma 1.61. Let W  V be a vector subspace of a bornological vector space; equip W and V =W with the subspace and quotient bornology, respectively. Then W is separated if V is separated, and V =W is separated if and only if W is closed in V . Suppose, in addition, that V is complete. Then W is complete if and only if W is closed. In this case, V =W is complete as well. Proof. See [50]. Theorem 1.62. Let V be a Fréchet space and let W  V be a closed subspace. Equip V =W with the quotient topology, so that it becomes a Fréchet space as well. The quotient bornology on V =W induced by Cpt.V / is equal to Cpt.V =W /. Proof. It is clear that precompact subsets of V map to precompact subsets of V =W . Conversely, Theorem 1.18 shows that any precompact subset of V =W is the image of a precompact subset of V because we may lift null-sequences in metrisable spaces. In contrast, there may be von Neumann bounded subsets of V =W that do not lift to von Neumann bounded subsets of V . Thus vN does not commute with quotients.

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31

1.3.2 Separated quotients and completions Lemma 1.63 ([50]). Let V be a bornological vector space. The closure of f0g is a closed vector subspace of V , and V is separated if and only if f0g D f0g. Definition 1.64. The separated quotient is the quotient space sep.V / WD V =f0g with the quotient bornology. This space is separated by Lemma 1.61. A bounded linear map from V to a separated bornological vector space W vanishes on f0g and hence descends to sep.V /. Thus Hom.V1 ; V2 / Š Hom.sep.V1 /; V2 / for all V1 22 Born1=2, V2 22 Born. Hence sep is a functor sep W Born1=2 ! Born that is left adjoint to the embedding functor Born ! Born1=2. The embedding Cborn ! Born has a left adjoint as well: the completion functor Born ! Cborn;

V 7! V c I

it is characterised by natural isomorphisms Hom.V1 ; V2 / Š Hom.V1c ; V2 /

(1.65)

for all V1 22 Born, V2 22 Cborn. Roughly speaking, any bounded linear map V1 ! V2 into a complete bornological vector space “extends” uniquely to the completion V1c . We avoid the word “extends” here because the map V1 ! V1c that induces the isomorphism (1.65) need not be injective. Example 1.66. There is a separated convex bornological vector space V ¤ f0g with Hom.V; W / D 0 for all complete bornological vector spaces W . Thus V c D f0g and the canonical map V ! V c is not injective. The following example is taken from [50]. Let V D t  CŒt  be the vector space of polynomials without constant coefficient. For all " > 0, let " .f / for f 2 V be the maximum of f on the interval Œ0; ". This is a norm for each " > 0. A subset of V is called bounded if it is " -bounded for some " > 0. This defines a separated bornology on V . Let l W V ! W be a bounded linear map with complete range W . We claim that l D 0. Let V" be the completion of V with respect to the norm " . By Weierstraß’Theorem, V" is isomorphic to C0 ..0; "/. Since W is complete, we can extend l to a bounded linear map l" W V" ! W . Pick a continuous function f 2 V" vanishing in a neighbourhood of 0. Then f is annihilated by the restriction map r";"0 W V" ! V"0 for suitably small "0 2 .0; ". However, l" D l"0 ı r";"0 factors through this restriction map. Hence l" .f / D 0. Since f .0/ D 0 for all f 2 V" , the space of functions vanishing in a neighbourhood of 0 is dense in V" . Thus l" D 0 and hence l D 0.

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The universal property determines V c and the canonical map V ! V c uniquely up to a canonical isomorphism. We still have to prove the existence of a completion. The simplest construction, which goes back to Henri Hogbe-Nlend [46], uses the relationship between bornological vector spaces and inductive systems. Therefore, we postpone the construction of completions until §1.5.5. An alternative construction of completions in [67] uses the familiar idea that the completion of V is the quotient of the space of Cauchy sequences in V by the subspace of null-sequences. Definition 1.67. A separated bornological vector space V is called subcomplete if the natural map V ! V c is a bornological embedding. This definition excludes the pathology of Example 1.66. We will say more about subcomplete spaces in §1.5.5 and §1.6.2–1.6.3. Remark 1.68. The embedding Cborn ! Born has a right adjoint functor as well, which equips a separated bornological vector space V with the bornology generated by Sc .V /. We do not use this functor in the following.

1.3.3 Kernels and cokernels First we recall the general categorical definition of kernels and cokernels (see [64]). It applies to all the settings that we shall consider: bornological algebras, ind-normed spaces, chain complexes of bornological vector spaces, and so on. Definition 1.69. A zero object in a category C is an object 0 such that C.0; A/ and C.A; 0/ have exactly one element for all A 22 C. This object is determined uniquely up to isomorphism, if it exists. A category C with zero object is called pointed. In a pointed category, the morphism spaces C.A; B/ contain a distinguished element called 0, namely, the unique map that factors through 0. Let C be a pointed category. A kernel for a morphism f W A ! B in C is defined by the universal property Hom.X; ker f / Š fh 2 Hom.X; A/ j f ı h D 0g: A cokernel for a morphism f W A ! B is defined by the universal property Hom.coker f; X / Š fh 2 Hom.B; X / j h ı f D 0g: Kernels and cokernels need not exist in general. If they do exist, they come equipped with canonical maps i

f

p

ker f !  A !B ! coker f satisfying f ıi D 0, p ıf D 0. This diagram is unique up to a canonical isomorphism. Furthermore, the map i is monic, that is, i ı h1 D i ı h2 implies h1 D h2 , and the map p is epic, that is, h1 ı p D h2 ı p implies h1 D h2 . All these assertions are easy to check.

1.3 Constructions with bornological vector spaces

33

Example 1.70. Let f W V1 ! V2 be a morphism in Born1=2, that is, a bounded linear map between two bornological vector spaces. Its kernel is the usual vector space kernel of f with the subspace bornology; its cokernel is V2 =f .V1 / with the quotient bornology. It is easy to check the universal properties. The same construction yields kernels for morphisms in Born and Cborn. Recall that ker f is closed whenever V2 is separated. Therefore, ker f is separated or complete if both V1 ; V2 are so by Lemma 1.61. The universal property follows because it already holds in the larger category Born1=2. The construction of coker f has to be modified for Born and Cborn because the quotient space V2 =f .V1 / need not be separated. Instead, the cokernels in Born and Cborn are   sep coker f Š sep V2 =f .V1 / Š V2 =f .V1 /: We denote this cokernel by sep coker f if we want to emphasise that it differs from the cokernel in Born1=2. By Lemma 1.61, sep coker f is complete if V2 is. Example 1.71. The functor Cpt from the category of Fréchet spaces to Cborn preserves cokernels of maps: if f W V ! W is a continuous linear map between two Fréchet spaces, then the cokernel of f is the quotient W =f .V /. Theorem 1.51 shows that the topological and bornological closures of f .V / agree. Hence the cokernel of Cpt.f / W Cpt.V / ! Cpt.W / is the same vector space W =f .V /, equipped with the quotient bornology from the precompact bornology. By Theorem 1.62, this quotient bornology is again the precompact bornology on the Fréchet space W =f .V /. Thus coker Cpt.f / D Cpt.coker f /. Both functors Cpt and vN commute with kernels, even on the whole category of locally convex topological vector spaces. Our explicit description shows that Forget W Born1=2 ! Vect preserves kernels and cokernels. When we restrict it to the subcategories Born and Cborn, this is no longer true because of the need to pass to separated quotients.

1.3.4 Extensions There is a general recipe for defining extensions in pointed categories. We will use this notion in various contexts. i

p

Definition 1.72. Let C be a pointed category. A diagram K !  E  ! Q in C with p ı i D 0 is called an extension if the maps K ! ker p and coker i ! Q induced by i and p are isomorphisms. i

p

We briefly write K  E  Q to denote that we have got an extension. p

i

Example 1.73. A diagram K !  E ! Q in Born1=2 is an extension if p and i induce Š

Š

! Q; here i.K/  E carries ! i.K/ and E= i.K/  bornological isomorphisms K 

34

1 Bornological vector spaces and inductive systems

the subspace bornology and E=i.K/ carries the quotient bornology. We get the same criterion for extensions in Born and Cborn. In these cases, Lemma 1.61 yields that the subspace i.K/  E is automatically closed because Q is separated. If we specialise further and consider categories of bornological algebras, bornological modules over some bornological algebra, or bornological chain complexes, then extensions are characterised by the same conditions. That is, the additional algebraic structure imposes no further constraints on extensions. (Of course, the maps i and p have to be algebra or module homomorphisms or chain maps). Usually, not every monomorphism i W A ! B in a pointed category can be embedded in an extension, and the same problem occurs for epimorphisms. In such cases, it is interesting to describe those monomorphisms and epimorphisms that can become part of an extension; they are called normal mono- and epimorphisms. In the category Born1=2 of bornological vector spaces, these are the bornological embeddings and the bornological quotient maps in the following sense: Definition 1.74. A bounded linear map f W V ! W between two bornological vector spaces is called a bornological embedding if it is an isomorphism onto its range equipped with the subspace bornology from W , and a bornological quotient map if the induced map V = ker f ! W is a bornological isomorphism. Many arguments involving extensions require some kind of section. This is formalised by the following definition: i

p

Definition 1.75. A bounded linear section for an extension K  E  Q in Born1=2 is a bounded linear map s W Q ! E with p ı s D idQ . An extension that admits such a section is called split or semi-split. The existence of a bounded linear section implies that E Š K ˚ Q as bornological vector spaces, so that such extensions deserve to be called split. But we are mainly interested in extensions of bornological algebras, modules, or chain complexes. They are only called split if the section is an algebra or module homomorphism or a chain map, respectively; if the section is merely bounded linear, we call them semi-split. Example 1.76. The functor Cpt is fully exact on the category of Fréchet spaces, that is, a diagram of Fréchet spaces is an extension of topological vector spaces if and only if Cpt maps it to an extension of bornological vector spaces. This follows from Theorem 1.62 and Example 1.58. Example 1.77. The fine bornology functor Fin W Vect ! Cborn  Born1=2 is fully exact, that is, a diagram of fine bornological vector spaces is a bornological extension if and only if it is an extension of vector spaces. The forgetful functor Born1=2 ! Vect is exact but not fully exact.

1.3 Constructions with bornological vector spaces

35

1.3.5 Direct sums and products P Definition 1.78. Let .Vi /i2I be a set of bornological vector spaces and let i2I Vi Q and i2I Vi be the vector space P direct sum and direct product of these spaces. The direct sum bornology on i2I Vi is the finest bornology for which the natural P maps j W Vj ! i2I Vi are bounded for all j 2 I ; equivalently, it is the bornology generated by the subsets of the form Q j .B/ for B 2 S.Vj / and j 2 I . The direct product bornology on i2I Vi is the coarsest bornology for which Q Q the projections j W i2I Vi ! Vj are bounded for all j 2 I ; equivalently, B  i2I Vi is bounded if and only if j .B/  Vj is bounded for all j 2 I . We always equip direct sums and products with these canonical bornologies. It is easy to check that the direct sums and products defined above yield coproducts and products in Born1=2 in the categorical sense (see [64]). P Q If the spaces Vi are separated or complete for all i 2 I , then so are i2I Vi and i2I Vi (see [50]). Hence we get coproducts and products in the full subcategories Cborn and Born of Born1=2 as well. By construction, the forgetful functor Born1=2 ! Vect and its restrictions to Born and Cborn commute with products and coproducts. Lemma 1.79. Products and coproducts preserve kernels and cokernels: if fi W Vi ! Vi0 is a family of bounded linear maps, then the natural maps Y  X  Y X ker ker.fi / ! ker fi ; fi ! ker.fi /; i2I

X i2I

i2I

coker.fi / ! coker

X i2I

 fi ; coker

i2I

Y



fi !

i2I

i2I

Y

coker.fi /

i2I

are bornological isomorphisms. Similar assertions hold for separated cokernels. Direct products and direct sums of extensions are again extensions. Proof. The assertions about kernels and cokernels are straightforward to check. The assertion about extensions follows because they are defined using kernels and cokernels. Direct sum and product coincide if I is finite. In particular, V1 ˚ V2 Š V1  V2 . Definition 1.80. A category C is called additive if it has the following properties: • the morphism spaces in C carry Abelian group structures such that the composition of morphisms is additive in each variable; • finite products and coproducts exist and agree in C. This includes the existence of a zero object, so that additive categories are pointed. A functor F W C1 ! C2 between additive categories is called additive if there are natural isomorphisms F .A1  A2 / Š F .A1 /  F .A2 / for all A1 ; A2 22 C1 .

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1 Bornological vector spaces and inductive systems

It is clear that the categories Born1=2, Born, and Cborn are additive. The addition of morphisms in an additive category is canonical. A functor F W C1 ! C2 is additive if and only if it induces group homomorphisms   F W Hom.A1 ; A2 / ! Hom F .A1 /; F .A2 / for all A1 ; A2 22 C1 : Remark 1.81. Abelian categories are additive categories with two additional properties: each morphism has a kernel and a cokernel, and maps that are both monic and epic are isomorphisms. This notion is of no use for us because the categories that we need are not Abelian. To see this for Cborn, let V be an infinite-dimensional Banach space. Then the identical map Fin.V / ! vN.V / is monic and epic but not invertible.

1.3.6 Tensor products and complete tensor products Each of the categories Born1=2, Born, and Cborn carries a canonical tensor product, which turns it into a closed symmetric monoidal category. Let V1 ; V2 22 Born1=2 and let V1 ˝ V2 be their vector space tensor product. We always equip V1 ˝ V2 with the (convex) bornology that is generated by the subsets B1 ˝ B2 WD fb1 ˝ b2 j b1 2 B1 ; b2 2 B2 g for B1 2 Sd .V1 /, B2 2 Sd .V2 /. Thus a subset of V1 ˝ V2 is bounded if and only if it is contained in the convex hull of B1 ˝ B2 for some B1 2 Sd .V1 /, B2 2 Sd .V2 /. Let \ W V1  V2 ! V1 ˝ V2 be the natural bilinear map. By the universal property of the vector space tensor product, any bilinear map f W V1  V2 ! W is of the form fN ı \ for a unique linear map fN W V1 ˝ V2 ! W . By construction, fN is bounded if and only if f is bounded. An analogous assertion holds for uniformly bounded families of maps. Thus we get a bornological isomorphism Hom.V1 ˝ V2 ; W / Š Hom.2/ .V1  V2 I W /

(1.82)

for all V1 ; V2 ; W 22 Born1=2. This determines V1 ˝ V2 uniquely up to isomorphism. Definition 1.83. Let V1 and V2 be complete bornological vector spaces. Their complete projective bornological tensor product is defined by y V2 WD .V1 ˝ V2 /c : V1 ˝ y V2 . By the We have a canonical bounded bilinear map \ W V1  V2 ! V1 ˝ universal properties of the completion and ˝, composition with \ induces a bornological isomorphism y V2 ; W / Š Hom.2/ .V1  V2 I W / Hom.V1 ˝ (1.84) y V2 uniquely up to isomorphism. for all V1 ; V2 ; W 22 Cborn. This determines V1 ˝ y Fin.V2 / Š Fin.V1 ˝ V2 / Example 1.85. We have Fin.V1 / ˝ Fin.V2 / Š Fin.V1 / ˝ for all vector spaces V1 and V2 .

1.3 Constructions with bornological vector spaces

37

Comparison with the topological tensor product. The complete projective topoy  W for complete locally convex topological vector spaces logical tensor product V ˝ y W ! X is defined by a similar universal property: continuous linear maps V ˝ correspond to jointly continuous bilinear maps V  W ! X . This tensor product is defined by Alexander Grothendieck in [36]. Let V and W be complete locally convex topological vector spaces. We have canonical maps y Cpt.W / ! Cpt.V ˝ y  W /; ˆV;W W Cpt.V / ˝ y vN.W / ! vN.V ˝ y W / ˆ0V;W W vN.V / ˝ y  W / and vN.V ˝ y  W / are complete and the canonical bilinear because Cpt.V ˝ y W / y  W / and vN.V /  vN.W / ! vN.V ˝ maps Cpt.V /  Cpt.W / ! Cpt.V ˝ are bounded. The issue is when these maps are bornological isomorphisms. We can only expect positive results for special topological vector spaces like Fréchet spaces. In this case, we can use results of Alexander Grothendieck ([36]). y W Example 1.86. Let V and W be Banach spaces. We recall the definition of V ˝ in this case and observe that ˆ0V;W is an isomorphism. Let B  V and D  W be the closed unit balls. Any bounded subset of vN.V / ˝ vN.W / is absorbed by .B ˝ D/} . Thus vN.V / ˝ vN.W / is a normed space equipped with its von Neumann bornology. Its norm is the gauge norm for .B ˝ D/} ; this is the maximal norm  with the property that .x ˝ y/  1 whenever x 2 B, y 2 D or, equivalently, .x ˝ y/  B .x/  D .y/

for all x 2 V , y 2 W .

The maximal norm with this property is, by definition, the projective tensor product norm of [36]. Since the completions for Banach and bornological vector spaces are y vN.W / Š vN.V ˝ y  W /. compatible, we get vN.V / ˝ Theorem 1.87. Let V and W be Fréchet spaces. The canonical continuous bilinear y  W induces a bornological isomorphism map \ W V  W ! V ˝ y Cpt.W / Š Cpt.V ˝ y  W /: Cpt.V / ˝

(1.88)

This is a consequence of the following results of [36]: y  W is of the form Theorem 1.89. Let V and W be two Fréchet spaces. Any x 2 V ˝ X xD .n/yn ˝ zn n2N

with null-sequences .yn / and .zn / in V and W and with  2 `1 .N/. Furthermore, if y  W , then there are null-sequences .yn / in V and .xk /k2N is a null-sequence in V ˝ .zn / in W , and a null-sequence k 2 `1 .N/ with X xk D k .n/yn ˝ zn : n2N

38

1 Bornological vector spaces and inductive systems

The first assertion is contained in [36, Théorème 1 in I.§2, no. 1, p. 51]; the second one is [36, Remarque 4 in I.§2, no. 1, p. 57]. Proof of Theorem 1.87. We must check that \ induces an isomorphism y  W /; X / Š Hom.2/ .Cpt.V /  Cpt.W /; X / Hom.Cpt.V ˝ y Cpt.W / uniquely. for all X 22 Cborn because this property characterises Cpt.V / ˝ Let f 2 Hom.2/ .Cpt.V /  Cpt.W /; X /. We want to show that f is of the form fQ ı \ y  W / ! X. for a unique bounded linear map fQ W P Cpt.V ˝ y Let x 2 V ˝ W and write x D n2N .n/  yn ˝ zn as in Theorem 1.89. Then X X X .n/fQ.yn ˝ zn / D .n/fQ ı \.yn ; zn / D .n/f .yn ; zn /: fQ.x/ D n2N

n2N

n2N

Hence fQ is unique if it exists. We claim that this formula for fQ.x/ is independent of P y W the infinite series representing x. Equivalently, .n/yn ˝ zn D 0 in V ˝ P if with .n/, yn , zn as in Theorem 1.89, thenP .n/f .yn ; zn / D 0. By assumption, the sequence xk WD nk .n/yn ˝ zn is a null-sequence in y  W . By the second half of Theorem 1.89, we can find other null-sequences .yn0 / V ˝ and .zn0 / in V and W and a null-sequence .0k / 2 `1 .N/ such that X X .n/yn ˝ zn D 0k .n/yn0 ˝ zn0 : n2N

nk

Since f is a bounded bilinear map, the set ff .yn0 ; zn0 / j n 2 Ng is bounded, so that X X X .n/f .yn ; zn / D lim .n/f .yn ; zn / D lim 0k .n/f .yn0 ; zn0 / D 0: n2N

k!1

nk

k!1

n2N

y  W ! X . It remains As a consequence, we have a well-defined linear map fQ W V ˝ y  W / to X . to prove that fQ is bounded as a map from Cpt.V ˝ y  W be precompact. By Theorem 1.18, there is a null-sequence Let S  V ˝ y .xk /k2N in V ˝ W such that S is contained in the complete disked hull of fxk g. As above, the second half of Theorem 1.89 yields that fQ.xk / is a bornological nullsequence in X. Hence the complete disked hull of ffQ.xk /g is bounded in X . Since this contains fQ.S/, we conclude that fQ is bounded. yW Corollary 1.90. Let V and W be Fréchet spaces and let S be the bornology on V ˝ consisting of all subsets S with S ~ B ˝ D for von Neumann bounded disks B  V , y  W induces D  W . Then the canonical continuous bilinear map \ W V  W ! V ˝ y vN.W / Š .V ˝ y  W; S/. an isomorphism vN.V / ˝ Proof. The complete disked hulls .B ˝ D/~ that are implicit in the definition of S y  W / is complete and all sets of the form B ˝ D are von make sense because vN.V ˝ y W . The only non-trivial Neumann bounded. We claim that S is a bornology on V ˝

1.3 Constructions with bornological vector spaces

39

y  W ; this follows from the first half of point to check is that fxg 2 S for all x 2 V ˝ Theorem 1.89. The bornology S is complete by construction. Hence we have a natural map y vN.W / ! .V ˝ y  W; S/: vN.V / ˝ If we are given a bounded bilinear map f W vN.V /  vN.W / ! X with X 22 Cborn, then we get a bounded bilinear map Cpt.V /  Cpt.W / ! X as well, so that y  W / ! X by Theorem 1.87. we can extend f to a bounded linear map fQ W Cpt.V ˝ y  W; S/ The bornology S is designed so that fQ remains bounded on S. Thus .V ˝ y satisfies the universal property that characterises vN.V / ˝ vN.W /. yW . In general, S may be strictly smaller than the von Neumann bornology on V ˝ This is equivalent to the negative answer to Grothendieck’s Problème des Topologies; this question is asked in [36, p. I.33–34] and answered negatively by Jari Taskinen in [100]. But we know many cases where S equals the von Neumann bornology. The following theorem of Grothendieck lists a few: y W Theorem 1.91. Let V and W be Fréchet spaces. The bornology S on V ˝ described in the statement of Corollary 1.90 is equal to the von Neumann bornology in the following cases: (1) if both V and W are Banach spaces; y  W can be identified (2) if V D L1 .X; / for some measure space; in this case, V ˝ with the space of W -valued, Lebesgue integrable functions .X; / ! W ; (3) if V is nuclear. We have already treated the first case in Example 1.86. The second one is contained in [36, Proposition 12 in I.§2, no. 2, p. 68–69], the third one in [36, Proposition 12 in II.§3, no. 1, p. 73–4]. Grothendieck’s memoir [36] contains several examples of tensor product computations for Fréchet spaces, which are examples in Cborn as well by Theorem 1.87. Warning 1.92. Projective tensor products are incompatible with supremum norms: the y W for a compact topological space X differs from C .X; W /. tensor product C .X / ˝ There are other Banach space tensor products that are compatible with Lp -norms for p ¤ 1 (see [30]). Next we consider LF-spaces instead of Fréchet spaces. This requires inductive limits, which we study carefully in §1.3.9. For LF-spaces, the most useful topological y  defined in [36, I.§3, tensor product is the inductive topological tensor product ˝ no. 1, p. 73–78]. It is defined to be universal for separately continuous bilinear maps. Whereas the projective topological tensor product commutes with projective limits, the inductive one commutes with inductive limits. For Fréchet spaces, the projective and inductive tensor products agree because there is no difference between joint and separate continuity for bilinear maps defined on Fréchet spaces.

40

1 Bornological vector spaces and inductive systems

Theorem 1.93. Let V and W be LF-spaces, and suppose that one of them is a strict inductive limit of nuclear Fréchet spaces. Then the canonical bilinear map induces an y Cpt.W / Š Cpt.V ˝ y  W /. isomorphism Cpt.V / ˝ Proof. Write V D lim Vn and W D lim Wn as inductive limits of Fréchet spaces. ! ! Recall that a precompact subset of an LF-space such as V is already contained in and precompact in a Fréchet subspace. Since the inductive topological tensor product commutes with inductive limits, we have y  Wn : y  W Š lim Vn ˝ V ˝ ! The nuclearity assumption ensures that this inductive system is again strict and therefore yields an LF-space. Hence   y  Wn / Š lim Cpt.Vn / ˝ y Cpt.Wn / y  W / Š lim Cpt.Vn ˝ Cpt.V ˝ ! ! y lim Cpt.Wn / Š Cpt.V / ˝ y Cpt.W /: Š lim Cpt.Vn / ˝ ! ! The first and last isomorphism follow from the description of precompact subsets of LF-spaces in Theorem 1.32; the second one follows from Theorem 1.87; the third one y commutes with inductive limits. follows because ˝ y  to define periodic cyclic homology In [4], Jacek Brodzki and Roger Plymen use ˝ for topological algebras with separately continuous multiplication whose underlying topological vector space is a nuclear LF-space. An important example is the Schwartz algebra of a reductive p-adic group. Theorem 1.93 shows that the theory in [4] is a special case of ours. Warning 1.94. If we allow arbitrary complete locally convex topological vector spaces, then the complete inductive tensor product need not be associative. A separately cony  Y /Z ! W need not give a separately continuous bilinear tinuous bilinear map .X ˝ y  Z/ ! W because this would involve extending a separately continuous map X .Y ˝ y  Z/. But bilinear map on the incomplete tensor product X  .Y ˝ Z/ to X  .Y ˝ the extended map need not remain separately continuous in the variable X . Example 1.95. Let M be a smooth manifold and let V be a bornological vector space. Then the space Cc1 .M / of compactly supported smooth functions on M is a nuclear y commutes with direct LF-space. We equip it with the precompact bornology. Since ˝ y V Š lim C01 .U / ˝ y VB , where B 2 Sc .V / and U runs limits, we have Cc1 .M / ˝ ! through the directed set of relatively compact open subsets of M . Theorem 1.93 yields y VB Š C01 .U / ˝ y  VB ; this is known to be isomorphic to C01 .U; VB /. C01 .U / ˝ Reversing the steps above, we get y V: Cc1 .M; V / Š Cc1 .M / ˝ As a special case, we obtain y Cc1 .M2 / Š Cc1 .M1  M2 / Cc1 .M1 / ˝ for two smooth manifolds M1 and M2 .

1.3 Constructions with bornological vector spaces

41

1.3.7 Functions of bounded variation and Stieltjes integrals Let V be a complete bornological vector space. We define the space BV .Œ0; 1; V / of functions of bounded variation Œ0; 1 ! V and the space A.Œ0; 1; V / of continuous functions of bounded variation Œ0; 1 ! V . Given f0 2 C.Œ0; 1; V0 / and f1 2 R1 y V1 and check some BV .Œ0; 1; V1 /, we define the Stieltjes integral 0 f0 ˝ df1 2 V0 ˝ simple identities. Functions of bounded variation Definition 1.96. A function f W Œ0; 1 ! V has bounded variation if there is a bounded disk S 2 Sd .V / such that f .t/ 2 S for all t 2 Œ0; 1 and n1 X

kf .ij C1 /  f .ij /kS  1

j D0

for any partition 0 D i0 < i1 <    < in D 1 of Œ0; 1. A set of functions has uniformly bounded variation if the same S works for all its elements. Definition 1.97. Let BV.Œ0; 1; V / be the space of functions Œ0; 1 ! V of bounded variation; a subset is bounded if its elements have uniformly bounded variation. We let A.Œ0; 1; V / D C .Œ0; 1; V / \ BV .Œ0; 1; V / be the space of continuous bounded variation functions Œ0; 1 ! V ; a subset of A.Œ0; 1; V / is bounded if it is both uniformly continuous and of bounded variation. By construction, BV .Œ0; 1; V / Š lim BV .Œ0; 1; VS /, and BV .Œ0; 1; VS / is a ! Banach space with the von Neumann bornology; hence BV .Œ0; 1; V / is a complete bornological vector space. Similarly, A.Œ0; 1; V / is a complete bornological vector space and A.Œ0; 1; V / Š lim A.Œ0; 1; VS /; but A.Œ0; 1; VS / does not carry the von ! Neumann bornology because C .Œ0; 1; VS / does not. It is known that a scalar-valued function f has bounded variation if and only if its derivative f 0 in the sense of distributions exists and is a Borel measure. It is easy to see that monotone functions have bounded variation. Conversely, any bounded variation function is a difference of two monotone functions. In contrast, bounded variation functions Œ0; 1 ! V need not have a derivative. A counterexample (even with vanishing variation) is the function Œ0; 1 ! L1 .Œ0; 1; dt / that maps x 2 Œ0; 1 to the characteristic function of Œ0; x (see [32]). There is a similar notion of bounded variation functions with values in a locally convex topological vector space. These two notions are compatible for Fréchet spaces:     Theorem 1.98. If V is a Fréchet space, then BV Œ0; 1; vN.V / D vN BV .Œ0; 1; V / . Proof. Clearly, bounded variation in the bornological sense is stronger than bounded variation in the topological sense. For a function f W Œ0; 1 ! V and a partition I, we define I .f / W N ! V by I .f /.n/ WD f .inC1 /  f .in /, extended by 0 outside

42

1 Bornological vector spaces and inductive systems

y  V . By the range of I. We view I .f / as an element of `1 .N; V / Š `1 .N/ ˝ definition, f has bounded variation in the topological sense if and only if the set of y  V . By functions I .f / for all partitions I is von Neumann bounded in `1 .N/ ˝ Theorem 1.91, any von Neumann bounded subset is contained in .D ˝ B/~ , where D is the unit ball of `1 .N/ and B  V is a von Neumann bounded closed disk. Thus f has bounded variation as a function to VB and hence to vN.V /. The same argument shows that the sets of functions with uniformly bounded variation are exactly the von Neumann bounded subsets of BV .Œ0; 1; V /. We can also define absolutely continuous functions by requiring lim

I!1

n1 X

kf .ij C1 /  f .ij /kS D 0

j D0

for some S 2 S.V /. In the scalar-valued case, this is equivalent to f 0 2 L1 .Œ0; 1; dt /. Although we do not use this notion in the sequel, we remark that absolute continuity is compatible with precompact bornologies on Fréchet / denotes the  spaces:  if AC.V   space of absolutely continuous functions, then AC Cpt.V / Š Cpt AC .V / . The Stieltjes integral Notation 1.99. Let f0 W Œ0; 1 ! V0 and f1 W Œ0; 1 ! V1 be functions into complete bornological vector spaces. For a partition I D .0 D i0 < i1 <    < in D 1/, let †I f0 ˝ df1 WD

n1 X

  y V1 : f0 .ij / ˝ f1 .ij C1 /  f1 .ij / 2 V0 ˝

j D0

The set of partitions of Œ0; 1 is a directed set with respect to the refinement partial order. The definition of convergence for sequences extends to functions from directed sets to complete bornological vector spaces: such a function f W I ! V converges to f1 if and only if there are i0 2 I , S 2 Sd .V /, and a function " W Ii0 ! RC with lim "i D 0 such that f1  f .i / 2 "i  S for all i 2 Ii0 ; equivalently, f .i / is eventually contained in VS and converges to f1 in this normed space. Lemma 1.100. The limit limI!1 †I f0 ˝ df1 over the directed sets of partitions y V1 if f0 2 C .Œ0; 1; V0 / and f1 2 BV .Œ0; 1; V1 / and is denoted by exists in V0 ˝ Z 1 Z 1 f0 ˝ df1 D f0 .t / ˝ f10 .t / dt: 0

0

The resulting map Z 1 y V1 ; W C .Œ0; 1; V0 /  BV .Œ0; 1; V1 / ! V0 ˝ 0

Z .f0 ; f1 / 7!

1

f0 ˝ df1 ; 0

is bilinear and bounded for the bornology of uniform boundedness on C.Œ0; 1; V0 /.

1.3 Constructions with bornological vector spaces

43

y V1 is complete, Proof. We check that .†I f0 ˝ df1 /I is a Cauchy net. Since V0 ˝ this ensures convergence. If f0 is continuous and f1 of bounded variation, we get S0 2 Sc .V0 / and S1 2 Sc .V1 / with the following properties: • f0 .t/ 2 S0 , f1 .t / 2 S1 for all t 2 Œ0; 1; • for any " > 0 there is a partition 0 D i0 < i1 <    < in D 1 with f0 .t /f0 .ij / 2 " S0 for all t 2 Œij ; ij C1 ; P 0 0 0 0 • jm1 D0 kf1 .ij C1 /  f1 .ij /kS1  1 for all partitions 0 D i0 <    < im D 1. Fix " > 0 and let I D .0 D i0 < i1 <    < in D 1/ be an appropriate partition as above. Let I 0 be a refinement of I; we write it as 0 D i0 D i00 < i01 <    < i0;`0 D i1 D i10 <    < in1;`n1 D in D 1 and use telescoping sums to compare †I f0 ˝ df1  †I0 f0 ˝ df1 D

n1 `X j 1 X

  f0 .ij / ˝ f1 .ij;kC1 /  f1 .ij;k /

j D0 kD0



n1 `X j 1 X

  f0 .ij k / ˝ f1 .ij;kC1 /  f1 .ij;k /

j D0 kD0

D

n1 `X j 1 X 

   f0 .ij /  f0 .ij k / ˝ f1 .ij;kC1 /  f1 .ij;k /

j D0 kD0

2

n1 `X j 1 X

" S0 ˝ kf1 .ij;kC1 /  f1 .ij;k /kS1 S1 } " S0 ˝ S1 :

j D0 kD0

Hence we have got a Cauchy net. The boundedness of

R1 0

is clear.

Z

1

Lemma 1.101. The Stieltjes integral satisfies Z 1 df D f .1/  f .0/ 0

for all f 2 BV .Œ0; 1; V / and Z 1 Z f0 ˝ d.f1 ˝ f2 / D 0

0

1

f0 ˝ f1 ˝ df2 C

f0 ˝ .df1 / ˝ f2 0

for f0 2 C.Œ0; 1; V0 /, f1 2 A.Œ0; 1; V1 /, f2 2 A.Œ0; 1; V2 /; here we flip tensor y V1 ˝ y V2 Š V0 ˝ y V2 ˝ y V1 to define the last integral. factors V0 ˝

44

1 Bornological vector spaces and inductive systems

R1 R1 Proof. We interpret 0 df D 0 1 df for the constant function 1. The formula for R1 0 df follows because †I df D f .1/  f .0/ for any partition I. It is easy to see that f1 ˝ f2 has bounded variation if both f1 and f2 have. Given a partition I, we compute †I f0 ˝ d.f1 ˝ f2 /  †I f0 ˝ f1 ˝ df2  †I f0 ˝ .df1 / ˝ f2 D

n1 X

  f0 .ij / ˝ f1 .ij C1 / ˝ f2 .ij C1 /  f1 .ij / ˝ f2 .ij /

j D0

   f0 .ij / ˝ f1 .ij / ˝ f2 .ij C1 /  f1 .ij / ˝ f2 .ij /    f0 .ij / ˝ f1 .ij C1 / ˝ f2 .ij /  f1 .ij / ˝ f2 .ij /

D

n1 X

    f0 .ij / ˝ f1 .ij C1 /  f1 .ij / ˝ f2 .ij C1 /  f2 .ij / :

j D0

Since f0 and f1 are continuous and f2 has bounded variation, similar estimates as in the proof of Lemma 1.100 show that this net converges to 0 for I ! 1. R1 If f1 is, say, continuously differentiable, then 0 f0 ˝ df1 agrees with the more y V1 . Hence the traditional integral of the continuous function f0 ˝ f10 W Œ0; 1 ! V0 ˝ R1 0 notation 0 f0 .t / ˝ f1 .t / dt for the Stieltjes integral creates no ambiguities. y A bounded bilinear map b W V0  V1 ! V extends to a bounded linear map V0 ˝ R1 V1 ! V , which we can apply to the Stieltjes integral 0 f0 ˝ df1 ; we also denote  R1 R1  the result by 0 b.f0 ; df1 / or 0 b f0 .t /; f10 .t / dt . In particular, this applies if V0 D V1 D V is a bornological algebra and b its multiplication.

1.3.8 Symmetric monoidal categories y  ˝ y Vn . It is Cyclic homology uses iterated tensor products V1 ˝    ˝ Vn or V1 ˝ crucial that such tensor products are associative and commutative and that our ground field acts as a unit object. This is formalised in the following definition: Definition 1.102. A symmetric monoidal category is a category C together with a bifunctor ˝ W CC ! C, an object 1 22 C called unit object, and natural isomorphisms .A ˝ B/ ˝ C Š A ˝ .B ˝ C /;

A ˝ B Š B ˝ A;

1 ˝ A Š A Š A ˝ 1;

called associativity, commutativity, and unit constraints. These natural transformations are subject to several conditions, which are listed carefully in [91]. An additive symmetric monoidal category is a symmetric monoidal category which is also additive and such that the functor ˝ is additive in each variable separately. Similarly, a K-linear symmetric monoidal category for a field K is a K-linear additive symmetric monoidal such that ˝ is K-bilinear.

1.3 Constructions with bornological vector spaces

45

Remark 1.103. The associativity, commutativity, and unit constraints are part of the structure of a symmetric monoidal category. It is possible to have non-equivalent choices here, which is relevant in some applications such as quantum field theory. But there are obvious choices for these maps in the cases that we are interested in, so that we usually do not even mention them. If .C; N˝/ is a symmetric monoidal category, then there is a well-defined tensor product i2I Xi for any finite set of objects .Xi /i2I , and there are natural transformations O O O Xi ˝ Xi Š Xi i2I1

i2I2

i2I

N for any disjoint union decomposition I D I1 t I2 . By convention, ; Xi D 1. The conditions on the natural transformations in Definition 1.102 are the minimal ones to ensure that this construction has the expected properties. We will frequently use tensor powers X ˝n for n 2 N, which satisfy X ˝0 D 1;

X ˝1 D X;

X ˝2 D X ˝ X;

X ˝mCn Š X ˝m ˝ X ˝n :

Examples 1.104. Classical examples of symmetric monoidal categories are the categories Sets and Ab of sets and Abelian groups with  and ˝ as tensor products; the unit objects are the one-point set in Sets and Z 22 Ab. The category Ab is additive, y are additive Sets is not. The categories .Born1=2; ˝/, .Born; ˝/, and .Cborn; ˝/ symmetric monoidal categories as well (we will prove a stronger statement below). The category of complete locally convex topological vector spaces is additive symmetric y  . The subcategories of Fréchet spaces and Banach spaces monoidal with respect to ˝ y  and therefore additive symmetric monoidal as well. are closed with respect to ˝ Algebras and modules. The framework of additive symmetric monoidal categories is ideal to study constructions that work for algebras of all kinds: rings, topological algebras, Banach algebras, bornological algebras, pro-algebras, and so on. Following Guillermo Cortiñas and Christian Valqui ([13]), we develop cyclic homology in this general context (some results require Q-linearity instead of additivity). This clarifies why the basic theory works in the same way for all reasonable kinds of algebras. Definition 1.105. An algebra in a symmetric monoidal category C is an object A 22 C together with a map m W A ˝ A ! A called multiplication or product, such that the diagram m˝idA /A˝A A˝3 idA ˝m

 A˝A

m m

 /A

commutes. This condition encodes the associativity of A; it uses the canonical isomorphism .A ˝ A/ ˝ A Š A ˝ .A ˝ A/ from the symmetric monoidal category structure.

46

1 Bornological vector spaces and inductive systems

A morphism f W A ! B between two algebras in C is called multiplicative or an algebra homomorphism if the diagram A˝A

m

/A

m

 /B

f ˝f

f

 B ˝B

commutes. We write Alg.A; B/ for the set of algebra homomorphisms A ! B; these form the morphisms of the category of algebras in C, which we denote by Alg.C/. Definition 1.106. An algebra is unital if it comes equipped with a map u W 1 ! A called unit such that the diagram idA ˝u

u˝idA

/A˝Ao A ˝ 1P 1˝A PPP nn PPŠP Š nnn m nnn can PPPP PP'  wnnnnn can A

commutes; the isomorphisms A ˝ 1 Š A Š 1 ˝ A come from the symmetric monoidal category structure. An algebra homomorphism f W A ! B between two unital algebras is unital if u

f

1!  A ! B agrees with the unit for B. We write AlgC .A; B/ for the set of unital algebra homomorphisms A ! B; these sets form the morphism spaces of the category of unital algebras in C, which we denote by AlgC .C/. The unit map u W 1 ! A is unique if it exists. Definition 1.107. Let A be an algebra in C. A (left) A-module in C is an object M 22 C together with a map mM W A ˝ M ! M , such that the diagram A˝A˝M

mA ˝idM

/A˝M

idA ˝mM

 A˝M

mM

 /M

mM

commutes; here we use the associativity constraint to identify A ˝ .A ˝ M / Š .A ˝ A/ ˝ M . Right modules are defined similarly. A map f W M1 ! M2 between two A-modules in C is called an A-module homomorphism if the diagram m / M1 A ˝ M1 idA ˝f

 A ˝ M2

m

f

 / M2

1.3 Constructions with bornological vector spaces

47

commutes. We write HomA .M1 ; M2 / for the space of A-module homomorphisms A ! B; these sets form the morphism spaces of the category of (left) A-modules in C, which we denote by Mod.A/. Definition 1.108. Let A be a unital algebra in C with unit u W 1 ! A. A (left) u˝idM

m

! M is the canonical A-module M is unital if the map 1 ˝ M ! A ˝ M  isomorphism 1 ˝ M ! M . The category of unital A-modules is a full subcategory of Mod.A/, which we denote by ModC .A/. The categories Mod.A/ and ModC .A/ are additive categories if C is additive; that is, morphism spaces are Abelian groups in a natural way and there exist finite direct products. An algebra homomorphism f W A ! B induces an additive functor f  W Mod.B/ ! Mod.A/ by composing the module structure B ˝ M ! M with f ˝ idM . Suppose that A and B are unital; then this functor restricts to a functor f  W ModC .B/ ! ModC .A/ if (and only if) f is unital. Example 1.109. The (unital) algebras in .Ab; ˝/ are exactly the (unital) rings. In the categories Born1=2, Born, or Cborn, we obtain the separated or complete bornological (unital) algebras in the sense of Definition 1.26. Closed symmetric monoidal categories Definition 1.110. A symmetric monoidal category .C; ˝/ is called closed if ˝ has a right adjoint functor Hom W Cop  C ! C, which is called the internal Hom functor. This functor is determined uniquely by the existence of natural isomorphisms   Hom.A ˝ B; C / Š Hom A; Hom.B; C / for all A; B; C 22 C, where Hom denotes spaces of morphisms in C. The internal Hom functor is not so crucial for the construction of the cyclic theories, but it is convenient for many other purposes. Proposition 1.111. The categories Born1=2 and Born with tensor functor ˝ and unit object C are closed symmetric monoidal categories. The category Cborn with tensor y and unit object C is a closed symmetric monoidal category as well. The functor ˝ internal Hom functor is the functor Hom introduced above in all cases. Proof. We first consider the very easy case of the incomplete tensor product. In this case, the associativity, commutativity, and unit constraints are the same as in the category of vector spaces. It is evident that they are bounded with respect to the appropriate tensor product bornologies. The various compatibility conditions hold because they hold in the category of vector spaces. To establish the adjointness between ˝ and Hom, we combine the universal property (1.82) of ˝ with the natural isomorphism   Hom.2/ .V1  V2 I V3 / Š Hom V1 ; Hom.V2 ; V3 /

48

1 Bornological vector spaces and inductive systems

  that sends a bilinear map f W V1  V2 ! V3 to the linear map x1 7! x2 7! f .x1 ; x2 / . This finishes the argument for the categories Born1=2 and Born. y in Cborn. We have Now we consider the complete tensor product ˝   y V2 ; V3 / Š Hom .V1 ˝ V2 /c ; V3 Hom.V1 ˝   Š Hom.V1 ˝ V2 ; V3 / Š Hom V1 ; Hom.V2 ; V3 / for all V1 ; V2 ; V3 22 Cborn. Notice that the functor Hom here is exactly the same as in the larger category Born1=2. The associativity, commutativity, and unit constraints on the tensor functor in a symmetric monoidal category are equivalent to certain properties of the internal Hom functor; we will discuss these properties below. Since the functors y and ˝ have essentially the same internal Hom functor and ˝ is symmetric monoidal, ˝ y has all required properties. we conclude that ˝ Remark 1.112. It is much harder to find a closed symmetric monoidal structure on the category of complete locally convex topological vector spaces. The complete projective y  yields a symmetric monoidal structure, but it cannot be topological tensor product ˝ y  would have to commute with closed: if there were an internal Hom functor, then ˝ inductive limits, which it does not. y  instead, which Given this, we may try the complete inductive tensor product ˝ does commute with inductive limits. However, it does not define a monoidal structure because of its lack of associativity pointed out in Warning 1.94. This problem is created y . by the completion that is inherent in ˝ If we drop completeness, then we finally obtain a closed symmetric monoidal category: the underlying category is the category of (possibly incomplete) locally convex topological vector spaces, the tensor product is the uncompleted inductive topological tensor product ˝ , and the internal Hom functor is obtained by equipping spaces of continuous linear maps with the topology of pointwise convergence. But this category does not seem so useful because many applications require completeness. The internal Hom functor in a closed symmetric monoidal category always has various nice properties (see [91]). Let .C; ˝; 1/ be a closed symmetric monoidal category with internal Hom functor Hom. We define spaces of multi-linear maps by Hom.n/ .X1      Xn I Y / WD Hom

n O

 Xi ; Y :

iD1

This reproduces the familiar functors Hom.n/ for categories of bornological vector spaces. A commutativity constraint is equivalent to natural isomorphisms Hom.n/ .X1      Xn I Y / Š Hom.n/ .X.1/      X.n/ I Y /

for all permutations  2 Sn . A unit constraint is equivalent to natural isomorphisms Hom.n/ .X1      Xn I Y / Š Hom.nC1/ .X1      Xj  1      Xn I Y /:

49

1.3 Constructions with bornological vector spaces

An associativity constraint is equivalent to a natural isomorphism   Hom.X ˝ Y; Z/ Š Hom X; Hom.Y; Z/ : This is stronger than the defining property of Hom because   Hom.X; Y / Š Hom 1; Hom.X; Y / : We have natural maps Hom.X; X / ˝ X ! X

(evaluation map),

Š

X ! Hom.1; X /; Hom.Y; Z/ ˝ Hom.X; Y / ! Hom.X; Z/ Hom.X1 ; Y1 / ˝ Hom.X2 ; Y2 / ! Hom.X1 ˝ X2 ; Y1 ˝ Y2 / 1 ! Hom.X; X /

(composition), (exterior product), (identity map). (1.113) We refer to [91] for more details. These maps lift the familiar maps on the level of Hom to Hom and inherit the expected properties. For instance, the composition product is associative and specialises to the evaluation map if we identify X Š Hom.1; X /. The maps in (1.113) turn Hom.X; X / for an object X 22 C into a unital algebra. A (unital) module structure on M is equivalent to a (unital) algebra homomorphism A ! Hom.M; M /. Similarly, a (unital) algebra structure can be encoded by a (unital) algebra homomorphism A ! Hom.A; A/ with suitable properties. In particular, this applies to our categories of bornological vector spaces. In this case, we can also replace morphisms A˝M ! M or A˝A ! A by bounded bilinear maps A  M ! M or A  A ! A. This shows that the categorical definitions of (unital) algebras and (unital) modules in Definitions 1.105–1.108 reproduce the familiar notions for the categories Born1=2, Born, and Cborn. An internal Hom functor in a pointed symmetric monoidal category is automatically compatible with kernels and cokernels in the sense that there are natural isomorphisms   ker f W Hom.X; A/ ! Hom.X; B/ Š Hom.X; ker f /;   (1.114) ker f  W Hom.B; X / ! Hom.A; X / Š Hom.coker f; X / for any f 2 C whose kernel or cokernel exists and any object X 22 C. This follows easily from the universal properties of kernels and cokernels and the adjointness between Hom and ˝. Similarly, we get compatibility with coproducts and products (if they exist in C): M  Y Hom Ai ; X Š Hom.Ai ; X /; i2I



Hom X;

Y i2I



Ai Š

i2I

Y i2I

for any set of objects .Ai /i2I and any X 22 C.

(1.115) Hom.X; Ai /

50

1 Bornological vector spaces and inductive systems

Symmetric monoidal functors Definition 1.116. A symmetric monoidal functor F W C1 ! C2 between two symmetric monoidal categories is a functor F with natural isomorphisms F .X / ˝ F .Y / Š F .X ˝ Y / that are compatible with the associativity, commutativity, and unit constraints in both categories (see [91]). These conditions are necessary N compatibility N to well-define natural isomorphisms F Xi Š F .Xi / for any finite set of objects .Xi /. We will only consider cases where the isomorphisms F .X / ˝ F .Y / Š F .X ˝ Y / are obvious, so that we do not care to specify them. sep

c

The functors Born1=2 ! Born  ! Cborn are symmetric monoidal functors in this sense, that is, we have natural isomorphisms y V2c Š .V1 ˝ V2 /c V1c ˝

sep.V1 / ˝ sep.V2 / Š sep.V1 ˝ V2 /;

(1.117)

for V1 ; V2 in the appropriate categories. As a consequence, we get natural isomorphisms Hom.n/ .sep.V1 /      sep.Vn /I VnC1 / Š Hom.n/ .V1      Vn I VnC1 /

(1.118)

if V1 ; : : : ; Vn 22 Born1=2, VnC1 22 Born and Hom.n/ .V1c      Vnc I VnC1 / Š Hom.n/ .V1      Vn I VnC1 /

(1.119)

if V1 ; : : : ; Vn 22 Born, VnC1 22 Cborn. The functor Cpt from Fréchet spaces to Cborn is symmetric monoidal as well. It is evident that the isomorphism in Theorem 1.87 is compatible with the associativity, commutativity, and unit constraints.

1.3.9 Direct and inverse limits Direct and inverse limits are general recipes for constructing objects of categories. We briefly recall these constructions and refer to [64] for more details. The data for both is a diagram in a category C, that is, a functor F W I ! C from some small category I (called indexing category) to C. Let F W I ! C be a diagram in a category C. Definition 1.120. A cone over F is an object C 22 C with morphisms 'i W C ! F .i / for all i 22 I , such that F .˛/ ı 'i D 'j for all i; j 22 I and all morphisms ˛ W i ! j in I (see the first diagram in (1.121)). Dually, a cone under F is an object C 0 22 C with morphisms 'i0 W F .i / ! C 0 for all i 22 I , such that 'j0 ı F .˛/ D 'i0 for all i; j 22 I and all morphisms ˛ W i ! j in I (see the second diagram in (1.121)). w C GGG GG'k ww w GG 'j ww GG w {ww  # / / F .k/ F .i/ F .j / 'i

F .˛/

F .ˇ /

F .i /

F .˛/ / F .j / F .ˇ / / F .k/ GG GG ww GG ww 'j0 w G w GG w 0 'i0 #  {ww 'k 0 C

(1.121)

1.3 Constructions with bornological vector spaces

51

Definition 1.122. The inverse limit of F is a universal    cone over F , that is, a cone C; .'i /i22I over F such that for any other cone D; . i /i22I over F there is a unique morphism f W D ! C with i D 'i ı f for all i 22 I . We denote the inverse limit by lim F .    Dually, the direct limit of F is a universal under F , that is, a cone C 0 ; .'i0 /i22I   0 cone under F such that for any other cone D ; . i0 /i22I under F there is a unique morphism f W C 0 ! D 0 with i0 D f ı 'i0 for all i 22 I . We denote the direct limit by lim F . ! D 9Š f

 j C D DD z DD zz D zz' z 'j DD i z " 

}z / F .j / F .i/ i

F .˛/

F .i / DD 0 DD'i DD DD " 0 i

/ F .j / y y yy y y y| y 'j0

C0

(1.123)

0 j

9Š f

  D0

F .˛/

Direct and inverse limits need not exist in general, but they do in many important categories like the categories of sets, Abelian groups, or topological spaces. If you already know what direct and inverse limits of sets are, you can reduce direct and inverse limits in other categories to them. The inverse limit lim F is an object of C  together with natural isomorphisms   (1.124) Hom.X; lim F / Š lim Hom X; F .i /   i22I

  for all X 22 C. The right hand side involves the diagram of sets i 7! Hom X; F .i / and hence is again a set. Similarly, the direct limit lim F is an object of C together ! with natural isomorphisms Hom.lim F; X / Š lim Hom.F .i /; X / ! 

(1.125)

i22I

for all X 22 C. These constructions contain some of our previous ones as special cases: f

• The kernel of a morphism f W A ! B is the inverse limit of the diagram A ! B 0. f

Dually, the cokernel of f is the direct limit of the diagram 0 A ! B. Q • The product i2I Ai of a set of objects .Ai /i2I is the inverse limit of the diagram I ! C, i 7! Ai , where we view I as a discrete category with only identity morphisms. P Dually, the coproduct i2I Ai is the direct limit of the same diagram; coproducts in additive categories are also called direct sums.

52

1 Bornological vector spaces and inductive systems

Conversely, general inverse and direct limits in additive categories can be reduced to kernels, products, cokernels, and coproducts. The direct limit of a diagram F W I ! C is the cokernel of the natural map M

M

A;B22I f 2I.A;B/

F .A/ !

M

F .C /;

C 22I

whose restriction to the summand F .A/ for f W A !L B is the difference of the inclusion L F .A/  F .C / and F .f / W F .A/ ! F .B/  F .C /. The inverse limit is the kernel of a similar map between direct products. We have already seen that any morphism in the categories Born1=2, Born, and Cborn has a kernel and a cokernel and that these categories have products and coproducts of arbitrary cardinality. Hence we conclude: Proposition 1.126. Any diagram in one of the categories Born1=2, Born, and Cborn has both a direct and an inverse limit. Definition 1.127. A category with this property is called bicomplete. For inverse limits, it makes no difference whether we work in Born1=2, Born, or Cborn: if X is a diagram of complete or separated bornological vector spaces, then the inverse limit of X in Born1=2 agrees with the inverse limit in Born or Cborn, respectively. The corresponding assertion for direct limits fails because we have to use sep coker in the categories Born and Cborn and coker in Born1=2. Therefore, we often write sep lim X for the direct limit of a diagram X in the categories Born and ! Cborn. It follows that the forgetful functor Born1=2 ! Vect commutes with inverse and direct limits in Born1=2. If we restrict to Born or Cborn, then this fails because the direct limits involve separated quotients. Let C be a closed symmetric monoidal category. Then we can lift the defining isomorphisms (1.124) and (1.125) to isomorphisms on the internal Hom functor:     Hom X; lim F .i / Š lim Hom X; F .i / ;   (1.128) Hom.lim F .i /; X / Š lim Hom.F .i /; X /: !  This follows from the adjointness between Hom and ˝. Equation (1.128) implies similar assertions about kernels, cokernels, products, and coproducts. Furthermore, the tensor product functor commutes with arbitrary direct limits, that is, 

   lim F .i / ˝ B Š lim F .i / ˝ B ; ! !

(1.129)

because it has a right adjoint. By Proposition 1.111, we can apply these general facts to our categories of bornological vector spaces. We are mainly interested in projective systems and inductive systems:

1.4 Categories of inductive systems

53

Definition 1.130. An inductive system is a diagram indexed by a directed set .I; /; thus it consists of objects .Ai /i2I for i 2 I and maps ˛ij W Ai ! Aj for all i; j 2 I with i  j , such that ˛jk ı ˛ij D ˛ik for i; j; k 2 I with i  j  k. A projective system is an inductive system in the opposite category, that is, we reverse the directions of the maps ˛ij .

1.4 Categories of inductive systems An inductive system B D .Bi ; ˇij /i2I in C yields a contravariant functor C ! Sets;

A 7! lim Hom.A; Bi /I !

we use the maps Hom.A; ˇij / W Hom.A; Bi / ! Hom.A; Bj / induced by the maps ˇij to turn i 7! Hom.A; Bi / into an inductive system of sets. We need this functor to define morphisms between inductive systems: Definition 1.131. If A D .Ai ; ˛ij / and B D .Bk ; ˇkl / are two inductive systems in the same category C, possibly with different indexing sets I and K, then we let Hom.A; B/ be the set of natural transformations between the associated functors lim Hom. ; Ai / ! lim Hom. ; Bk /: ! ! !  We let C be the category whose objects are the inductive systems and whose morphisms are given by Hom.A; B/. More explicitly, we have Hom.A; B/ Š lim lim Hom.Ai ; Bk /:  ! i

(1.132)

k

Any morphism is represented by a family of maps fi W Ai ! Bk.i/ for a function k W I ! K, such that for any i; j 2 I with i  j there is k 2 K with k  k.i /; k.j / for which the diagram Ai

fi

/ Bk.i/

j ˛i

 Aj

fj

/ Bk.j /

k ˇk.i / k ˇk.j /

 / Bk

commutes. The explicit description shows that these morphisms form a set, not just a class. We leave it to the reader to formulate explicitly when two such families represent the same morphism A ! B. It is possible to describe the composition of morphisms in terms of such families. Since this gets rather messy, we prefer the abstract Definition 1.131.

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1 Bornological vector spaces and inductive systems

Remark 1.133. We can often replace a morphism of inductive systems by a morphism of diagrams, that is, a family of maps fi W Ai ! Bi satisfying the obvious compatibility condition. As above, we represent a morphism of inductive systems by a compatible k family of maps fi W Ai ! Bk.i/ . For k  k.i /, let fik WD ˇk.i/ ı fi W Ai ! Bk . Let M be the set of pairs .i; k/ with k  k.i /. It becomes a directed set for the partial order .i; k/  .j; l/ ” i  j , k  l, and ˇkl ı fik D fjl ı ˛ij . By construction, we get a commuting diagram A

f

/B

Š

 j .Ai ; ˛i /.i;k/2M

Š

.fik /.i;k/2M

 / .Bk ; ˇ l /.i;k/2M k

of inductive systems whose bottom row is a morphism of diagrams. We view objects of C as constant inductive systems. The functor Cop ! Sets;

A 7! lim Hom.A; Bk / ! k

that is used in Definition 1.131 to define morphisms of inductive systems is equivalent !  to Hom.A; B/. Hence we get a fully faithful functor C ! C . If A D .Ai ; ˛ij /, then we have Hom.A; B/ Š lim Hom.Ai ; B/. By the definition (1.125) of a direct limit, i this means .Ai /i2I D lim Ai : ! i

! ! ! We are mainly interested in the categories Norm1=2, Norm, and Ban of inductive systems of semi-normed spaces, normed space, and Banach spaces. By definition, ! ! ! Ban is a full subcategory of Norm, which in turn is a full subcategory of Norm1=2. Example 1.134. Let .Ai ; ˛ij /i2I be an inductive system indexed by a directed set .I; / and let S  I be cofinal. Then the embedding S ! I induces an isomorphism of inductive systems Š

.Ai ; ˛ij /i2S  ! .Ai ; ˛ij /i2I : To construct the inverse, choose a function m W I ! S with m.i /  i for all i 2 I ; this is possible because S is cofinal. This function and the maps ˛im.i/ W Ai ! Am.i/ represent a morphism of inductive systems. It is easy to check that it is the required inverse. A more elegant proof shows first that both inductive systems yield equivalent notions of cone (Definition 1.120) and hence equivalent notions of direct limit.

1.4 Categories of inductive systems

55

1.4.1 Direct and inverse limits Recall that a category is bicomplete if any diagram has a direct and an inverse limit. ! ! ! Proposition 1.135. The categories Ban, Norm, Norm1=2 are bicomplete. ! ! The embeddings Ban ! Norm preserve all direct and inverse limits. The embed! ! ding Norm ! Norm1=2 preserves direct sums, direct limits of inductive systems, and inverse limits; it does not preserve general direct limits. !  Proof. Let C be a category and let F W I ! C be a diagram. Then we can form contravariant functors lim F W Cop ! Sets; ! lim F W Cop ! Sets; 

  A 7! lim Hom A; F .i / ; ! i2I   A 7! lim Hom A; F .i / :  i2I

These are the direct and inverse limit of F in the (large) category of contravariant functors C ! Sets (this is not quite a category because the morphism spaces may !  be classes instead of sets). In order to verify that C is bicomplete, we must realise these functors lim F and lim F by inductive systems, which then provide the direct !  !  and inverse limit of the diagram in C . We only have to consider kernels, cokernels, products, and coproducts because other direct and inverse limits are constructed out of these special cases if C is additive. !  In general, we claim that the existence of kernels and cokernels in C follows if !  kernels and cokernels exist in C. We represent a morphism in C by a morphism of diagrams, that is, a compatible family of maps fi W Ai ! Bi as in Remark 1.133; this may involve changing the indexing categories of source and target. Once we have a morphism of diagrams, we may form inductive systems .ker fi / and .coker fi /; these !  realise the kernel and cokernel of f in C . ! ! This explicit construction also shows that the embedding Ban ! Norm preserves ! ! kernels and cokernels and that the embedding Norm ! Norm1=2 preserves kernels; the latter functor does not preserve general cokernels because the cokernels in Norm1=2 and Norm differ by taking separated quotients. !  Next we construct coproducts. Again, this works in C for any category C with finite coproducts. Let A.k/ , k 2 K, be a set of inductive systems. Write A.k/ D j;.k/ .A.k/ /, with indexing sets I .k/ . We consider the partially ordered set of pairs i ; ˛i .F; '/, where F  K is a finite subset and ' selects an element of I .k/ for each k 2 F ; the partial order is defined by .F; '/  .F 0 ; ' 0 / ” F  F 0 and '.k/  ' 0 .k/ for all k 2 F .

56

1 Bornological vector spaces and inductive systems

L This partially ordered set is directed. Let k2K A.k/ be the inductive system AF ;' WD L .k/ k2F A'.k/ , where the structure maps AF ;' ! AF 0 ;' 0 are induced by the maps L .k/ !  ' 0 .k/;.k/ ˛'.k/ . One checks easily that this realises the coproduct A in C . ! ! ! This yields coproducts in Norm1=2, Norm, and Ban and shows that the embeddings between these categories preserve coproducts. The construction of direct products is slightly more subtle because the categories Norm1=2, Norm, and Ban are not closed underQ products. Let A.k/ be a set of inductive systems in Norm1=2, say. The direct product k2K A.k/ is indexed by the set of all pairs of functions '1 W K ! I , '2 W K ! N1 with '1 .k/ 2 I .k/ for all k 2 K, and with the obvious partial order .'1 ; '2 /  .'10 ; '20 / ” '1 .k/  '2 .k/ and '10 .k/  '20 .k/ for all k 2 K. For such a pair ˆ WD .'1 ; '2 /, we let n ˇ Q ˇ Aˆ WD .xk /k2K 2 k2K A.k/ '1 .k/

kxk k '2 .k/

o remains bounded :

This is a semi-normed space in an obvious fashion, and A WD .Aˆ / isQan inductive system of semi-normed spaces. One checks that it realises the product k2K A.k/ in ! ! ! Norm1=2. The same construction also works in the subcategories Norm and Ban. ! ! ! Finally, we must check that the embeddings Ban ! Norm ! Norm1=2 preserve inductive limits. This is not clear from !  the above construction. But there is a direct construction of inductive limits in C for any category C: an inductive system of inductive systems yields a one-variable inductive system by merging the indexing sets appropriately. This merged inductive system realises the inductive limit, and it is clear ! ! that this construction is compatible with the embedding Norm ! Norm1=2.

1.4.2 Separated quotients and completions The separated quotient functor for bornological vector spaces restricts to a functor sep W Norm1=2 ! Norm; which is again left adjoint to the embedding Norm ! Norm1=2. The induced functor   ! ! sep W Norm1=2 ! Norm; .Ai / 7! sep.Ai / ; is the separated quotient functor for inductive systems. It is left adjoint to the embed! ! ding functor Norm ! Norm1=2: Hom.sep.A/; B/ Š lim lim Hom.sep.Ai /; Bk /  ! i

k

Š lim lim Hom.Ai ; Bk / Š Hom.A; B/  ! i

k

1.4 Categories of inductive systems

57

! ! for all A 22 Norm1=2, B 22 Norm. The completion functor Norm ! Ban, V 7! V c , is defined in the usual way. Since this completion satisfies Hom.V c ; W / Š Hom.V; W / for all W 22 Cborn, it is the restriction of the completion functor Born1=2 ! Cborn. It also extends to a completion functor ! ! c  W Norm ! Ban; .Vi / 7! .Vic /: The same computation as for the separated quotient functor shows that it is left adjoint ! ! to the embedding functor Ban ! Norm. Composing both left adjoints, we get a left ! ! adjoint functor Norm1=2 ! Ban, A 7! .sep A/c for the embedding functor. The separated quotient and completion functors are left adjoints and hence commute with arbitrary direct limits. (This is not very informative for separated quotients because ! the direct limit functor in Norm directly involves separated quotients.) One can check that both functors also commute with products.

1.4.3 Tensor products of inductive systems !  If .C; ˝; 1/ is a symmetric monoidal category, then C inherits a symmetric monoidal category structure: define ˝ for inductive systems entrywise by .Ai /i2I ˝ .Bk /k2K WD .Ai ˝ Bk /.i;k/2I K : ! ! ! y in Ban. This defines tensor products ˝ in Norm1=2, Norm and ˝ !  Proposition 1.136. This tensor product turns C into a symmetric monoidal category, whose unit is the constant inductive system 1. Suppose that C is closed with internal !  !  Hom functor Hom, and suppose that C is bicomplete. Then C is closed as well, with 



Hom .Ai /; .Bk / WD lim lim Hom.Ai ; Bk /:

 !

i

k

! ! ! In particular, this applies to our categories Norm1=2, Norm, and Ban. Proof. It is straightforward to check that the tensor product on inductive systems remains associative and commutative and that the constant inductive system 1 acts as a !  unit object. Thus C becomes a symmetric monoidal category. It remains to check that the above definition of Hom yields an internal Hom functor. The universal property of

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1 Bornological vector spaces and inductive systems

projective limits yields natural isomorphisms     Hom A; Hom.B; C / D Hom A; lim lim Hom.Bk ; Cm /  ! k m     Š lim Hom A; lim Hom.Bk ; Cm / Š lim lim Hom Ai ; lim Hom.Bk ; Cm /    ! ! m m i k k    Š lim lim lim Hom Ai ; Hom.Bk ; Cm / Š lim lim lim Hom Ai ˝ Bk ; Cm /   !   ! m m i i k k  Š lim lim Hom Ai ˝ Bk ; Cm / Š Hom.A ˝ B; C /  ! i;k

m

!  for any A; B; C 22 C . Now the general machinery of closed symmetric monoidal categories pays off because various useful assertions about internal Hom functors follow from the general theory, as explained in §1.3.8. For instance, we get natural maps Hom.B; C /  Hom.A; B/ ! Hom.A; C /

that lift the composition of morphisms. In particular, this provides Hom.A; A/ with an ! algebra structure for any object A 22 Ban. We also get compatibility results between the internal Hom functor and direct and inverse limits. Any symmetric monoidal category has a canonical forgetful functor Forget W C ! Sets;

X 7! Hom.1; X /I

if C is additive or K-linear for some field K, then the same construction yields a forgetful functor to Abelian groups or K-vector spaces, respectively. We always have Forget Hom.A; B/ Š Hom.A; B/. For categories of bornological vector spaces, this yields the underlying vector space. For categories of inductive systems, the canonical forgetful functor maps .Ai /i2I to the direct limit lim Ai in the category of vector spaces. ! ! ! ! ! The functors sep W Norm1=2 ! Norm and c W Norm ! Ban are symmetric monoidal because this holds for the corresponding functors Norm1=2 ! Norm ! ! ! Ban. The embedding Norm ! Norm1=2 is symmetric monoidal as well, but the ! ! embedding Ban ! Norm is not. ! !  Warning 1.137. The categories Alg.C/ and Alg. C / are different; this is clear from Theorem 3.70 below. !  In contrast, the category of A-modules in C for A 22 Alg.C/ is equivalent to the category of inductive systems of A-modules in C. The proof is similar to the proof of the corresponding statement for chain complexes (Theorem 2.15).

1.5 Dissecting bornological vector spaces

59

1.5 Dissecting bornological vector spaces Let V be a bornological vector space. Recall that the set of bounded disks in V with the relation of absorption is a directed set Sd .V / (Lemma 1.8). Definition 1.138. The semi-normed spaces VB for B 2 Sd .V / form an inductive system in Norm1=2 because we have injective bounded linear maps VB ! VD for B; D 2 Sd .V / with B  D. This construction is functorial: a bounded linear map f W V ! W induces a map of directed sets Sd .V / ! Sd .W / and bounded linear maps fB W VB ! Wf .B/ for B 2 Sd .V /, which combine to a morphism of inductive ! systems. The resulting functor diss W Born1=2 ! Norm1=2 is called dissection functor. If V is separated, then VB is a normed space for all B 2 Sd .V /, so that the ! dissection restricts to a functor diss W Born ! Norm. If V is complete, then we ! get an isomorphic object of Norm if we replace Sd .V / by the cofinal subset Sc .V / ! (Lemma 1.8). Hence we can further restrict to a functor diss W Cborn ! Ban. Thus we get a commuting diagram of functors Cborn





diss

diss

 ! Ban

/ Born



 / ! Norm

/ Born1=2 diss



 ! /  Norm1=2 .

In the opposite direction, we have three inductive limit functors ! ! lim W Norm1=2  Born1=2 ! Born1=2; ! ! ! sep lim W Norm ! Born; Ban ! Cborn: !

1.5.1 Properties of the dissection functor Theorem 1.139. The dissection functor has the following properties: (1) We have natural isomorphisms lim diss.V / Š V for all V 22 Born: ! ! (2) The functor diss W Born ! Norm is fully faithful. (3) It commutes with inverse limits and direct sums. (4) It commutes with Hom and spaces of multi-linear maps, that is,   Hom diss.V1 /; diss.V2 / Š diss Hom.V1 ; V2 /;   Hom.n/ diss.V1 /  diss.Vn /I diss.VnC1 / Š diss Hom.n/.V1   Vn I VnC1 /:

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1 Bornological vector spaces and inductive systems

! ! (5) The functor sep lim W Norm ! Born is left adjoint to diss W Born ! Norm; ! even more, we have natural isomorphisms diss Hom.sep lim V; W / Š Hom.V; diss W / ! ! for all V 22 Norm, W 22 Born, and similar isomorphisms for spaces of multi-linear maps. ! (6) The functor sep lim W Norm ! Born is symmetric monoidal. ! ! (7) Assertions (1)–(6) remain true for if we replace the triple .Born; Norm; sep lim/ ! ! ! by .Born1=2; Norm1=2; lim/ or .Cborn; Ban; sep lim/. ! ! ! ! (8) The functors diss W Born ! Norm and diss W Born1=2 ! Norm1=2 are sym! metric monoidal as well, but the functor diss W Cborn ! Ban is not. Proof. Let V be a bornological vector space. Then the vector space direct S limit of diss V may be identified with V because VB  V for all B 2 Sd .V / and VB D V . Thus we get a vector space isomorphism lim diss.V / Š V . A subset of lim diss.V / is ! ! bounded if and only if it is absorbed by some B 2 Sd .V /. Hence the vector space isomorphism lim diss.V / Š V is bornological. Of course, if V is separated, then this ! implies sep lim diss.V / Š V , finishing the proof of (1). ! Next we prove (5). We claim that Hom.V; diss W / WD lim Hom.V; WB / Š diss Hom.V; W /

!

(1.140)

B

for any V 22 Norm, W 22 Born; it is crucial here that V is just a normed space. Equation 1.140 is equivalent to the fact that a set of linear maps V ! W is uniformly bounded if and only if it is uniformly bounded as a set of maps V ! WB ! for some B 2 Sd .W /. Applying the forgetful functor Norm ! Vect, we also get Hom.V; diss W / D Hom.V; W /. ! Now let V D .Vi ; ˛ij / 22 Norm. We have already observed that an inductive system is naturally isomorphic to its own inductive limit: V Š lim.Vi /. Hence ! Hom.V; diss W / Š lim Hom.Vi ; diss W / Š lim Hom.Vi ; W / Š Hom.sep lim V; W /:  !  i

i

The last isomorphism is just the defining property of inductive limits (1.125). As a ! ! result, the functor sep lim W Norm ! Born is left adjoint to diss W Born ! Norm. ! It is straightforward to verify that the dissection functor commutes with direct sums. To get (3), notice that right adjoint functors such as diss and Hom commute with inverse

1.5 Dissecting bornological vector spaces

61

limits, whereas left adjoint functors such as lim commute with direct limits. Combining ! this with (1.140), we may copy the above argument for the internal Hom functors: Hom.V; diss W / Š lim Hom.Vi ; diss W / Š lim diss Hom.Vi ; W /





i

i

Š diss lim Hom.Vi ; W / Š diss Hom.sep lim V; W /  ! i

! for all V 22 Norm, W 22 Born. We get corresponding statements for the higher internal Hom functors Hom.n/ because we can describe them by iterating Hom. This finishes the proof of (5). Combining this adjointness property with sep lim diss.V / Š V , we get ! diss Hom.V; W / Š diss Hom.sep lim diss V; W / Š Hom.diss V; diss W / ! for all V; W 22 Born. Applying the forgetful functors, this contains (2). The same arguments also work for spaces of multi-linear maps, yielding (4). Since the tensor product functor in Born is a left adjoint functor, it commutes with inductive limits in both variables. This implies easily that sep lim commutes with tensor ! products. The resulting isomorphism sep lim V1 ˝ sep lim V2 Š sep lim .V1 ˝ V2 / is ! ! ! compatible with the associativity, commutativity, and unit constraints as well, yielding (6). ! All arguments so far carry over literally to the situations Born  Norm and ! Cborn  Ban, yielding (7). Next we check that diss commutes with incomplete tensor products. Recall that the bornology on W1 ˝ W2 is generated by subsets of the form .B1 ˝ B2 /} with Bj 2 Sd .Wj / for j D 1; 2. These disked hulls are the unit balls of the corresponding tensor products .W1 /B1 ˝ .W2 /B2 . Since these special disks form a cofinal subset in Sd .W1 ˝ W2 /, we get   diss.W1 ˝ W2 / Š .W1 /B1 ˝ .W2 /B2 B 2S .W / D diss W1 ˝ diss W2 : j

d

j

That is, diss commutes with the tensor product ˝. Again, these natural isomorphisms are compatible with the associativity, commutativity, and unit constraints, yielding (8). ! y is false because completions in Born1=2 and Norm The corresponding assertion for ˝ may be different.

1.5.2 Functional analysis with inductive systems Let V D .Vi /i2I be an inductive system of normed spaces. We do not have enough elements of V in this case (see §A.2.1) and, therefore, cannot talk of convergent sequences or Cauchy sequences in V . But the corresponding spaces for Vi are well-defined normed

62

1 Bornological vector spaces and inductive systems

spaces and form inductive systems. Thus we get well-defined inductive system of convergent and Cauchy sequences in V . ! Similarly, if V 22 Ban, then we get inductive systems C0 .X; V / for a locally compact space X , C k .M; V / for k 2 N [ f1; !g and a compact C k -manifold M , ! and BV.Œ0; 1; V / and A.Œ0; 1; V /. Each of these functors on Ban extends the corresponding functor on Cborn in the sense that we have natural isomorphisms C0 .X; diss V / Š diss C0 .X; V / for all V 22 Cborn; and so on. In this way, we carry over the basic constructions of functional analysis to categories of inductive systems.

1.5.3 The essential range of the dissection functor Definition 1.141. A bornological vector space V is the direct union of an inductive system .Vi / of bornological vector spaces if V D lim .Vi /, Vi  V for all i , and the ! structure maps Vi ! Vj are identical inclusions for all i  j . The inductive systems that arise in this fashion are reduced in the following sense: Definition 1.142. An inductive system .Ai ; ˛ij / in a category C is called reduced if all the maps ˛ij are monomorphisms (in our applications, this means: injective). It is called essentially reduced if it is isomorphic to a reduced inductive system. Definition 1.143. The essential range of a functor F is the set of all objects that are isomorphic to F .X / for some X . Proposition 1.144. Let V D .Vi ; fij / be an inductive system of normed spaces. Then the following assertions are equivalent: (1) V is essentially reduced; (2) for all i 2 I there is j 2 Ii with ker fij D ker fik for all k 2 Ij ; (3) V Š diss.sep lim V /; ! (4) V Š diss.W / for some W 22 Born, that is, V belongs to the essential range of diss. ! ! Analogous statements hold for .Norm1=2; Born1=2; lim/ or .Ban; Cborn; sep lim/ in ! ! ! stead of .Norm; Born; sep lim/. ! Moreover, if V is an essentially reduced inductive system in Born, then lim V is ! separated, so that sep lim V D lim V . ! !

1.5 Dissecting bornological vector spaces

63

Proof. (1) , (2). This equivalence holds for inductive systems in any category C. We only prove (2) ) (1) because the other implication follows from the other characterisations. Fix j.i / D j 2 Ii as in (2) and let VQi WD Vi = ker fij.i/ . We have .fik /1 .ker fkj.k/ / D ker fij.i/ for all i; k 2 I with j  i . Hence the map fij descends to a monomorphism fQij W VQi ! VQj . This yields a reduced inductive system VQ D .VQi ; fQij /. The quotient maps and the maps fij.i/ define morphisms of inductive systems V ! VQ ! V that are inverse to each other. Hence V Š VQ is essentially reduced. (1) , (3). We may assume that V is itself reduced. Then the direct limit lim V ! contains Vi as a vector subspace for all i 2 I , and a subset of lim V is bounded if ! and only if it is absorbed by the closed unit ball of Vi for some i 2 I . Hence these closed unit balls form a cofinal subset of Sd .lim V /, and lim V is separated. Thus ! ! diss.sep lim V / Š V . ! (3) , (4) is trivial. (4) , (2). Represent the isomorphism V ! diss W by a compatible family of bounded linear maps ˛i W Vi ! WB.i/ . Since the maps WB ! WD for B  D are all injective, the maps ˛i are compatible in the much stronger sense that ˛j ı fij and ˛i agree as maps to W for all i; j 2 I with i  j . Therefore, if ˛i .x/ ¤ 0 for x 2 Vi , then fij .x/ ¤ 0 for all j 2 Ii , that is, ker fij  ker ˛i for all j 2 Ii . The inductive system ker.˛i / is isomorphic to 0 because .˛i / is an isomorphism. Hence there is j  i with ker fij D ker ˛i ; any such j 2 Ii will do for (2).

1.5.4 Dissection of extensions Recall that a functor between pointed categories is called fully exact if it preserves and detects extensions. Proposition 1.145. The functors in the following diagram are fully exact: Cborn



diss

 ! Ban

/ Born



diss

diss



 / ! Norm

/ Born1=2



 ! /  Norm1=2 .

Proof. Our explicit description of extensions in categories of bornological vector spaces implies that the embeddings Cborn ! Born ! Born1=2 are fully exact. All functors that occur in our statement are fully faithful and right adjoints by Theorem 1.139. Hence they preserve inverse limits and, in particular, kernels. Therefore, the condition i Š ker.p/ is preserved and detected by all of our functors.

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1 Bornological vector spaces and inductive systems

  It remains to check what happens to the condition p Š coker.i / Š coker ker.p/ . We claim that all the functors F in our diagram satisfy   F coker ker.f / Š coker ker F .f / for any morphism f W V ! W . Being fully faithful as well, they preserve and detect the property p Š coker ker.p/. ! ! The claim is trivial for the embedding Ban ! Norm: kernels and cokernels are constructed by exactly the same recipe in both categories. ! ! To verify the claim for the embedding Norm ! Norm1=2, we have to look again at the explicit construction of kernels and cokernels in these categories during the proof of Proposition 1.135. This involves taking entrywise kernels and cokernels. The ! only difference is that for Norm we have to take separated quotients of the entrywise cokernels. But for a map of the form ker.f / ! V the entrywise ranges are already closed, so that there is no need to take separated quotients. Therefore, the embedding ! ! Norm ! Norm1=2 preserves coker ker.f /. Finally, we turn to the dissection functors. It suffices to check the assertion for ! the dissection functor Born1=2 ! Norm1=2. We have coker ker.f / D V = ker.f /. The inductive system diss ker.f / is isomorphic to .VB\ker.p/ /B2Sd .V / . Hence the cokernel of the map diss ker.f / ! diss V is isomorphic to the inductive system  .VB =VB\ker.p/ /B2Sd .V / . This is isomorphic to diss V = ker.f / as desired. Lemma 1.146. Let C be a pointed category in which each morphism has a kernel and !  a cokernel. Then any extension K  E  Q in C is isomorphic to the inductive limit of a diagram of extensions Ki  Ei  Qi in C. Proof. Replace the map K  E by a morphism of diagrams fi W Ki0 ! Ei as in Remark 1.133. Let Qi WD coker fi ; and Ki WD ker.Ei ! Qi /. The universal property of the kernel provides canonical maps Ki0 ! Ki . Now the universal property of the cokernel shows that Qi is a cokernel for the map Ki ! Ei as well. Thus we have an extension Ki  Ei  Qi . Since it is constructed naturally out of fi , these extensions form a diagram. Since .fi / represents the map K  E and Q is its cokernel, we have Q Š .Qi /. The same argument for the maps Ei  Qi shows that K Š .Ki /. Thus the given extension is the inductive limit of the diagram of extensions Ki  Ei  Qi .

1.5.5 Separated quotients and completions once again sep

c

The separated quotient and completion functors Born1=2 ! Born  ! Cborn commute with arbitrary direct limits and, in particular, with inductive limits because they are left adjoints. This does not give much information about separated quotients because

65

1.5 Dissecting bornological vector spaces

they already enter in the definition of inductive limits in Born. For the completion functor, Theorem 1.139 and compatibility with inductive limits imply V c Š .sep lim diss V /c Š sep lim .diss V /c : ! !

(1.147)

This provides a construction of the completion for bornological vector spaces, using ! ! the straightforward completion functor Norm ! Ban. The same construction is used by Henri Hogbe-Nlend in [46]. Theorem 1.139 implies   y Š sep lim diss.V /˝sep y y V ˝W lim diss.W / Š sep lim diss.V /˝diss.W / (1.148) ! ! ! y W is the for all V; W 22 Cborn. In more concrete terms, (1.148) means that V ˝ y WD , where B separated direct limit of the inductive system of Banach spaces VB ˝ and D run through the directed sets Sc .V / and Sc .W /, respectively. The inductive systems in (1.147) and (1.148) need not be reduced because the completion does not preserve injectivity. This is why the map V ! V c may fail to be injective for V 2 Born (see Example 1.66). In contrast, the canonical map V ! V c ! is a monomorphism for any V 22 Norm. If V 22 Born, then the canonical map V ! V c induces a map diss V ! diss.V c /, which induces a map .diss V /c ! diss.V c / by the universal property of completions ! because diss.V c / 22 Ban. If this map is an isomorphism, it makes no difference ! whether we complete V in Cborn or Ban. The next proposition gives several equivalent conditions for this: Proposition 1.149. Let V be a separated bornological vector space. Then the following conditions are equivalent: (1) V is subcomplete, that is, the natural map V ! V c is a bornological embedding; (2) there is a bornological embedding of V into a complete bornological vector space; ! (3) the natural map .diss V /c ! diss.V c / is an isomorphism in Ban; ! (4) .diss V /c belongs to the essential range of diss W Cborn ! Ban; (5) for any B 2 Sd .V / there is D 2 Sd .V / with D  B, such that any Cauchy sequence in VB that is a null-sequence in V already is a null-sequence in VD . More equivalent characterisations of subcomplete spaces can be found in [67]. Proof. The implications (1) ) (2) and (3) ) (4) are trivial. (2) ) (3). Let V ! W be a bornological embedding with W 22 Cborn. Then the set of bounded disks of the form B \ V with B 2 Sc .W / is cofinal in Sd .V /. Hence diss V is isomorphic to the inductive system .VB\V /B2Sc .W / . The completion

66

1 Bornological vector spaces and inductive systems

of VB\V is a subspace of the Banach space WB and hence injects into W . Therefore, c the inductive system .VB\V /B2Sc .W / is reduced. By Proposition 1.144 and (1.147), this implies   .diss V /c Š diss sep lim.diss V /c Š diss.V c /: ! (4) , (5). We verify that (5) is equivalent to condition (2) of Proposition 1.144 for the inductive system diss V . Let B 2 Sd .V /. Any x 2 VBc is the limit of a Cauchy sequence .xn / in VB . The sequence .xn / is a null-sequence in V if and only if it is a null-sequence in VD for some D 2 Sd .V /. We may assume D  B, so that we get an induced map VBc ! VDc . Of course, x D lim xn is annihilated by this map if and only if .xn / is a null-sequence in VD . Thus (5) is equivalent to the requirement that for any B 2 Sd .V / there is D 2 Sd .V / with D  B such that ker.VBc ! VDc / is maximal, as desired. (4) , (1). Suppose that .diss V /c Š diss W for some W 22 Cborn. Then Theorem 1.139 and (1.147) yield W Š sep lim diss W Š sep lim .diss V /c D lim .diss V /c D V c : ! ! ! We use the last assertion in Proposition 1.144 to get rid of the separated quotients here. Since lim is exact on the category of vector spaces and the map diss V ! .diss V /c is ! monic, the map V ! V c is injective. Let S  V c be bounded and contained in V . It remains to prove that S 2 S.V /. There is B 2 Sd .V / such that S is contained in the unit ball of VBc . Let D 2 Sd .V / be as in (5) for the disk B. If x 2 S , then x D lim xn for some Cauchy sequence .xn / in VB . Then xn  x is a Cauchy sequence in VB and a null-sequence in V . Thus it is a null-sequence in VD by construction of D. Hence x D .x  xn / C xn 2 "n D C B  .1 C "n /D for all n 2 N, where lim "n D 0. This yields x 2 D because D is a disk. Therefore, S  D is bounded already in V , as desired. Example 1.150. Let V be a locally convex topological vector space, and let V c be its completion as a topological vector space. Then the natural map V ! V c is a topological embedding. It induces bornological embeddings Cpt.V / ! Cpt.V c / and vN.V / ! vN.V c /. Since Cpt.V c / and vN.V c / are complete, Proposition 1.149 yields that Cpt.V / and vN.V / are subcomplete. By the way, the completion of Cpt.V / is isomorphic to Cpt.V c / if V is metrisable (see [67]). The corresponding assertions for the von Neumann bornology or for nonmetrisable spaces only hold under additional hypotheses. Corollary 1.151. Let V; W 22 Cborn. The natural map y .diss W / ! diss.V ˝ y W/ .diss V / ˝ is an isomorphism if and only if V ˝ W is subcomplete. Proof. Theorem 1.139implies that diss.V ˝ W / Š .diss V / ˝ .diss W / and hence c  y y / Š diss .V ˝W /c . .diss V /˝.diss W / Š diss.V ˝W / . Furthermore, diss.V ˝W Now the assertion follows from Proposition 1.149.

1.6 Metrisability and the approximation property

67

1.6 Metrisability and the approximation property 1.6.1 Bornological metrisability We have seen that the von Neumann and precompact bornologies on topological vector spaces are particularly well-behaved for Fréchet spaces. The notion of bornological metrisability captures many of the special properties of Fréchet spaces from the bornological point of view. This notion is already rather old: Henri Hogbe-Nlend attributes it to George Mackey. We treat it as an instance of a general notion of category theory. I am grateful to Clark Barwick for explaining this to me. The results about metrisable spaces in [67] can be explained in this general context. Definition 1.152. Let  be a cardinal. A partially ordered set .X; / is called -filtered if any subset S of X with jSj <  has an upper bound. We have no need for the more general notion of a -filtered category. By definition, being @0 -filtered is the same as being directed. Definition 1.153. A bornological vector space V is called .bornologically/ metrisable if .Sd .V /; / is @1 -filtered, that is, any countable set of bounded disks .Bn /n2N is absorbed by some bounded disk. Theorem 1.154. Let V be a metrisable .locally convex/ topological vector space. Then Cpt.V / and vN.V / are bornologically metrisable. Proof. Let .Sn /n2N be a sequence of precompact (or von Neumann) bounded subsets. Let .Un / be a decreasing sequence of closed convex neighbourhoods of the origin that we may find "n > 0 defines the topology of V . Since S Sn is von Neumann bounded, } with "n Sn  Un . Let T WD 1 " S . It is clear that T absorbs Sn for all n 2 N. n n nD1 S " S [ U for all N 2 N; this implies that T is precompact We have T  N N C1 nD1 n n (or von Neumann); hence so is T } . The following result from category theory explains why filtered categories are useful. Proposition 1.155. Let I1 and I2 be directed sets .or more general categories/, and let  be a cardinal number. Suppose that jI1 j <  and that I2 is -filtered. Then there is a natural isomorphism lim lim F Š lim lim F !   ! I2

I1

I1

I2

for any functor F W I1  I2 ! Sets, that is, the direct and inverse limits commute in !  this case. A similar statement holds for functors to C for any category C. Proof. For each i2 22 I2 , we have a canonical map lim F .i1 ; i2 / ! lim  !

i1 22I1

lim F .i1 ; j2 /: 

j2 22I2 i1 22I1

(1.156)

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1 Bornological vector spaces and inductive systems

These combine to a canonical map lim lim F ! lim lim F , which we claim is !I2 I1 I1 !I2 invertible for suitable C. First assume that F is a functor to Sets because we want to use the concrete description of direct and inverse limits in Sets. An element of lim lim F is represented by an element of lim F . ; i2 / for some !I2 I1 I1 (fixed) i2 22 I2 ; two such elements represent the same class if they are identified by applying some morphisms in I2 . Unravelling the projective limit, we get a family of elements x.i1 ; i2 / 2 F .i1 ; i2 / for all i1 22 I1 that satisfy appropriate compatibility conditions. An element of lim lim F is a family of elements of lim F .i1 ; / for all i1 22 I1 I1 !I2 !I2 that satisfy appropriate compatibility conditions. Unravelling thedirect limit,  we get a function ˛ W Ob.I1 / ! I2 and for each i1 22 I1 an element of F i1 ; ˛.i1 / , satisfying appropriate compatibility conditions. The only difference is that now the second entry i2 is allowed to vary, whereas it was fixed for elements of lim lim F . !I2 I1 But the filteredness and cardinality assumptions ensure that there is an object i2 22 I2 that receives morphisms from ˛.i1 / for all i1 22 I1 . Hence in the second case we can find another representative that belongs to F .i1 ; i2 / with a fixed i2 22 I2 for all i1 22 I1 . Thus the canonical map constructed above is surjective. A similar argument establishes injectivity. This yields the assertion for functors with values in sets. !  Now consider a functor with values in C . A morphism of inductive systems f W V ! W is an isomorphism if and only if it induces bijections Hom.X; V / ! Hom.X; W / for all X 22 C. We have Hom.X; lim lim F / Š lim Hom.X; lim F / Š lim lim Hom.X; F /; !  !  !  I2

I1

I2

I1

I2

I1

Hom.X; lim lim F / Š lim Hom.X; lim F / Š lim lim Hom.X; F /:  !  !  ! I1

I2

I1

I2

I1

I2

Hence the assertions for the Sets-valued functors Hom.X; F / imply the assertion for F itself. Let V be bornologically metrisable and let C be a countable category. A functor F W C  .Sd .V /; / ! Cborn is nothing but a family of functors FB W C ! Cborn for B 2 Sd .V / together with natural transformations FB ! FD for B  D that satisfy obvious compatibility conditions for B1  B2  B3 . In this situation, we have lim !

lim FB .i / Š lim  

B2Sd .V / i2C

lim !

FB .i /:

i2C B2Sd .V /

We consider some useful applications. Proposition 1.157. Let M be a compact smooth manifold. If V 22 Born is bornologically metrisable, then C 1 .M; V / Š lim C k .M; V /, that is, a function M ! V is k smooth if and only if it is C k for all k 2 N.

1.6 Metrisability and the approximation property

69

Proof. Let C be the partially ordered set .N; / viewed as a category. Let FB .k/ WD C k .M; VB / for k 2 N, B 2 Sd .V /. The identical inclusions for k  k 0 , B  B 0 turn this into a functor F W .N; /  .Sd .V /; / ! Ban  Cborn. It is well known that we have lim C k .M; VB / D C 1 .M; VB /  k

for all B 2 Sd .V /. By definition, lim C k .M; VB / D C k .M; V / for k 2 N [ f1g. !B Hence Proposition 1.155 yields exactly the assertion we want. Definition 1.158. A bornological vector space is countably generated if its bornology contains a cofinal sequence, which we can take to be increasing. A countably generated (complete) bornological vector space is called a Silva space if its bornology is relatively compact. A Silva algebra is a bornological algebra whose underlying bornological vector space is a Silva space. Silva spaces are studied in [33], [50], [99]. The main theorem asserts that any Silva space is the dual of a Fréchet–Schwartz space, equipped with the equicontinuous bornology. This duality is fully faithful, that is, bounded maps between Silva spaces are in bijection with continuous maps between the dual Fréchet–Schwartz spaces. Proposition 1.159. Let V be a countably generated space and W a metrisable bornological vector space. Then Hom.V; W / Š

Hom.V; WB /:

lim !

B2Sd .W /

Thus any bounded linear map V ! W factors through the semi-normed space VB for some bounded disk B  W . Proof. Let .Bn / be a cofinal increasing sequence in Sd .V / and abbreviate Vn WD VBn . We have V D lim Vn and compute ! Hom.V; W / Š lim Hom.Vn ; W / Š lim





n2N

Š

lim !

lim !

Hom.Vn ; WB /

n2N B2Sd .W /

lim Hom.Vn ; WB / Š 

B2Sd .W / n2N

lim !

Hom.V; WB /;

B2Sd .W /

where the first and last isomorphisms are the internal Hom versions of the universal property of inductive limits, the second isomorphism follows from (1.140), and the third isomorphism follows from Proposition 1.155. Notation 1.160. Let A be an object of a closed symmetric monoidal category. Its dual is A0 WD Hom.A; 1/:

70

1 Bornological vector spaces and inductive systems

Corollary 1.161. In the situation of Proposition 1.159, assume in addition that W is y W , so that any bounded linear map V ! W is nuclear. Then Hom.V; W / Š V 0 ˝ nuclear. Proof. We use some standard facts about nuclear spaces and operators. By assumption, W is complete, so that we may replace Sd .W / by Sc .W /. By definition, a bornological vector space W is nuclear if and only if the canonical map y W Š Hom.X; C/ ˝ y Hom.C; W / ! Hom.X; W / X0 ˝ is a bornological isomorphism for any Banach space X . If W is nuclear, then this map is at least injective for any X 22 Cborn. It remains to show that the map y W ! Hom.V; W / is a bornological quotient map. Proposition 1.159 yields that V0˝ any bounded subset of Hom.V; W / is the image of a bounded subset in Hom.V; WB / for some B 2 Sc .W /. Since the canonical injection WB ! W is nuclear, we get a y W , which maps to V ˝ y W. y WB0 ˝ bounded subset of Hom.V; WB / ˝

1.6.2 Subcompleteness via metrisability Definition 1.162. Let V be a bornological vector space. Let Ssd .V / be the set of all B 2 Sd .V / for which VB is separable, that is, contains a dense sequence. We call V locally separable if Ssd .V / is cofinal in Sd .V /, that is, any bounded disk is contained in another one that belongs to Ssd .V /. Notice that a separable Banach space usually contains many bounded complete disks that are not separable. Proposition 1.163. Let V be a metrisable locally convex topological vector space. The precompact bornology on V is always locally separable. If V is separable then vN.V / is locally separable. Proof. This result is proved in [67], using a generalisation of Theorem 1.36 for uniformly continuous functions defined on metric spaces. We do not reprove this result here. Theorem 1.164. If a bornological vector space is bornologically metrisable and locally separable, then it is subcomplete. Proof. Let V be metrisable and locally separable. We verify the condition (4) of Proposition 1.149. Let Ssd .V / be the set of bounded disks for which VB is separable. By hypothesis, this is a cofinal subset in Sd .V /, so that the inductive system .VB /B2Ssd .V / c is isomorphic to diss V . For B; D 2 Ssd .V / with B S  D, let KBD  VB be the c c kernel of the canonical map VB ! VD . Let KB WD DB KBD . We have to check that there is D 2 Ssd .V / with D  B such that KB D KBD . Since VB is separable, so is VBc . Hence the subspace KB  VBc contains a countable dense subset X  KB . Elements of VBc are limits of Cauchy sequences in VB . Each

1.6 Metrisability and the approximation property

71

x 2 X is contained in KBD for some D  B. We write x as a limit of a Cauchy sequence .xn / in VB . Since x becomes 0 in VDc , this Cauchy sequence is a nullsequence in VDc and hence in VD . Since V is metrisable, we can find a single bounded disk D that works for all .xn /, that is, the map VBc ! VDc annihilates X . By continuity, it annihilates the closure of X as well, so that KB D KBD . Corollary 1.165. If V and W are locally separable and bornologically metrisable, y diss.W / Š diss.V ˝ y W /. then V ˝ W is subcomplete and diss.V / ˝ Proof. The tensor product V ˝ W is locally separable and bornologically metrisable as well. Theorem 1.164 yields that V ˝ W is subcomplete, and Corollary 1.151 yields the last assertion. Now we apply this to tensor products of Fréchet spaces: Theorem 1.166. Let V and W be Fréchet spaces. The canonical continuous bilinear y  W induces an isomorphism map \ W V  W ! V ˝ y diss Cpt.W / Š diss Cpt.V ˝ y  W /: diss Cpt.V / ˝ y diss vN.W / ! diss.V ˝ y  W; S/ is an isomorThe corresponding map diss vN.V / ˝ phism as well provided V and W are separable. Proof. Theorem 1.154 and Proposition 1.163 assert that the relevant bornological vector spaces are metrisable and locally separable (for the von Neumann bornologies, we need the requirement that V and W are separable). Now Corollary 1.165 shows that y .diss W /Š diss.V ˝ y W /. V ˝ W is subcomplete. Corollary 1.151 yields .diss V / ˝ Finally, the assertions follow from Theorem 1.87 and Corollary 1.90. We can extend Theorem 1.166 to tensor products of LF-spaces provided one factor is nuclear (see Theorem 1.93).

1.6.3 Subcompleteness of tensor products via the approximation property The following is a bornological analogue of Grothendieck’s approximation property for topological vector spaces (see [36]): Definition 1.167. A separated bornological vector space V has the .strong local/ approximation property if, for each bounded disk B  V , there is D 2 Sd .V /B and a sequence of finite-rank bounded linear maps VB ! VD that converge uniformly on precompact subsets of VB to the identical map VB ! VD . This notion is slightly stronger than the local approximation property studied in [67, Section 5.3] because we require the same sequence of finite rank maps VB ! VD to work for all precompact subsets of B. Both notions obviously agree if V carries a precompact bornology. This is the case of greatest interest to us, anyway. I am indebted to Christian Voigt for pointing out that my original definition does not quite suffice to prove Theorem 1.169.

72

1 Bornological vector spaces and inductive systems

Theorem 1.168 ([67, Theorem 5.11]). Let V be a Fréchet space. Then Cpt.V / has the strong local approximation property if and only if V has Grothendieck’s approximation property as a topological vector space .see [36]/. Theorem 1.169. Let V; W 22 Cborn and suppose that one of them has the strong local approximation property. Then V ˝ W is subcomplete, so that y W / Š .diss V / ˝ y .diss W /: diss.V ˝ The proof depends on the following Lemma: Lemma 1.170. Consider injective bounded linear maps between Banach spaces i12

i23

! V2  ! V3 ; V1 

j12

j23

W1 ! W2 ! W3

and put i13 WD i23 ı i12 and j13 WD j23 ı j12 . Suppose that i12 can be approximated uniformly on compact subsets by finite-rank bounded linear maps ˛k W V1 ! V2 . Then     y j12 W V1 ˝ y W1 ! V2 ˝ y W2 D ker i13 ˝ y j13 W V1 ˝ y W1 ! V3 ˝ y W3 : ker i12 ˝ In particular, if V1 has the approximation property, then we may take i12 and j12 to be identity maps, and the lemma shows that a pair of injective bounded linear maps y W1 ! V3 ˝ y W3 . V1 ! V3 , W1 ! W3 induces an injective bounded linear map V1 ˝ y W1 with Proof. The inclusion  is clear. For the converse containment, take 2 V1 ˝ y 13 /. / D 0. We claim that .i12 ˝j y 12 /. / D 0. Since the map V1 ! V3 is injective, .i13 ˝j the transpose map V30 ! V10 has dense range in the topology of uniform convergence on compact subsets. Therefore, we may approximate any finite-rank linear map V1 ! V2 by restrictions of maps V3 ! V2 and assume that the maps ˛k are of the form ˛Q k ı i13 y j13 /. / D 0 for for finite-rank bounded linear maps ˛Q k W V3 ! V2 . This implies .˛k ˝ 3 y j12 /. / D 0 all k 2 N. Since the range of ˛k is finite and j2 is injective, we get .˛k ˝ for all k 2 N as well. y .W1 /T for compact disks S  V1 Theorem 1.87 implies that belongs to .V1 /S ˝ and T  W1 . Therefore, the uniform convergence of ˛k on compact subsets implies y j12 /. / D lim .˛k ˝ y j12 /. / D 0: .i12 ˝ k!1

Proof of Theorem 1.169. The approximation property means that for each S 2 Sc .V /, there is S1  S such that the identical map VS ! VS1 may be approximated uniformly on compact subsets by finite-rank bounded linear maps. Now Lemma 1.170 shows that y WT ! VS1 ˝ y WT and VS ˝ y WT ! VS2 ˝ y WT2 have the same kernel for the maps VS ˝ y .diss W / is essentially all S2  S1 , T2  T . Hence the inductive system .diss V / ˝ reduced by Proposition 1.144. Thus V ˝ W is subcomplete by Proposition 1.149. The second assertion now follows from Corollary 1.151.

1.6 Metrisability and the approximation property

73

1.6.4 Maps with approximably dense range The approximation property allows us to gain linearity for maps with uniformly dense range. Definition 1.171. Let W and V be separated bornological vector spaces. A bounded linear map f W W ! V has approximably dense range if, for any B 2 Sd .V /, there is a sequence of bounded linear maps n W VB ! W such that f ı n .x/ converges towards x uniformly for all x 2 B. Approximably and uniformly dense range (Definition 1.52) only differ by the linearity requirement in Definition 1.171. Proposition 1.172. If f .W / is locally dense and if V has a precompact bornology and the local approximation property, then f has approximably dense range. Proof. Fix B 2 Sd .V /. We have to construct approximate local sections VB ! W for f that are linear. By the local approximation property and precompactness of the bornology, there is D 2 Sd .V /B and a sequence of finite-rank maps 'n W VB ! VD that converges in the norm-topology towards the identical map VB ! VD . Passing to a subsequence, if necessary, we can achieve that 'n .x/  x 2 2n D for all n 2 N, x 2 B. We already know from Proposition 1.53 that f has uniformly dense range, so that we get a sequence of bounded non-linear maps n W D ! W such that lim f ı n .x/ D x uniformly for x 2 D. This uniform convergence happens in some bounded disk E V. For each n 2 N, we choose a basis x1n ; : : : ; xkn 2 D for the range of 'n . Specifying the values on the basis, we define a bounded linear map Q m W 'n .VB / ! W;

xjn 7! m .xjn / for j D 1; : : : ; k:

Then f ı Q m converges uniformly on the bounded subset 'n .B/ towards the identity map. Hence there is m.n/ 2 N such that f ı Q m.n/ .x/  x 2 2n E for all x 2 'n .B/. Now the sequence of maps n WD Q m.n/ ı'n W VB ! W has the required properties.

Chapter 2

Relations between entire, analytic, and local cyclic homology

We define entire, analytic, and local cyclic homology and cohomology and compare these theories. We show that analytic and local cyclic homology coincide for some classes of algebras; these positive results are purely a matter of functional analysis because they do not involve the boundary maps on the chain complexes that compute these theories. This is why we can discuss them at this early stage. We also compute periodic, entire, analytic, and local cyclic (co)homology for certain convolution algebras on a compact Lie group K in §2.4. Although these algebras are all biprojective and hence of finite bidimension, periodic, entire, and analytic cyclic homology and cohomology may differ for them. Unlike analytic and local cyclic homology, the periodic cyclic homology depends on the choice of the dense subalgebra in C  .K/. The proofs of the statements in §2.4 require a lot of machinery that we develop later in this book. Therefore, you probably cannot follow them when you read this book for the first time. Nevertheless, you should browse through the results because they exhibit important differences between the analytic and local theories on the one hand and the periodic and entire theories on the other hand. First we recall an equivalent definition of Alain Connes’ entire cyclic cohomology for Banach algebras. Since the growth condition that defines entire cyclic cochains only depends on the von Neumann bornology of the algebra, it is obvious how to generalise it to complete bornological algebras. In this context, we also get a canonical (pre-)dual homology theory; for Banach algebras with the von Neumann bornology, this dual theory is already constructed along the way by Ezra Getzler and András Szenes in [34], but without much discussion. We may choose another bornology on a topological algebra than the von Neumann bornology. In fact, our default choice is the bornology of precompact subsets because this yields considerably better results. To avoid confusion with the existing entire theories, we call our theories analytic cyclic homology and cohomology and denote them by HA and HA . Both are defined for complete bornological algebras. When we apply the analytic theories to topological algebras, we always use the precompact bornology, whereas entire cyclic homology and cohomology use the von Neumann bornology. Another reason to change notation is that the apparent importance of entire functions for entire cyclic cohomology is an artefact created by looking only at a cohomology theory. The relevant function algebra is the algebra C..t // of analytic power series, that is, power series with non-zero radius of convergence. The space of entire functions only appears because it is the dual space of C..t //.

2 Relations between entire, analytic, and local cyclic homology

75

The chain complex that computes the analytic cyclic homology HA .A/ is denoted by HA.A/ and called the analytic cyclic chain complex. The bivariant analytic cyclic homology HA .A; B/ for two complete bornological algebras A and B is the space of homotopy classes of chain maps HA.A/ ! HA.B/Œ. The univariant theories HA .B/ and HA .A/ are special cases of the bivariant theory because HA.1/ is chain homotopy equivalent to 1 by Theorem 5.63, so that HA .1; B/ Š HA .B/;

HA .A; 1/ Š HA .A/I

here 1 D R or 1 D C depending the category of bornological vector spaces we use. We prefer to work with the chain complex HA.A/ instead of the associated homology theories because the passage to homology forgets part of the information and some important phenomena cannot be described in terms of homology. ! The construction of HA.A/ carries over literally to algebras in Ban. We denote the resulting chain complex by HL and call it the local cyclic chain complex of A. Its homology is called local cyclic homology and denoted by HL .A/. Whereas the chain complexes HA and HL and the homology theories HA and HL have almost identical definitions, local cyclic cohomology HL .A/ and bivariant local cyclic homology HL .A; B/ are much harder to define than the corresponding analytic theories; the complicated definitions ensure that the local theory has nice properties. One of the most crucial results about the local cyclic theory is that it is invariant under passage to suitable dense subalgebras. Technically, this holds because the inclusion A ! B of such a subalgebra induces a local chain homotopy equivalence HL.A/ ! HL.B/. A functor on the category of chain complexes is called local if it maps local chain homotopy equivalences to isomorphisms. Whereas homology is local, cohomology and bivariant homology are not. To repair this, we replace cohomology and bivariant homology by suitable localisations in the definition of HL .A/ and HL .A; B/. Roughly speaking, the localisation of a functor is the best local approximation to it. More explicitly, the localisation replaces direct limits by homotopy direct limits. The localisation does not change the underlying chain complex HL.A/, so that we can ignore it as long as we make statements about HL.A/. We define local cohomology and local bivariant homology here for the sake of completeness, but later we treat these theories as a black box and never use more than some formal properties. For many purposes, we can work equally well with the following poor man’s versions of the theories:   poor HLpoor .A/ WD Hom.HL .A/; 1/; HL .A; B/ WD Hom HL .A/; HL .B/ I (2.1) here we view HL .A/ just as a vector space, and Hom denotes spaces of linear maps. The Universal Coefficient Theorem in §7.2 yields that there are many algebras A with poor HL .A/ Š HLpoor .A/ and HL .A; B/ Š HL .A; B/ for all B. ! Since we may view complete bornological algebras as algebras in Ban using the dissection functor, we may apply both local and analytic cyclic homology to them. The only difference between diss HA.A/ and HL.diss A/ is the category in which we

76

2 Relations between entire, analytic, and local cyclic homology

! ! perform the completion. Whereas the completion functor Norm ! Ban is local, this fails for its bornological counterpart; hence many results about HL like invariance under passage to isoradial subalgebras break down for HA . The good news is that the problem with completions does not occur in applications: we will see in §2.2 that HA .A/ and HL .A/ agree if A is metrisable and locally separable or has the strong local approximation property; the first case covers Fréchet algebras with the precompact bornology, the second one also covers nuclear bornological algebras. In fact, I do not know of any example where HA and HL are different. In contrast, we need rather strong hypotheses to identify HA .A/ and HL .A/ (see §2.3.7). There is a fourth related theory, called asymptotic cyclic cohomology, defined by Michael Puschnigg in [83]. My impression is that this theory is superseded by the more recent local cyclic cohomology. Similarly, analytic cyclic cohomology supersedes entire cyclic cohomology. The analytic and local cyclic homology theories agree in most cases, but in general the local theory is more powerful than the analytic one. Hence it suffices to study only local cyclic homology; but if you are not at home with inductive systems, then you may work with the analytic cyclic theory without losing much. Since the chain complexes HA.A/ and HL.A/ that we use in this chapter do not explain the properties of the theories, we will replace them by other, chain homotopy equivalent complexes in Chapter 5.

2.1 Several definitions 2.1.1 Entire and analytic cyclic cohomology We begin by defining the entire cyclic cohomology of a Banach algebra A. The cochain complex used by Alain Connes in [7] only works if A is unital. This is repaired by Masoud Khalkhali in [62], who checks that the following construction extends entire cyclic cohomology to non-unital Banach algebras. Let C n .A/ be the space of continuous n C 1-linear maps AnC1 ! 1. We define the cyclic rotation map  W C n .A/ ! C n .A/ and the 0th face map d 0 W C n .A/ ! C nC1 .A/ by .'/.x0 ; : : : ; xn / WD .1/n '.xn ; x0 ; : : : ; xn1 /; .d 0 '/.x0 ; : : : ; xnC1 / WD '.x0  x1 ; x2 ; : : : ; xnC1 /:

(2.2)

Notice that nC1 D idC n .A/ on C n .A/. We define d j W C n .A/ ! C nC1 .A/ for j D 0; : : : ; n C 1 by d j WD j d 0 j , that is, .d j '/.x0 ; : : : ; xnC1 / D .1/j '.x0 ; : : : ; xj 1 ; xj  xj C1 ; xj C2 ; : : : ; xnC1 /; 0  j  n; .d nC1 '/.x0 ; : : : ; xnC1 / D .1/nC1 '.xnC1  x0 ; : : : ; xn /:

77

2.1 Several definitions

We also define Q W C n .A/ ! C n .A/ and b 0 ; b W C n .A/ ! C nC1 .A/ by Q WD

n X

j ;

b 0 WD

j D0

n X

dj;

b WD

j D0

nC1 X

dj:

j D0

It is well known that these operators satisfy b ı b D 0; Q ı b 0 D b ı Q; b 0 ı b 0 D 0; .1  / ı b D b 0 ı .1  /:

.1  / ı Q D 0; Q ı .1  / D 0;

Thus the diagram in Figure 2.1 is a bicomplex, called cyclic bicomplex; it is 2-periodic :: :O 

:: :O

/ C 3 .A/ .1// C 3 .A/ O O / C 2 .A/ O

1

/ C 2 .A/ O

/ C 1 .A/ O

.1/

/ C 1 .A/ O



/ C 0 .A/

Q

Q

1

/ C 0 .A/

/ C 2 .A/ O / C 1 .A/ O / C 0 .A/

Q

/ C 3 .A/.1/ /    O

b0 1

/ C 2 .A/ O

b Q

/ C 2 .A/ O

b 0 .1/

/ C 1 .A/ O

1

/ C 0 .A/

1

/ 

b Q

/ C 1 .A/.1/ /    O

b0

b Q

:: :O

/ C 3 .A/ .1// C 3 .A/ O O

b

b0

b

:: :O

b

b 0

b



Q

b0

b



:: :O

b Q

/ C 0 .A/

1

/ 

.1  /'.x0 ; : : : ; xn / D '.x0 ; : : : ; xn /  .1/n '.xn ; x0 ; : : : ; xn1 /; Q'.x0 ; : : : ; xn / D

n X

.1/j n '.xj ; : : : ; xn ; x0 ; : : : ; xj 1 /;

j D0

b 0 '.x0 ; : : : ; xnC1 / D

n X

.1/j '.x0 ; : : : ; xj 1 ; xj  xj C1 ; xj C2 ; : : : ; xnC1 /;

j D0

b'.x0 ; : : : ; xnC1 / D b 0 '.x0 ; : : : ; xnC1 / C .1/nC1 '.xnC1  x0 ; x1 ; : : : ; xn /: Figure 2.1. The cohomological cyclic bicomplex.

in the horizontal direction. To be specific, the cyclic bicomplex .C pq ; ıv ; ıh / has ( C q .A/ for q  0, pq C WD 0 for q < 0,

78

2 Relations between entire, analytic, and local cyclic homology

the horizontal and vertical boundary maps ıh W C pq ! C pC1;q and ıv W C pq ! C p;qC1 are ( ( 1   if p is even, b if p is even, pCq pCq  ıv WD .1/  ıh WD .1/ Q if p is odd, b 0 if p is odd. There are L the total complex of such a bicomplex: one Qtwo standard ways to take has entries pCqDn C pq , the other pCqDn C pq ; the boundary map is ıh C ıv in L1 C p .A/ or either case. Both complexes are 2-periodic with the same space pD0 Q1 p pD0 C .A/ in both even and odd degree. Hence we may view them as Z=2-graded chain complexes. The direct product-total complex is contractible because the cyclic bicomplex has contractible rows. In contrast, the direct sum-total complex computes the periodic cyclic cohomology HP .A/. A subtle intermediate choice between these two total complexes leads to entire cyclic cohomology. Q 2 n2N C n .A/ Definition 2.3. Let A be a Banach algebra. P1A sequence .'n /n2N satisfies the entire growth condition if z 7! nD0 bn=2cŠk'n kz n is an entire function, that is, for every " > 0 there is a c" > 0 such that k'n k 

c "  "n bn=2cŠ

for all n 2 N;

here bn=2c is n=2 if n is even and .n1/=2 if n is odd. The entire growth condition is preserved if we multiply .'n / by a scalar-valued function of at most exponential growth. Hence the boundary maps b; b 0 ; 1  ; Q in the cyclic bicomplex (Figure 2.1) preserve the entire growth condition, so that we get a subcomplex C " .A/ of the direct product-total complex. Definition 2.4. The entire cyclic cohomology of A is defined as the cohomology of the Z=2-graded cochain complex C " .A/; we denote its even and odd part by HE0 .A/ and HE1 .A/; the Z=2-graded group they form is denoted by HE .A/. We remark that C " .A/ is a chain complex of Fréchet spaces, whose topology is defined by the increasing sequence of semi-norms   k .'n / WD sup k n bn=2cŠ  k'n k n2N

for k 2 N. This is a special feature of Banach algebras (and Silva algebras). Using the definition of the norm of a multi-linear map, we see that the entire growth condition can be rewritten as follows: for any r > 0, the set of scalars fbn=2cŠ  'n .x0 ; : : : ; xn / j n 2 N and x0 ; : : : ; xn 2 A with kxj k  r for all j g remains bounded. Equivalently, this holds for n 2 N and x0 ; : : : ; xn 2 S for each von Neumann bounded subset S of A.

2.1 Several definitions

79

This formulation in terms of von Neumann bounded subsets makes sense for all locally convex topological algebras, as already observed by Alain Connes. But since the entire growth condition depends only on the von Neumann bornology, it is better to assume right away that A is a bornological algebra. As explained above, we also change the name of the theory, as follows: Definition 2.5. Let A be a bornological algebra. We let C " .A/ be the subcomplex of Q the direct-product total complex that consists of cochains .'n /n2N 2 n2N C n .A/ that satisfy the following growth condition: the set of scalars fbn=2cŠ  'n .x0 ; : : : ; xn / j n 2 N; x0 ; : : : ; xn 2 S g   is bounded for each S 2 S.A/. The homology H C  .A/ is denoted HA .A/ and called analytic cyclic cohomology of A. It is clear that Definitions 2.3 and 2.5 agree if A is a Banach algebra with the von Neumann bornology. But we have additional freedom and may equip Banach and topological algebras with, say, the precompact bornology. Surprisingly, this innocentlooking change of bornology significantly changes the cohomology. Various seemingly arbitrary but crucial choices in Definition 2.5 – like the factors bn=2cŠ – will be explained in Chapter 5 by another approach towards analytic cyclic cohomology. The linear functionals 'n extend to the completion Ac , and the entire growth condition holds on A if and only if it holds on Ac . Therefore, we assume that A is complete in the following.

2.1.2 Analytic cyclic homology Now we realise C " .A/ as the dual complex of a certain chain complex of bornological vector spaces HA.A/. We first define its underlying bornological vector space. Since it does not depend on the structure, we define it for any bornological vector L algebra ˝n space V . Let B.V / WD 1 V . If S 2 S.V /, we let nD1 hhSiiŠ WD

[

bn=2cŠ  S ˝n

}

 B.V /I

(2.6)

here S ˝n  V ˝n denotes the set of elementary tensors x1 ˝  ˝xn with x1 ; : : : ; xn 2 S. Notation 2.7. Let S" .V / be the bornology on B.V / generated by the subsets hhS iiŠ . Let HA0 .V / D HA1 .V / be the completion of B.V / with respect to this bornology, and let HA.V / WD HA0 .V / ˚ HA1 .V / be the resulting Z=2-graded bornological vector space. A linear functional ' W B.V / ! 1 is equivalent to a sequence of multi-linear functionals 'n W V nC1 ! 1; these satisfy the growth condition in Definition 2.5 if and only

80

2 Relations between entire, analytic, and local cyclic homology

if ' is bounded on S" .V /. Thus C " .V / is naturally isomorphic to the dual space of HA.V /. The signed cyclic rotation operator  W V ˝nC1 ! V ˝nC1 ;

x0 ˝    ˝ xn 7! .1/n xn ˝ x0 ˝    ˝ xn1 ;

extends to a bounded operator on HA.V /, whose dual is, up to signs, the corresponding operator in Figure 2.1. The resulting operator Q is bounded as well. To get the other maps for the cyclic bicomplex, we need a bornological algebra A instead of V ; then d0 W A˝nC1 ! A˝n ;

x0 ˝    ˝ xn 7! x0  x1 ˝ x2 ˝    ˝ xn

makes sense. Now we get bounded operators dj , 1  , Q, b and b 0 as above, see Figure 2.2. This yields a predual for the cyclic bicomplex in Figure 2.1. An equivalent approach to this bicomplex using differential forms is explained in §A.6.2. :: :  o  o

:: :

 A˝4 o

1

 A˝4 o

A˝3 o 

1

b

 o

A˝2 o

 o

 Ao

Q



Q

A˝3 o 

b 0

1

A˝2 o

1

 Ao

Q

b 0

b

:: :

 A˝4 o

b 0

b



:: : 1

A˝3 o 

1

b

A˝2 o  Ao

Q

 A˝4 o

b 0



A˝3 o 

Q

b 0

1

A˝2 o

1

 Ao

Q

b 0

b

Q

 A˝4 o

b



:: : 

b



A˝3 o 



b

A˝2 o



b

Q

 Ao



.1  /.x0 ˝    ˝ xn / D x0 ˝    ˝ xn  .1/n xn ˝ x0 ˝    ˝ xn1 ; Q.x0 ˝    ˝ xn / D

n X

.1/j n xj ˝    ˝ xn ˝ x0 ˝    ˝ xj 1 ;

j D0

b 0 .x0 ˝    ˝ xnC1 / D

n X

.1/j x0 ˝    ˝ xj 1

j D0

˝ xj  xj C1 ˝ xj C2 ˝    ˝ xnC1 ; b.x0 ˝    ˝ xnC1 / D b .x0 ˝    ˝ xnC1 / 0

C .1/nC1 xnC1  x0 ˝ x1 ˝    ˝ xn :

Figure 2.2 The homological cyclic bicomplex.

2.1 Several definitions

81

It is manifest that the cohomological cyclic bicomplex in Figure 2.1 is dual to the chain complex in Figure 2.2. Now we form the direct sum-total complex of the homological cyclic bicomplex; this yields a 2-periodic chain complex with B.A/ WD L1 ˝n A in both even and odd degrees, which we view as a Z=2-graded chain nD0 complex. It is straightforward to check that the boundary maps on this total complex extend to bounded linear maps HA0 .A/  HA1 .A/, so that the latter becomes a Z=2-graded chain complex of complete bornological vector spaces. Definition 2.8. We denote this chain complex by HA.A/. Its homology HA .A/ is called the analytic cyclic homology of A. If A and B are complete bornological algebras, then the space of bounded linear maps HA.A/ ! HA.B/ becomes a chain complex in a canonical way (see §A.1); its homology is the bivariant analytic cyclic homology HA .A; B/. The dual of HA.A/ is the chain complex of bounded linear functionals HA.A/ ! 1 with the induced boundary map @.f / D .1/jf j f ı @. This is exactly the chain complex that is used in Definition 2.5 to define analytic cyclic cohomology HA .A/. Definition 2.9. An algebra homomorphism f W A ! B induces a chain map HA.f / W HA.A/ ! HA.B/;

which defines a class HA.f / 2 HA0 .A; B/. We call f an HA-equivalence if HA.f / is invertible, that is, there is g 2 HA0 .B; A/ with f ı g D idB and g ı f D idA . Equivalently, f induces a chain homotopy equivalence HA.A/ ! HA.B/. An HA-equivalence induces isomorphisms HA .A/ Š HA .B/, HA .B/ Š HA .A/, HA .D; A/ Š HA .D; B/, and HA .B; D/ Š HA .A; D/ for all D 22 Cborn.

2.1.3 Local cyclic homology Now we translate the construction of HA and HA from bornological algebras to inductive systems. This is rather technical and only notationally complicated. First we ! ! construct the underlying object HL0 .V / D HL1 .V / in Ban for V 22 Ban. Then we use an algebra structure to define the boundary map. Let V D .Vi /i2I for some directed set I . We construct the indexing set of HL .V /. Let J be the set of all pairs .i; D/, where i 2 I and D  Vi is an absorbing disk. We 0 partially order J by .i; D/  .i 0 ; D 0 / if i  i 0 and the structure map fii W Vi ! Vi 0 0 maps D to D . It is easy to see that J is again directed. Of course, .i; D/ 7! Vi is an inductive system of Banach spaces that is isomorphic to V . Equation 2.6 defines absorbing disks hhDiiŠ  B.Vi / for .i; D/ 2 J . We let HL0 .V /i;D be the completion of B.Vi / for the gauge norm of hhDiiŠ . It is clear that these Banach spaces form an ! inductive system indexed by J ; this is the desired object HL0 .V / D HL1 .V / of Ban.

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2 Relations between entire, analytic, and local cyclic homology

The map  on V ˝n makes sense for any V and yields endomorphisms 1   and Q of the inductive system HL .V /. We get a morphism of inductive systems d0 ! as well if we replace V by an associative algebra A in Ban. We claim that this yields morphisms of inductive systems 1  ; Q; b; b 0 . This is rather easy for the first two maps. The multiplication map m W A ˝ A ! A is specified locally by bounded bilinear   maps   mi W Ai  Ai ! Am .i/ for some function m W I ! I . We let m .i; D/ D m .i/; 2  mi .D  D/ for .i; D/ 2 J . Then we get bounded linear maps b; b 0 W HL0 .V /i;D ! HL0 .V /m .i;D/ because the factor 2 in the definition of m .i; D/ generates a factor of 2n on n-forms, which outweighs the number of summands in b and b 0 . Thus b and b 0 are morphisms ! in Ban. The associativity of A implies that the homological cyclic bicomplex (Figure 2.2) is indeed a bicomplex. The necessary relations extend to the completions, so that we ! get a Z=2-graded chain complex HL0 .A/  HL1 .A/ in Ban (we say more about chain complexes in additive categories in §A.1). ! Definition 2.10. We denote the chain complex in Ban constructed above by HL.A/. Its homology is denoted by HL .A/ and called local cyclic homology of A. This homology is defined by the following general recipe. First, we apply the canonical forgetful functor Hom.1; / to HL.A/; this yields the chain complex of vector spaces lim HL.A/ (without separated quotients); then we take the homology of the chain ! complex of vector spaces lim HL.A/. !

2.2 Comparison of analytic and local cyclic homology Let A be a complete bornological algebra. On the one hand, we can apply the dissection ! functor diss to HA.A/ and get a Z=2-graded chain complex in Ban. On the other hand, we can consider HL.diss A/. We want to compare these twochain complexes.    First we observe that HA .A/ D H HA.A/ D H diss HA.A/ because the ! ! composition of the standard forgetful functor Ban ! Vect with diss W Cborn ! Ban is the standard forgetful functor Cborn ! Vect. As a result, diss HA.A/ Š HL.diss A/ H) HA .A/ Š HL .diss A/: In the following proposition, all direct limits are taken in Cborn, so that lim WD ! sep lim. ! ! Proposition 2.11. Let A be an algebra in Ban. Then lim HL.A/ Š HA.lim A/ ! !

2.2 Comparison of analytic and local cyclic homology

83

as chain complexes in Cborn. We have diss HA.lim A/ Š HL.A/ if and only if the ! inductive system HL.A/ is essentially reduced. Proof. For the first statement, it suffices to show that the spaces of bounded linear maps lim HL.A/ ! X and HA.lim A/ ! X are naturally isomorphic for any X 22 Cborn ! ! and that the isomorphism is compatible with the boundary maps. This depends on Theorem 1.139. Let AQ WD lim A. By the universal property of the completion, bounded linear maps ! L Q ! X are equivalent to linear maps 1 AQ˝n ! X that are bounded with HA0 .A/ nD1 Q Such a map is described by a sequence of multi-linear respect to the bornology S" .A/. maps 'n W AQn ! X that satisfy an appropriate growth condition. Since inductive limits commute with tensor products, we may replace 'n by a compatible family of multilinear maps 'n;i W Ani ! X S for all i 2 I , where A D .Ai /i2I . The growth condition n n means in this situation that 1 nD1 b =2cŠ  'n;i .D / is bounded for all .i; D/ 2 J . We may recombine these maps differently and consider 'n;i for n 2 N as a linear map 'i W B.Ai / ! X that remains bounded on hhDiiŠ for all D 2 Sd .Ai /. Since X is complete and HL0 .A/i;D is the completion of B.Ai / with respect to the norm generated by hhDiiŠ , we may extend 'i to a bounded linear map HL0 .A/i;D ! X . Finally, these extensions are compatible and fit together to a map HL0 .A/ ! X . This yields the Q ! X. desired bijection between bounded linear maps lim HL0 .A/ ! X and HA0 .A/ ! It is easy to check compatibility with the boundary maps, so that we get the first assertion. This implies diss HA.lim A/ Š diss lim HL.A/. By Proposition 1.144, this ! ! is isomorphic to HL.A/ if and only if HL.A/ is essentially reduced. For any A 22 Cborn, we have a canonical map HL.diss A/ ! diss lim HL.diss A/ Š diss HA.A/

!

by Theorem 1.139 and Proposition 2.11. The issue is whether it is an isomorphism. Proposition 2.12. Let A be a bornological algebra. We have diss HA.A/ Š HL.diss A/ if and only if .B.A/; S" / is subcomplete, if and only if the spaces A˝n are uniformly subcomplete in the following sense: for each S 2 Sd .A/, there is T 2 Sd .A/S such that y y y y ˝n ˝n ˝n ker.A˝n S ! AT / D ker.AS ! AU / for all U 2 Sd .A/T , n 2 N; equivalently, any Cauchy sequence in A˝n S for some n 2 N that becomes a null-sequence in A˝n for some U already becomes a nullU ˝n sequence in AT . Proof. Since HL.diss A/ is the completion of diss.B.A/; S" /, it is essentially reduced if and only if .B.A/; S" / is subcomplete (Proposition 1.149). We claim that this is equivalent to the uniform subcompleteness of the tensor powers A˝n for all n 2 N. To check this, consider the projection Pn W B.A/ ! A˝n onto the nth direct summand. Let S 2 Sd .A/ and let SQ WD hhSiiŠ 2 S" .A/. The projections Pn restrict

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2 Relations between entire, analytic, and local cyclic homology

to projections Pn W B.A/SQ ! A˝n S for all S 2 Sd .A/, which then extend to proy

c c 0 jections Pn0 W B.A/cQ ! A˝n S . The projection Pn maps ker.B.A/ Q ! B.A/ Q / onto y

y

S

S

T

˝n ˝n subcompleteness of ker.VS ! VT /, and similarly for the pair S  U . Therefore, ˝n of A for n 2 N. B.A/; S" .A/ implies uniform subcompletenessP For the converse, observe that the projections jnD1 Pn are equibounded and converge pointwise to the identity map on B.A/SQ for all S. This remains true after completL y y ing. It follows that ker.B.A/cQ ! B.A/cQ / is the closure of n2N ker.VS˝n ! VT˝n /. S T   Since the same holds for the pair S  U , we get subcompleteness of B.A/; S" .A/ .

Theorem 2.13. Let A be a bornological algebra. We have diss HA.A/ Š HL.diss A/ in the following cases: (1) if A is a Banach algebra with the von Neumann bornology; (2) if A is metrisable and locally separable; (3) if A has the strong local approximation property. Proof. We verify the uniform subcompleteness condition of Proposition 2.12 in each case. This is trivial if A is a Banach algebra with the von Neumann bornology. If A is metrisable and locally separable, the subcompleteness of the tensor powers A˝n follows from Theorem 1.164; this produces countably many disks Tn , which can be absorbed by a single one by metrisability. This yields the second case. Finally, in the last case we use Lemma 1.170 as in the proof of Theorem 1.169 to get uniform subcompleteness. The last two conditions cover Fréchet algebras with the precompact bornology and nuclear bornological algebras.

2.3 The local homotopy category of chain complexes We assume that the reader is familiar with the basic results about chain complexes that are recalled in §A.1. Since we are only concerned with Z=2-graded chain complexes here, we abbreviate Kom.C/ WD Kom.CI Z=2/ and

HoKom.C/ WD HoKom.CI Z=2/

for the category of Z=2-graded chain complexes in an additive category C and its !  homotopy category. We are going to prove that the categories Kom. C / of chain ! complexes of inductive systems and Kom.C/ of inductive systems of chain complexes are equivalent if C has cokernels. We are mainly interested in C D Ban.

2.3 The local homotopy category of chain complexes

85

Then we turn to the local homotopy category of chain complexes, which we describe in terms of homotopy direct limits and use to define local cyclic cohomology and bivariant local cyclic homology. Finally, we show that analytic and local cyclic cohomology agree for a (rater restrictive) class of algebras, namely, those whose underlying bornological vector space is a Silva space with the approximation property. The crucial notion in this section is that of a local chain homotopy equivalence. You may ignore everything else if you are happy with the poor man’s versions of HL .A/ and HL .A; B/ defined in (2.1).

2.3.1 Inductive systems of chain complexes The goal of this section is to prove the following theorem about the local structure of ! chain complexes in Ban or Cborn. Theorem 2.14. The category Kom.Cborn/ of chain complexes in Cborn is equivalent to the category of reduced inductive systems of chain complexes of Banach spaces. This equivalence extends to an equivalence of categories ! ! Kom.Ban/ Š Kom.Ban/: ! In particular, any chain complex in Ban is isomorphic to an inductive limit of chain complexes of Banach spaces. ! There are corresponding statements for chain complexes in Born, Born1=2, Norm, ! and Norm1=2. The bornological part of the theorem is quite easy to prove. The proof for inductive systems is more technical. Let C 2 Kom.Cborn/. We denote the projections onto the even and odd part of C by P0 and P1 , and the boundary map by @. Let S@ .C /  Sc .C / be the subset of complete disks S  C with P0 .S /; P1 .S /; @.S /  S. Equivalently, the Banach space CS  C inherits the structure of Z=2-graded chain complex. The subset S@ .C /  Sc .C / is cofinal because any S 2 Sc .C / is absorbed by P0 .S / C P1 .S / C @P0 .S/ C @P1 .S / 2 S@ .C /. Therefore, Theorem 1.139 yields diss.C / Š .CS /S2S@ .C / ;

C Š

lim !

CS :

S2S@ .C /

That is, C is an inductive limit of a reduced inductive system of Z=2-graded chain complexes of Banach spaces. As in the proof of Theorem 1.139 and Proposition 1.144, this provides an equivalence of categories between Kom.Cborn/ and the full subcate! gory of reduced inductive systems in Kom.Ban/. In addition, the functors sep lim and ! ! diss between Kom.Cborn/ and Kom.Ban/ are adjoint as in Theorem 1.139. The assertion about inductive systems in Theorem 2.14 is a special case of the following general result:

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2 Relations between entire, analytic, and local cyclic homology

Theorem 2.15. Let C be an additive category in which each morphism has a cokernel. !  Then any Z=2-graded chain complex in C is an inductive limit of Z=2-graded chain complexes in C, and this yields an equivalence of categories ! !  Kom.C/ Š Kom. C /: There are similar theorems for Z=p-graded chain complexes for any p 2 N; we leave it to the reader to extend the proof. The proof of Theorem 2.15 will finish the proof of Theorem 2.14. Category theory provides some general techniques for deciding whether a given category C0 is equivalent to a category of inductive systems. A necessary condition is !  that any inductive system in C0 has a direct limit; this is clearly the case for Kom. C /. Definition 2.16. An object A of a category C0 is finitely presented if the functor C0 .A; / commutes with inductive limits, that is, the natural map lim C0 .A; / ! C0 .A; lim / ! !

(2.17)

is an isomorphism for any inductive system in C0 . We let C0fp  C0 be the full subcategory of finitely presented objects. It is easy to see that the inductive limit functor restricts to a fully faithful functor ! lim W C0fp ! C0 I !

(2.18)

even more, C0fp is the maximal subcategory with this property. We say that C0 has enough finitely presented objects if the functor in (2.18) is an equivalence of categories; equivalently, any object of C0 is (isomorphic to) an inductive limit of objects of C0fp . Example 2.19. Let C be any category (for example, C D Ban or C D Kom.Ban/), !  and let C0 D C . Then the image of C in C0 consists of finitely presented objects, and !  the functor in (2.18) is the identical functor C ! C0 . Conversely, any finitely presented object in C0 is a retract of one from C: write A D lim .Ai /i2I with Ai 2 C and use ! idA 2 lim Hom.A; Ai / to get a retraction A ! Ai for some i . If C is already closed ! under retracts (like Ban or Kom.Ban/), then C D C0fp . Thus we can reconstruct C !  !  from C : it is the subcategory of finitely presented objects in C . ! Whereas Banach spaces are finitely presented in Ban, they are not finitely presented in Cborn because the inductive limit functor involves a separated quotient, which invalidates (2.17). We only have (2.17) for (essentially) reduced inductive systems in Cborn. The analogue of (2.17) for (essentially) reduced inductive systems characterises finitely generated objects; the issue is that we cannot successively add relations in a reduced inductive system.

2.3 The local homotopy category of chain complexes

87

!  Proof of Theorem 2.15. Let C0 WD Kom. C /. It is easy to see that objects of Kom.C/  ! C0 are finitely presented. Hence the direct limit functor Kom.C/ ! C0 is fully faithful. It remains to show that any object of C0 is an inductive limit of objects of Kom.C/. !  !  !  !  The forgetful functor Forget W Kom. C / ! C has a left adjoint F W C ! Kom. C / by Remark A.3. The units of this adjunction provide a canonical chain  map C W F ı Forget.C / ! C and a map C W Forget.C / ! Forget F ı Forget.C / that is a section for C . Therefore, C is a semi-split surjection (this is also easy to see directly from the explicit description of F ). Let KC WD ker C . We conclude that C is isomorphic to the cokernel of the map KC



F ı Forget.KC / ! KC  ! F ı Forget.C /:

(2.20)

Since F is a left adjoint functor, it commutes with direct limits. Writing Forget.C  / D .Ci /i2I for an inductive system in C, we get F ıForget.C / Š lim F .Ci /. Here F .Ci / ! is an inductive system in Kom.C/ because F restricts to a functor C ! Kom.C/. Thus ! F ı Forget.C / and F ı Forget.KC / belong to Kom.C/. ! !  Since the functor Kom.C/ ! Kom. C / is fully faithful, the map in (2.20) is ! a morphism in Kom.C/. Using Remark 1.133, we may replace it by a morphism of diagrams, which consists of a compatible family of chain maps fi W Ai ! Bi in Kom.C/. The entrywise cokernels coker.fi / inherit a structure of Z=2-graded ! chain complex, and the inductive system .coker fi / is a cokernel for .fi / in Kom.C/. Since lim commutes with direct limits, it maps this cokernel to a cokernel for the map ! in (2.20). Thus C Š lim.coker fi / is isomorphic to an inductive limit of Z=2-graded ! chain complexes as desired. Remark 2.21. The above theorem still works for p-periodic chain complexes with p ¤ 2. For Z-graded chain complexes, some changes are necessary because only bounded chain complexes in C are finitely presented in Kom.C/. Letting Kom.C/b be the category of bounded chain complexes in C, we get ! !  Kom. C / Š Kom.C/b :

2.3.2 Local homotopy equivalences As in Theorem 2.15, we allow C to be any additive category with cokernels, although we are mainly interested in the case C D Ban. We have a canonical functor ! !  Forget W HoKom. C / ! HoKom.C/

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2 Relations between entire, analytic, and local cyclic homology

! because the maps Ai ! A for A D .Ai /i2I in Kom.C/ induce a natural map     H0 .Ai /i2I ; .Bj /j 2J ! lim H0 Ai ; .Bj /j 2J Š lim lim H0 .Ai ; Bj /   ! i i j   Š Hom! .Ai /; .Bj / : HoKom.C/

This forgetful functor is not faithful; this leads to the following notions: Definition 2.22. A chain complex C is locally .chain/ contractible if Forget.C / Š 0 ! in HoKom.C/. A chain map f is a local .chain/ homotopy equivalence if Forget.f / ! is invertible in HoKom.C/. !  Lemma 2.23. Let C be a chain complex in C , written as C Š lim.Ci /i2I for an ! inductive system of chain complexes Ci in C by Theorem 2.15. The following are equivalent: • C is locally contractible; • H .X; C / D 0 for all X 2 Kom.C/; • for all i 2 I , the canonical map Ci ! C is null-homotopic; • for all i 2 I , there is j 2 Ii such that the structure map Ci ! Cj is nullhomotopic. !  Local contractibility is hereditary for inductive limits in Kom. C /: if A D lim.Aj /j 2J ! with locally contractible Aj , then A is locally contractible. ! !  Proof. The spaces of morphisms X ! C in HoKom. C / and HoKom.C/ coincide with lim H .X; Ci / if X 22 Kom.C/. Hence H .X; C / D 0 if C is locally contractible. ! In particular, H .Ci ; C / D 0 for all i 2 I , so that the canonical map Ci ! C is null-homotopic. Then the canonical map Ci ! Cj is null-homotopic for some j  i because H .Ci ; C / Š lim H .Ci ; Cj /: ! j

Thus the identity map on C represents the zero element in lim lim H0 .Ci ; Cj /, which i !j ! ! is the group of endomorphisms of C in HoKom.C/. Thus C Š 0 in HoKom.C/, that is, C is locally contractible. This finishes the proof that all our conditions characterise local contractibility. The last condition is manifestly hereditary for inductive limits. The following lemma is proved similarly: ! !  Lemma 2.24. Let f W A ! B be a morphism in Kom. C / Š Kom.C/. We use Remark 1.133 to write f D .fi /i2I for a morphism of diagrams fi W Ai ! Bi in Kom.C/. The following are equivalent:

2.3 The local homotopy category of chain complexes

89

• f is a local homotopy equivalence; • H .X; f / is invertible for all X 2 Kom.C/; • for each i 2 I there are maps gi W Bi ! A;

hA i W Ai ! A;

hB i W Bi ! B;

B where gi is a chain map, and hA i and hi are chain homotopies between gi ı fi W Ai ! A and f ı gi W Bi ! B and the canonical maps Ai ! A and Bi ! B, respectively;

• for each i 2 I there are j 2 Ii and maps gi W Bi ! Aj ;

hA i W Ai ! Aj ;

hB i W Bi ! Bj ;

B where gi is a chain map, and hA i and hi are chain homotopies between gi ı fi W Ai ! Aj and fj ı gi W Bi ! Bj and the canonical maps Ai ! Aj and Bi ! Bj , respectively.

Being a local homotopy equivalence is hereditary for inductive limits of chain maps. Corollary 2.25. A chain map f is a local homotopy equivalence if and only if cone.f / is locally contractible. A chain complex C is locally contractible if and only if the zero map 0 ! C is a local homotopy equivalence. Proof. Combine the Puppe Exact Sequence in Theorem A.18 and the second characterisations in Lemmas 2.23 and 2.24. Viewing chain complexes of complete bornological vector spaces as reduced inductive systems in Kom.Ban/, we can also apply these notions to chain complexes and chain maps in Cborn. It is easy to specialise the above criteria to this case. For instance, a chain complex C in Cborn is locally contractible if and only if for each S 2 S@ .C / there is a bounded linear map h W CS ! C of degree 1 such that Œ@; h W CS ! C is the identical inclusion. Equivalently, the map CS ! C is null-homotopic. !  !  Definition 2.26. A functor from HoKom. C / or Kom. C / to another category is called local if it maps local homotopy equivalences to isomorphisms. A chain complex C is !  called local if the functor H .C; / from HoKom. C / to the category of Z=2-graded Abelian groups is local. !  A local functor automatically descends to HoKom. C / because the latter is the !  localisation of Kom. C / at the class of chain homotopy equivalences. If a functor on !  HoKom. C / is a triangle functor or homological or cohomological, then being local is equivalent to annihilating locally contractible chain complexes by Corollary 2.25. Example 2.27. Any chain complex C in C is local by Lemma 2.24. If C is a symmetric !  monoidal category with unit object 1, then C has the same unit object. Since the homology of a chain complex is defined by H .B/ WD H .1; B/, it is a local functor.

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2 Relations between entire, analytic, and local cyclic homology

2.3.3 Homotopy direct limits Our next goal is to replace an arbitrary functor by a local one. This requires homotopy direct limits. We recall this standard construction from algebraic topology. Let I be a directed set. We let n .I / be the set of all chains i0  i1      in in I . These sets for n 2 N form a simplicial set in a standard way: the face and degeneracy maps omit or double one entry, respectively. Let .Ai /i2I be an inductive system in Kom.C/ and define A.i0 ;:::;in / WD Ai0

for all .i0 ; : : : ; in / 2  .I /

if  is a face of  , then their first entries are related by i0 . /  i0 . /, so that the structure ./ map ˛ii00./ provides a natural map A ! A . This defines a coefficient system on the simplicial set  .I / with values in Kom.C/. The homotopy direct limit ho-lim Ai of the diagram .Ai /i2I is the chain complex ! that computesLthe homology of  .I / with coefficients in A. Explicitly, this chain complex has  2 n .I / A in dimension n; the boundary map M M @n W A ! A  2 n .I /

2 n1 .I /

./ is described by the block matrix .@ / with @ WD .1/k ˛ii00./ if  is the kth face of  for some k, and @ WD 0 otherwise. This defines a chain complex, that is, @n ı @nC1 D 0. It depends on the diagram .Ai /i2I , so that isomorphic objects in ! Kom.C/ rarely have isomorphic homotopy direct limits. In our case, the A are themselves chain complexes, so that ho-lim Ai has two ! commuting differentials. We add signs to the internal differentials A ! A with odd jj to make the differentials anti-commute and then equip ho-lim Ai with the ! total complex structure. That is, we add the internal and external Z=2-grading and ! !  differential. Thus ho-lim Ai becomes an object of Kom. C / Š Kom.C/. L ! The maps Ai ! A for i 2 I yield an augmentation map ˛ W i2 0 .I / Ai ! A, which vanishes on the range of @1 . Thus we get a canonical chain map

˛ W ho-lim Ai ! lim Ai D A: ! ! The dual construction of homotopy projective limits uses products instead of direct sums. We have Hom.ho-lim Ai ; C / Š ho-lim Hom.Ai ; C / !  i2I

i2I

!  for all C 22 Kom. C /. The homology of this complex can be computed by a spectral sequence because we are dealing with a bicomplex. The spectral sequence need not converge in general (this is a typical problem for cohomological spectral sequences). Nevertheless, it can be used to prove that ho-lim Bi is exact if Bi is exact for all i2I i 2 I . We omit this computation.

2.3 The local homotopy category of chain complexes

91

Proposition 2.28. The map ˛ W ho-lim Ai ! lim Ai is a local homotopy equivalence. ! ! We have H .ho-lim Ai ; B/ Š 0 if B is locally contractible, that is, ho-lim Ai !i2I !i2I is local. Hence ho-lim Ai is contractible .globally/ if .Ai /i2I is locally con!i2I tractible. S Proof. Write I D m2I Im , then   ho-lim Ai Š lim ho- lim Ai : ! ! ! i2I

m2I

i2Im

The canonical map ho-lim A ! lim A Š Am is a homotopy equivalence !i2Im i !i2Im i because m is a maximal element of Im ; the proof uses the contracting homotopy .i0 ; : : : ; in / 7! .i0 ; : : : ; in ; m/ on the level of simplicial sets. Therefore, ˛ is an inductive limit of homotopy equivalences and hence a local homotopy equivalence by Lemma 2.24. If B is locally contractible, then Hom.Ai ; B/ is contractible for all i 2 I . As we remarked above, this implies that Hom.ho-lim Ai ; B/ Š ho-lim Hom.Ai ; B/ is exact,  ! that is, H .ho-lim Ai ; B/ D 0. ! If .Ai / itself is locally contractible, so is ho-lim Ai because ˛ is a local homotopy ! equivalence. Hence H .ho-lim Ai ; ho-lim Ai / D 0, so that ho-lim Ai is contractible. ! ! ! Corollary 2.29. Any locally contractible chain complex is chain homotopy equivalent to an inductive limit of contractible chain complexes. Proof. Let A D .Ai / be locally contractible. Then ho-lim Ai is contractible. By the ! Puppe Exact Sequence (Theorem A.18), A is homotopy equivalent to the mapping cone of ˛ W ho-lim Ai ! A. The latter is an inductive limit of contractible chain complexes ! because ˛ is an inductive limit of homotopy equivalences.

2.3.4 Localisation of the homotopy category !  Now we are ready to pass from the homotopy category HoKom. C / to the local homo!  topy category HoKom. C /loc . This category is characterised by a universal property: it is the target of the universal local functor !  !  loc W HoKom. C / ! HoKom. C /loc ; that is, loc is local and any local functor factors uniquely through loc. We can describe this category quite explicitly; this also proves its existence. !  Let A 22 Kom. C / and write A D lim.Ai /i2I as in Theorem 2.15. Let AQ WD ! ho-lim Ai . Then we have a canonical local homotopy equivalence ˛ W AQ ! A by ! Proposition 2.28. Hence F .˛/ is invertible for any local functor F .

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2 Relations between entire, analytic, and local cyclic homology

Lemma 2.30. Construct local homotopy equivalences ˛A W AQ ! A and ˛B W BQ ! B !  for A; B 22 Kom. C / as above. Then any chain map A ! B lifts to a chain map AQ ! Q which is unique up to chain homotopy. Thus the homotopy direct limit construction B, !  Q B/ Q Š H .A; Q B/. becomes functorial on HoKom. C /. Furthermore, we have H .A; Proof. Since ˛B is a local homotopy equivalence, its mapping cone is locally con Q cone.˛B / D 0. By the Puppe Exact Sequence (TheoremA.18), tractible, so that H A; the map Q B/ Q ! H .A; Q B/ H .A; Q Functoriality is bijective. This yields the unique lifting of maps A ! B to AQ ! B. follows from uniqueness. Theorem 2.31. The local homotopy category exists and has spaces of morphisms Hloc .A; B/ WD H .ho-lim Ai ; ho-lim Bj / Š H .ho-lim Ai ; B/ ! ! ! if A Š lim Ai and B Š lim Bj with inductive systems in Kom.C/. The universal ! ! local functor is given by the functoriality of homotopy direct limits .Lemma 2.30). !  The category HoKom. C /loc is equivalent to the full subcategory of local chain !  complexes in HoKom. C /. Proof. Lemma 2.30 shows that there is a category with morphism spaces Hloc .A; B/ !  and a functor from HoKom. C / to this category, which we temporarily denote by ho-lim. We must check that it is the universal local functor. ! Let f W A ! B be a local homotopy equivalence and represent f by a morphism of diagrams fi W Ai ! Bi , i 2 I , as in Remark 1.133. We must show that ho-lim fi is a chain homotopy equivalence. Since the mapping cone construction is ! !  natural on Kom. C /, we have cone.ho-lim fi / Š ho-lim cone.fi /. Since f is a local ! ! homotopy equivalence, the inductive system cone.f / is locally contractible, so that ho-lim cone.fi / Š cone.ho-lim fi / is contractible by Proposition 2.28. This means ! ! that ho-lim fi is a homotopy equivalence, so that the functor ho-lim is local. ! ! If F is any local functor, then F .ho-lim A/ Š F .A/, so that elements of Hloc .A; B/ ! induce maps F .A/ Š F .ho-lim A/ ! F .ho-lim B/ Š F .B/: ! ! This establishes the universality of ho-lim. ! An inductive system of chain complexes .Ai /i2I is local if and only if the canonical map ho-lim.Ai / ! lim.Ai / is a chain homotopy equivalence. Hence the restriction ! ! of ho-lim to local objects is fully faithful. Since any object is locally chain homotopy ! !  equivalent to a local object, we conclude that HoKom. C /loc is equivalent to the full !  subcategory of local chain complexes in HoKom. C /.

2.3 The local homotopy category of chain complexes

93

!  If a functor F W HoKom. C / ! C0 for some category C0 is not yet local, we can enforce locality by taking LF .A/ WD F .ho-lim Ai /: ! This new functor comes equipped with a natural transformation LF ! F . An argument similar to the proof of Theorem 2.31 shows that LF is local and that any natural transformation from a local functor to F factors uniquely through LF . The functor F is called the localisation of F ; synonyms for localisation are .total/ left derived functor and left Kan extension. These notions from algebraic topology, algebraic geometry, and category theory become equivalent in the context of triangulated categories. !  The general theory of triangulated categories shows that HoKom. C /loc inherits a !  triangulated category structure from HoKom. C / such that the functor !  !  loc W HoKom. C / ! HoKom. C /loc is a triangle functor (see [74]). In our case, this is easy to see because ho-lim is a ! !  triangle functor on HoKom. C /. Let C be symmetric monoidal with unit object 1. Since the homology functor is already local, it agrees with its localisation Hloc .A/ WD Hloc .1; A/ (Example 2.27). In contrast, the cohomology H .A/ WD H .A; 1/ may differ from the local cohomology  Hloc .A/ WD Hloc .A; 1/ D H .ho-lim A; 1/: ! Now consider the special case C D Ban. An interesting feature of homotopy direct limits is that they are automatically reduced. Hence the construction of homotopy direct limits can be carried over to the full subcategory Kom.Cborn/. Since any local chain complex is homotopy equivalent to a homotopy direct limit, we conclude ! that the subcategories of local objects in Kom.Cborn/ and Kom.Ban/ are equivalent. Therefore, Theorem 2.31 and its analogue for Kom.Cborn/ yield an equivalence of categories ! HoKom.Cborn/loc Š HoKom.Ban/loc :

This result is due to Fabienne Prosmans and Jean-Pierre Schneiders ([82]). There are similar equivalences of categories for Born1=2 and Born. ! ! Remark 2.32. The completion functor Norm ! Ban is manifestly local, but the corresponding functor Born ! Cborn is not. If we identify ! HoKom.Born/loc Š HoKom.Norm/loc ;

! HoKom.Cborn/loc Š HoKom.Ban/loc

as explained above, then the completion functors on both categories have the same localisation because ho-lim V is always reduced. Since the completion for inductive ! systems is already local, we may view it as the localisation of the completion functor for bornological vector spaces.

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2 Relations between entire, analytic, and local cyclic homology

2.3.5 The definition of bivariant local cyclic theory Finally, we can define local cyclic cohomology and the corresponding bivariant theory. ! Definition 2.33. If A; B 22 Alg.Ban/, then we define      HL .A; B/ WD Hloc HL.A/; HL.B/ ; HL .A/ WD Hloc HL.A/ : These are the bivariant local cyclic theory and the local cyclic cohomology of A. There is no need to change   our definition  of HL .A/ because the homology functor is local, that is, H HL.A/ Š Hloc HL.A/ . ! Definition 2.34. An algebra morphism f W A ! B in Alg.Ban/ induces a chain map HL.f / W HL.A/ ! HL.B/;

which determines a class HL.f / 2 HL0 .A; B/. We call f an HL-equivalence if HL.f / is invertible, that is, there is g 2 HL0 .B; A/ with f ıg D idB and g ıf D idA . Equivalently, f induces a local chain homotopy equivalence HL.A/ ! HL.B/. An HL-equivalence induces isomorphisms HL .A/ Š HL .B/, HL .B/ Š HL .A/, HL .D; A/ Š HL .D; B/, and HL .B; D/ Š HL .A; D/ for all complete bornological algebras D. 

2.3.6 Locality as an exactness property The local homotopy category can be viewed as the derived category of an exact category. This is interesting because it shows that locality of functors is an exactness property. Exact categories are an ideal setting for homological algebra with topological or bornological algebras because the relevant module categories are never Abelian. Recall that an exact category is an additive category together with a class of admissible extensions, satisfying suitable axioms. These axioms ensure that any small exact category is a full, fully exact subcategory of an Abelian category. Following work of Alex Heller ([42]), Daniel Quillen defined exact categories in [90] and stated the embedding theorem without proof. Bernhard Keller simplified Quillen’s definition and wrote down the proof of the embedding theorem in [59] (see also [60] for a good summary). Many important constructions in homological algebra still work in exact categories. In particular, such categories have a derived category, and derived and total derived functors work as expected. Unfortunately, this generalisation is rarely treated in textbooks on homological algebra. !  Definition 2.35. An extension K  E  Q in C is called locally split if Hom.X; K/  Hom.X; E/  Hom.X; Q/ is a short exact sequence for all X 22 C.

2.3 The local homotopy category of chain complexes

95

Equivalently, any map X ! Q lifts to a map X ! E. The corresponding notion in the category Cborn is that for any bounded disk S 2 Sd .Q/, there is a bounded linear map QS ! E that is a section on QS for the projection map E  Q. From now on, we assume that morphisms in C have a kernel and a cokernel to avoid technical problems that are irrelevant for our applications. Definition 2.36. A chain complex .C; @/ in an exact category (with kernels) is called exact if ker @  C  ker @ is an admissible extension; here the map ker @ ! C is the identical inclusion, the map C ! ker @ is @. A chain map is a quasi-isomorphism if its mapping cone is exact. The derived !  category is the localisation of HoKom. C / at the quasi-isomorphisms. !  Proposition 2.37. Consider C as an exact category using locally split extensions. The exact chain complexes and quasi-isomorphisms are the locally contractible chain complexes and local chain homotopy equivalences, respectively, and the derived category !  is HoKom. C /loc . Proof. Let C be locally contractible. Equipping X 22 CZ=2 with the zero boundary map, we view maps f W X ! ker @ as chain maps X ! C . Since C is locally contractible, any such map is null-homotopic; equivalently, it lifts to fO W X ! C with @ @ ı fO D f . Therefore, ker @ ! C ! ker @ is a locally split extension. Conversely, suppose that this diagram is a locally split extension. Let A 2 Kom.C/ and let f W A ! C be a chain map. Its restriction f0 W ker @A ! ker @C lifts to a map 0 W ker @A ! C with @C ı 0 D f0 . Let '0 WD f  0 ı @A W A ! C , then @C ı '0 D @C ı f  @C ı 0 ı @A D f ı @A  f0 ı @A D 0: Hence we may lift '0 to a map h0 W A ! C with @C ı h0 D '0 . Let f 0 WD f  Œ@; h0 , so that f 0 f . We compute f 0 D f  @C h0  h0 @A D f  '0  h0 @A D .0  h0 /@A : Thus f 0 ı @A D 0. Since f 0 is a chain map, @C ı f 0 D 0 as well, so that f 0 factors through B WD coker @A 22 C and ker @C . The resulting map B ! ker @ lifts to a map h1 W A ! B ! C with @C h1 D f 0 and h1 @A D 0. Thus Œ@; h1  D f 0 , so that !  f f 0 0. Since f is arbitrary with A 22 Kom. C /, we conclude that C is locally contractible. Now Corollary 2.25 shows that the quasi-isomorphisms are the local chain homo!  topy equivalences. Thus the derived category is HoKom. C /loc . !  Definition 2.38. An additive functor F W C ! C0 to some exact category C0 is called locally split-exact if it maps locally split extensions to distinguished extensions in C0 . !  Any additive functor F induces a functor F W Kom. C / ! Kom.C0 /. Proposition 2.37 shows that F is locally split-exact if and only if F is local.

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2 Relations between entire, analytic, and local cyclic homology

2.3.7 Local versus non-local cohomology ! Now we compare the local and non-local cohomology of chain complexes in Ban. We  will find a class of chain complexes for which the canonical map H .C / ! Hloc .C /  is invertible. We view Hloc .C / as the localisation of the contravariant functor on chain ! complexes induced by the dual space functor V 7! V 0 WD Hom.V; 1/ on Ban. If K  E  Q is a (locally split) extension, then the sequence 0 ! Q0 ! 0 E ! K 0 is always exact, but the map E 0 ! K 0 need not be surjective. Surjectivity means that bounded linear functionals K ! 1 extend to bounded linear functionals E ! 1. For topological vector spaces, the Hahn–Banach Theorem ensures that this is possible. Since this fails for bornological vector spaces, the dual space functor on !  .C / may differ. Cborn or Ban is not locally split-exact and H .C / and Hloc If the underlying bornological vector space of C is, say, a Fréchet space with the precompact or von Neumann bornology, then the Hahn–Banach Theorem holds for C because the notions of bounded and continuous linear functional agree for subspaces of C . But the construction of ho-lim C leaves the subcategory of Fréchet spaces and ! produces bornological vector spaces about which we know very little. More precisely, the underlying bornological vector space of ho-lim C is always a direct sum of Banach ! spaces. Such spaces do not satisfy the Hahn–Banach Theorem for all subspaces. Therefore, the following question seems to be open:  Question 2.39. Is H .C / Š Hloc .C / if C is a chain complex of Fréchet spaces with the precompact or von Neumann bornology?

This question is irrelevant for analytic and local cyclic cohomology because the underlying vector space of HA.A/ is never Fréchet. The following vague question seems more relevant:  Question 2.40. Is the isomorphism problem for H .C / and Hloc .C / related to reflexivity?

Since the homotopy projective limit construction is rather intractable for uncountable directed sets, we restrict attention to countable inductive systems now. Let C D .Cn /n2N be a countable inductive system of chain complexes of Banach spaces with structure maps n W Cn ! CnC1 . In this case, we can choose a much simpler model for the homotopy direct limit of the form M n2N

idS

Cn !

M

Cn ;

n2N

where S is the shift map that sends the L direct summand Cn to CnC1 via n ; the natural maps Cn ! lim Cn combine to a map n2N Cn ! lim Cn , whose composition with ! ! id  S vanishes. Hence we get a canonical map cone.id  S / ! C . It is easy to see that this map is a local chain homotopy equivalence and that cone.id  S / is local; for

2.3 The local homotopy category of chain complexes

97

the first assertion, notice that M

idS

Cn !

n2N

M n2N

Cn ! lim Cn !

(2.41)

is an inductive limit of split extensions. Hence cone.id  S / is chain homotopy equivalent to the homotopy direct limit of C . We shall use this model to compute the local cohomology. The dual space functor maps lim Cn and ho-lim Cn to lim Cn0 and the bicomplex ! !  Y Y 0 idS Cn0 ! Cn0 ; n2N

n2N

respectively; both are chain complexes of vector spaces. The kernel and cokernel of id  S 0 are, by definition, lim Cn0 and lim1 Cn0 . Hence the natural map H .C / !    .C / is an isomorphism if and only if lim1 Cn0 is exact. It is easier to decide whether Hloc  lim1 Cn0 D 0, that is, whether id  S 0 is surjective. The following criterion is well  known: Theorem 2.42 ([77], [78]). Let .Vn /n2N be a projective system of complete metrisable topological vector spaces and suppose that the structure maps VnC1 ! Vn have dense range for all n 2 N. Then lim1 Vn D 0 and the canonical maps lim Vn ! Vm have   dense range for all m 2 N. ! ! Corollary 2.43. Let C be a chain complex in Ban whose underlying object in Ban is isomorphic to a countable, reduced inductive system .Cn /n2N of reflexive Banach  .C /. spaces. Then H .C / Š Hloc We do not need the Banach spaces Cn to be chain complexes. 0 ! Cn0 is again Cn ! CnC1 because Cn Proof. The transpose of the map CnC1 and CnC1 are reflexive, and the latter map is injective because the inductive system is 0 reduced. Hence the map CnC1 ! Cn0 has dense range in the norm topology. (Without reflexivity, we only get dense range in the weak topology.) Now Theorem 2.42 yields  lim1 Cn0 D 0, so that the canonical map H .C / ! Hloc .C / is an isomorphism.  Instead of reflexivity, it suffices if the maps Cn ! CnC1 are weakly compact, that is, the unit ball of Cn becomes relatively compact in the weak topology on CnC1 . This is equivalent to .Cn / being a direct union of reflexive Banach spaces (see [51]). Hence Corollary 2.43 applies if C is a Silva space. We can use Corollary 2.43 to compare the analytic and local cyclic cohomology ! for a class of bornological algebras. Let A be an algebra in Ban and let AQ WD lim A; ! this is an algebra in Cborn. Proposition 2.11 yields       Q D H HA.lim A/ D H lim HL.A/ D H HL.A/ : HA .A/ ! !

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2 Relations between entire, analytic, and local cyclic homology

   This differs from HL .A/ D Hloc HL.A/ only by the localisation of the cohomology  Q if and only if H and Hloc functor. Hence HL .A/ Š HA .A/ agree for HL.A/. Theorem 2.44. Let A be a complete bornological algebra whose underlying bornological vector space is a Silva space with the approximation property. Then HL .diss A/ Š HA .A/. Proof. For Silva spaces, the local, strong local, and global approximation properties coincide. The strong local approximation property ensures diss HA.A/ Š HL.diss A/ by Theorem 2.13. Furthermore, HA.A/ inherits the property of being a Silva space. By the remark after Corollary 2.43, we can write HA.A/ as an inductive limit of reflexive Banach spaces. Hence the assertion follows from Corollary 2.43. Silva spaces have not yet been used much in non-commutative geometry. Theorem 2.44 suggests to pay more attention to them. Many classical examples of Fréchet algebras contain an interesting dense Silva subalgebra (see §3.6.1). Nevertheless, Silva spaces are orthogonal to Fréchet spaces in the sense that there are very few bounded linear maps from Silva spaces to Fréchet spaces by Proposition 1.159. Example 2.45. The algebra C ! .M / of real-analytic functions on a real-analytic manifold is a Silva algebra. So is the convolution algebra S ! .G/ of functions on a discrete group with subexponential decay (see §3.5.4). Both algebras have the approximation property.

2.4 Some counterexamples with compact Lie groups This section deals with some surprisingly simple counterexamples, which exhibit differences between periodic, entire, analytic, and local cyclic homology and cohomology. Most proofs depend on results that will only be proved much later in this monograph. Let K be an infinite compact Lie group such as the circle group K D T 1 ; we exclude finite groups because they do not create counterexamples. We study the cyclic homology of the following (C-valued) convolution algebras on K: • the group C  -algebra C  .K/ (equipped with the precompact bornology); • the group Banach algebra L1 .K/ (equipped with the precompact bornology); • the nuclear Fréchet algebra of smooth functions C 1 .K/; its precompact and von Neumann bornologies coincide; • the nuclear Silva algebra of real-analytic functions C ! .K/; • the algebra R.K/ of matrix coefficients of finite-dimensional representations of K, equipped with the fine bornology.

2.4 Some counterexamples with compact Lie groups

99

Recall that a matrix coefficient of a representation  W K ! Aut.V / is a function of the form x 7! l.x  v/ for v 2 V , l 2 V 0 . L The Peter–Weyl Theorem identifies R.K/ Š 2KO Md./ .C/, where KO is the set of irreducible representations of K and d./ is the dimension of . This extends to an isomorphism between C  .K/ and the C  -algebraic direct sum, which consists of C0 -sequences .x /2KO with lim!1 kx k D 0, where kx k denotes the C  -norm on Md./ .C/. The other convolution algebras are completions of R.K/ for other norms. We are going to compute various cyclic (co)homology theories for these convolution algebras, as far as possible. We begin with the local cyclic theory, where the results are most satisfactory. It turns out that the obvious maps R.K/ ! C ! .K/ ! C 1 .K/ ! L1 .K/ ! C  .K/ are HL-equivalences and that the Chern–Connes character K .A/ ˝Z C ! HL .A/ is an isomorphism if A is one of these algebras. Of course, we implicitly apply the ! dissection functor to view our algebras as algebras in Ban. For periodic cyclic homology, the situation is rather different: all four algebras have different periodic cyclic homology, and the only one for which we have HP .A/ Š HL .A/ or HP .A/ Š HL .A/ is R.K/. The failure of such an isomorphism for C 1 .K/ and C ! .K/ is striking because they are quasi-free. They cannot be analytically quasi-free because then their X-complexes would compute both HP and HL , which is not the case. We prove a theorem of Masoud Khalkhali ([62]) which shows that entire and periodic cyclic cohomology agree for Banach algebras of finite bidimension such as L1 .K/ and C  .K/. In contrast, HP and HA differ for them; thus HA .A/ depends on the bornology on A. Since HP and HE differ for C 1 .K/ and C ! .K/, Khalkhali’s Theorem cannot extend to quasi-free Fréchet algebras or Silva algebras.

2.4.1 Local cyclic homology Proposition 2.46. The obvious injections R.K/ ! C ! .K/ ! C 1 .K/ ! L1 .K/ ! C  .K/ are HL-equivalences. The Chern–Connes character K .A/ ˝Z C ! HL .A/ Š HA .A/ is an isomorphism if A is one of these algebras. Explicitly, we have O O K .A/ Š ZŒK; HA .A/ Š HL .A/ Š CŒK;   O O K C  .K/ Š ZK ; HA .A/ Š HL .A/ Š C K ;

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2 Relations between entire, analytic, and local cyclic homology

where KO is the set of irreducible representations of K and O D CŒK

M

O

C; C K D

2KO

O D ZŒK

M

Y

O C/; C Š HomC .CŒK;

2KO

Z;

Z

KO

D

Y

O Z/: Z Š HomZ .ZŒK;

2KO

2KO

    The Chern–Connes character K C  .K/ ˝Z C ! HL C  .K/ is not invertible. Proof. By the Peter–Weyl Theorem, R.K/ is a direct sum of matrix algebras; the other convolution algebras are suitable completions. We can map our convolution algebras back to R.K/ by multiplication with an approximate unit consisting of central projections in R.K/. This shows that the maps above have approximably dense range. Since the fine bornology is the only separated bornology on a finite-dimensional vector space, R.K/ is isoradial in the other convolution algebras by Proposition 3.40. Theorem 6.21 shows that all our convolution algebras are HL-equivalent to R.K/ and have the same local cyclic homology and cohomology. For the same reason, they all have the same K-theory. By the way, the other maps such as C 1 .K/ ! C  .K/ are isoradial as well. We omit the proof because we do not need this. The C  -algebra C  .K/ belongs to the bootstrap category because it is a C  -algebraic direct sum algebras. Hence Theorem 7.7 shows that the Chern–Connes    of matrix  character K C  .K/ ˝Z C ! HL C  .K/ is an isomorphism. Theorem 5.70 and the stability of HL yield  M   M  O HL.A/ HL R.K/ HL Md .C/ C D CŒKI 2KO

2KO

 our formulas for HL and  HL .A/ follow. Similar computations yield the formulas   .A/   for K .A/ and K C .K/ . These computations yield another proof that the Chern– Connes character K .A/ ˝Z C ! HL .A/ Š HA .A/ is an isomorphism. In contrast, the character in K-homology

    O O ZK ˝Z C Š K C  .K/ ˝Z C ! HL C  .K/ Š C K O

O

is injective but not surjective: .a / 2 ZK belongs to ZK ˝Z C if and only if there is O a finitely generated subgroup G  C with a 2 G for all  2 K.

2.4.2 Hochschild, cyclic, and periodic cyclic homology These homology theories are extremely easy to compute because the algebras we consider are biprojective (see §A.7). It is far more difficult to treat non-compact Lie groups (see [75]).

2.4 Some counterexamples with compact Lie groups

101

Lemma 2.47. The map y C ! .K/ Š C ! .K  K/;  W C ! .K/ ! C ! .K/ ˝

f .g0 ; g1 / WD f .g0  g1 /;

is a C ! .K/-bimodule section for the multiplication map y C ! .K/ ! C ! .K/; m W C ! .K/ ˝

f1 ˝ f2 7! f1  f2 :

Thus C ! .K/ is biprojective. We have   C ! .K/=Œ ;  Š Z C ! .K/ D ff 2 C ! .K/ j f .xyx 1 / D f .y/ for all x; y 2 Kg: y C ! .K/ Š C ! .K  K/. It is clear that  is Proof. We use the isomorphism C ! .K/ ˝ a well-defined bounded R linear map. It is a section for m because the convolution takes the form m.h/.g/ D K h.gg1 ; g11 / dg1 on C ! .K  K/. The bimodule structure on y C ! .K/ is defined by left and right convolution, the bimodule structure on C ! .K/ ˝ C ! .K/ is defined by left convolution on the left and right convolution on the right factor. These actions of C ! .K/ are integrated forms of group actions of K on these spaces, defined by g f .g1 / WD f .g 1 g1 /, g h.g0 ; g1 / WD h.g 1 g0 ; g1 /, g f .g1 / D f .g1 g/ and g h.g0 ; g1 / WD h.g0 ; g1 g/. It is trivial to check that  is equivariant with respect to these group actions. Hence it is a bimodule map with respect to their integrated forms. y ! We consider C ! .K/ as a direct of C ! .K/ via  ım in order to  summand  C ˝C .K/ ! ! ! y C .K/ =Œ ;  Š C ! .K/ because C ! .K/ compute C .K/=Œ ; . We have C .K/C ˝ is free as a left module. On commutator quotients, the map  ı m induces Z ! ! P W C .K/ ! C .K/; Pf .g/ WD f .h1 gh/ dh: K

On the one hand, its range is C ! .K/=Œ ; , on the other, it is the subspace C ! .K= Ad K/ of conjugation-invariant functions. This is equal to the centre of C ! .K/. y R.K/ Š Lemma 2.47 carries over to R.K/, C 1 .K/, and L1 .K/ because R.K/ ˝ y C 1 .K/ Š C 1 .K  K/, and L1 .K  K/ Š L1 .K/ ˝ y R.K  K/, C 1 .K/ ˝ L1 .K/; these isomorphisms use Example 1.95 and Theorem 1.91. The proof for C  .K/ y C  .K/ 6Š C  .K  K/ (see [39, Theorem 9]). requires more work because C  .K/ ˝ In each case, the commutator quotient A=Œ ;  agrees with the centre Z.A/. Now Proposition A.131 completely describes the Hochschild, cyclic, and periodic cyclic (co)homology for our five convolution algebras. In particular, HP0 .A/ Š Z.A/ and HP1 .A/ Š 0 for all of them.   O this is a vector space with The Peter–Weyl Theorem yields Z R.K/ Š CŒK;   a countable basis equipped with the fine bornology. Similarly, Z C  .K/ is the   O The centre Z C 1 .K/ is a nuclear Fréchet space; arguing as C  -algebra C0 .K/. O of rapidly decreasing functions KO ! C; in [81], we may identify it with the space S.K/ to define rapid decay, we map irreducible representations to their highest weights; this identifies KO with a discrete subset of TO d Š Zd for a maximal torus T d  K.

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2 Relations between entire, analytic, and local cyclic homology

  Similarly, we can identify Z C ! .K/ , which is a nuclear Silva space, with the space   O of sequences of subexponential decay. Finally, Z L1 .K/ is the Banach space S ! .K/ L1 .K= Ad K/ for a suitable measure on the space K= Ad K of conjugacy classes in K. The Fourier transform of this space is not so easy to describe. Notice that HP0 .A/ Š Z.A/ is different in all cases. Hence the Chern–Connes character K .A/ ˝Z C ! HP .A/ is only invertible if A D R.K/.

2.4.3 Analytic and entire cyclic homology Analytic and local cyclic homology agree for all five convolution algebras because they have the strong local approximation property (Theorem 2.13). Theorem 2.44 shows that analytic and local cyclic cohomology agree for R.K/ and C ! .K/ because they are Silva algebras with the approximation property. We can only get partial information about the analytic cyclic cohomology of the other three algebras C 1 .K/, L1 .K/, and C  .K/. Before we discuss this, we notice that we can even compute the bivariant analytic cyclic theory for R.K/ because Proposition 5.72 yields  M   M  O HA R.K/ HA Mdim  .C/ C Š CŒK: 2KO

2KO

Therefore, R.K/ satisfies the Universal Coefficient Theorem     HA .R.K/; B/ Š Hom HA R.K/ ; HA .B/ for all complete bornological algebras B. Proposition 2.48. Let A be C 1 .K/, L1 .K/, or C  .K/. Then the natural map from HA .A/ to HL .A/ is surjective. O

Proof. We have HL1 .A/ D 0 and HL0 .A/ Š C K by Proposition 2.46. Recall O that K0 .A/ Š ZK (Proposition 2.46). The Chern–Connes character ch W K 0 .A/ ! 0 HA .A/ that we will construct in §7.5 provides a map     O ch W ZK ˝Z C Š K0 C  .K/ ˝Z C ! HA0 C  .K/ : O

O

Its composition with the natural map to HL0 is the canonical map ZK ˝Z C ! C K . Recall that its range consists of all maps KO ! C whose images are contained in some finitely generated subgroup of C.   Traces on C  .K/ generate another subspace of HA0 C  .K/ isomorphic to O C/ Š `1 .K/: O Hom.C  .K/=Œ ; ; C/ Š Hom.C0 .K/; Since C contains a dense finitely generated subgroup, any sequence in C can be decomposed as .an / C .bn / with a sequence .an/ whose entries liein a finitely  generated O The map HA0 C  .K/ ! HL0 C  .K/ is surjective subgroup and .bn / 2 `1 .K/. because its range contains both .an / and .bn /.

2.4 Some counterexamples with compact Lie groups

103

It seems plausible that analytic and local cyclic cohomology should agree for all three algebras in Proposition 2.48. However, I do not know how to prove this. In any case, our partial results show that periodic and analytic cyclic cohomology differ for them. Proposition 2.49. The algebras C ! .K/ and C 1 .K/ are quasi-free and biprojective, but not analytically quasi-free. Proof. Alexei Pirkovskii shows in [81] that C 1 .K/ has cohomological dimension 1, so that it is quasi-free. We claim that the same method applies to C ! .K/. The first step in the proof is to reduce the assertion to the corresponding statement for the centre ([81, Theorem 5.2]); this works for both C ! .K/ and C 1 .K/. Hence we are led to study the function algebras S.N/ and S ! .N/ of rapidly decreasing sequences or sequences with subexponential decay; the multiplication is pointwise. By a theorem of Alexander Helemski˘ı (Theorem A.129), a biprojective algebra A is quasi-free if and only if there is a bimodule section for the monomorphism y A ! .AC ˝ y A/ ˚ .A ˝ y AC /; A˝

x ˝ y 7! .x ˝ y; x ˝ y/:

y AC This is easy to accomplish if A is S.N/ or S ! .N/. We view elements of AC ˝ 2 as functions on N . Multiplication by the characteristic functions of the sets fx  yg y A ! A˝ y A and A ˝ y AC ! A ˝ y A, and fx > yg defines bimodule maps AC ˝ 1 ! respectively; together they generate the required section. Thus C .K/ and C .K/ are quasi-free. Periodic and analytic cyclic (co)homology coincide for analytically quasi-free algebras because the X-complex computes both by Corollaries 4.33 and 5.48. But this is not the case for C ! .K/ and C 1 .K/. Finally, we consider entire cyclic (co)homology. For the algebras R.K/, C ! .K/, and C 1 .K/, the precompact and the von Neumann bornology coincide, so that there is no difference between the analytic and entire theory. For the Banach algebras L1 .K/ and C  .K/, we can use the following special case of a theorem of Masoud Khalkhali, ([61, Theorem 5.2]). Theorem 2.50. Let A be a Banach algebra of finite bidimension with the von Neumann bornology, so that HA .A/ D HE .A/. The canonical map HA.A/ ! HP.A/ is a chain homotopy equivalence, so that HL .A/ Š HA .A/ Š HE .A/ Š HP .A/; HL .A/ Š HA .A/ Š HE .A/ Š HP .A/: Since analytic and periodic cyclic (co)homology differ for the quasi-free algebras C ! .K/ and C 1 .K/, Theorem 2.50 only works for Banach algebras and already fails for Fréchet algebras or Silva algebras. Proof. We can rewrite the cyclic bicomplex in Figure 2.2 using the boundary map bCB on non-commutative differential forms (see §A.6.2). Thus HA.A/ is the completion

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2 Relations between entire, analytic, and local cyclic homology

S n n of .A/ for the bornology generated by 1 nD1 b =2cŠ hS i.dS / for S 2 S.A/ with boundary map B C b. Now we recall some ideas from the proof of Theorem A.123. The finite bidimension assumption means that k .A/ is a projective bimodule for some k 2 N. Hence it supports a right bimodule connection r W k .A/ ! kC1 .A/. We extend it to a map k .A/ ! kC1 .A/ of degree 1 as usual, so that r.x  !/ D x  r.!/ for x 2 A, ! 2 k .A/, and r.!  / D r.!/   C .1/m !  d./ for all ! 2 m .A/,  2 n .A/ with m  k. We extend r to .A/ by letting rj m .A/ D 0 for m < k. The following computations hold in m .A/ for m > k. Using the right connection property and the definition of b via commutators, we compute r ıb Cb ır D id. Hence Œb CB; r D idCŒB; r, so that id ŒB; r. Iterating, we get id .1/n ŒB; rn for all n 2 N; the chain homotopy for this is n1 X .1/j r ı ŒB; rj : hn WD j D0

P The resulting infinite series h1 WD j1D0 .1/j r ıŒB; rj automatically converges in   Q HP.A/ and contracts the subcomplex n>k n .A/  b kC1 .A/ ; therefore, HP.A/ is chain homotopy equivalent to the truncated complex X

.k/

.A/ WD

k1 Y

k .A/ ; n .A/   kC1 b  .A/ nD0

whose boundary map is induced by b C  B.  If the series h1 converges in Hom HA.A/; HA.A/ as well, then we get HA.A/ X .k/ .A/ HP.A/. Now we use that A is a Banach algebra with the von Neumann bornology. The norm of r on n .A/ is the same for all n  k; the norm of B on n .A/ is n; hence rŒB; rj has norm n.n C 2/    .n C 2j  2/ on n .A/. Since factors bn=2cŠ also appear in the definition of our bornology, the series h1 is convergent on HA.A/. Let A be one of the Banach algebras L1 .K/ and C  .K/. Since A is biprojective, it has bidimension 2 by Theorem A.129, so that Theorem 2.50 applies. We get HE .A/ Š HP .A/;

HE .A/ Š HP .A/:

Hence analytic and entire cyclic (co)homology differ for A. Thus the analytic cyclic (co)homology depends on the bornology on A.

Chapter 3

The spectral radius of bounded subsets and its applications

We are going to use spectral radius estimates in order to define two classes of bornological algebras: analytically nilpotent algebras and local Banach algebras. In the bornological framework, it is very natural to extend the familiar notion of spectral radius for a single algebra element to a joint spectral radius for bounded subsets. This joint spectral radius is the crucial ingredient for all results in this section. The spectral radius of a single element does not yet see the non-commutativity of the algebra because it only involves the subalgebra generated by a single element and its resolvent, which is commutative. Inspired by the work of Michael Puschnigg ([88]), we introduce isoradial homomorphisms; these are bounded algebra homomorphisms between locally multiplicative bornological algebras that preserve the spectral radius of bounded subsets and have dense range in an appropriate sense. Their important role for local cyclic homology will become apparent in Chapter 6. In the setting of topological algebras, there is no totally satisfactory notion of “smooth subalgebra.” The most sophisticated one due to Bruce Blackadar and Joachim Cuntz ([3]) covers most applications of interest, but lacks some desirable formal properties, and has a rather complicated definition. In contrast, the notion of an isoradial homomorphism has an obvious definition, covers even more important examples, and gives rise to a theory with very satisfactory formal properties. We introduce the joint spectral radius and establish its basic properties in §3.1. We also introduce the closely related notion of a power-bounded subset. We discuss the special case of commutative bornological algebras, where the joint spectral radius often contains no more information than the spectral radius for single elements. We study locally multiplicative bornological algebras in §3.2. Their defining property is that all bounded subsets have finite spectral radius. A bornological algebra has this property if and only if it is an inductive limit of semi-normed algebras. Therefore, a complete locally multiplicative bornological algebra is nothing but an inductive limit of Banach algebras. This important observation already dates back to the work of Henri Hogbe-Nlend and Lucien Waelbroeck ([48]) in the 1970s. We show that the class of locally multiplicative bornological algebras is closed under various constructions, including extensions. We analyse what it means for a Fréchet algebra with the precompact bornology to be locally multiplicative, reproducing a result of Michael Puschnigg. A bornological algebra is called analytically nilpotent if the spectral radius vanishes for all bounded subsets. This class of algebras is introduced in §3.3. It is crucial for analytic cyclic homology because we use it to characterise the analytic tensor

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3 The spectral radius of bounded subsets and its applications

algebra. In fact, all examples of analytically nilpotent algebras that we shall need are derived from analytic tensor algebras. We show that the class of analytically nilpotent bornological algebras is closed under extensions and several other constructions. We define isoradial homomorphisms in §3.4 and study their behaviour for extensions and tensor products. Several results about isoradial homomorphisms are essentially due to Michael Puschnigg; some others have appeared previously in [67]. We are mainly interested in injective isoradial homomorphisms, that is, isoradial subalgebras. We consider some important examples of isoradial subalgebras in §3.5 and compare our notion with notions of smooth Fréchet subalgebras in Banach algebras by Bruce Blackadar and Joachim Cuntz ([3]) and by Larry Schweitzer ([92]). Given a dense subalgebra of a locally multiplicative algebra that is not yet isoradial, we construct the isoradial subalgebra that it generates in §3.6. Roughly speaking, all elements of this isoradial hull are generated by non-commuting power series in the given dense subalgebra that converge in the ambient algebra. We also show that we can often find an isoradial subalgebra that is Silva and has the approximation property. In §3.7, we examine briefly how these notions extend to algebras in the category ! ! ! Ban. This category of algebras Alg.Ban/ is strictly bigger than the category Alg.Ban/ of inductive systems of Banach algebras: the latter is the subcategory of locally multiplicative algebras. Since inductive systems have neither elements nor subsets, we have to modify our notion of spectral radius in this context. The characterisation of locally ! multiplicative algebras in Ban requires more work than the corresponding result for bornological algebras. Once this is accomplished, analytically nilpotent algebras and isoradial homomorphisms create no further difficulties.

3.1 The spectral radius Notation 3.1. Let A be an algebra. We let S1  S2 WD fx1  x2 j x1 2 S1 ; x2 2 S2 g for S1 ; S2  A. Using this, we define S n  A for S  A and n 2 N1 . We also let S

1

WD

1 [

S n:

nD1

Definition 3.2. A subset S  A is called submultiplicative if S 2  S . Clearly, S 1 is the smallest submultiplicative subset containing S. Definition 3.3. Let A be a bornological algebra. A bounded subset S  A is called power-bounded if S 1 is bounded. Definition 3.4. Let A be a bornological algebra and let S 2 S.A/. We define the spectral radius %.S / D %.SI A/ of S as the infimum of the numbers r 2 R>0 for which r 1 S is power-bounded. If no such r exists, we put %.S / D 1.

3.1 The spectral radius

107

If A is complete and S  A is power-bounded, then .S 1 /~ is bounded as well; this is the smallest complete submultiplicative disk containing A. By Lemma 1.43, its elements are given by power-series X X n xn;1    xn;l.n/ with jn j  1 and xn;j 2 S . n2N

n2N

The spectral radius is local in the following sense: if A is the union of a directed set of bornological subalgebras .Ai /i2I then %.S I A/ D lim %.SI Ai / D inf %.SI Ai / for all S 2 S.A/. i2I

(3.5)

i2I

Lemma 3.6. Let A be a bornological algebra. If S1  S2 and S2 2 S.A/, then %.S1 /  %.S2 /. If S 2 S.A/, then %.S / D %.S } /, %.cS / D jcj%.S / for all scalars c, and %.S n / D %.S /n for all n 2 N1 . Let f W A1 ! A2 be  a bounded homomorphism between two bornological algebras. Then % f .S/I A2  %.SI A1 / for all S 2 S.A1 /. Proof. We only prove %.S n /  %.S /n , the remaining assertions are obvious. Write Sn1 1 S D j D0 S j  .S n /1 . Hence S 1 is bounded once .S n /1 is. If S has just one element x 2 A, then its spectral radius is the spectral radius of x in the usual sense familiar from operator theory. If S is arbitrary, then we necessarily have %.S /  supf%.x/ j x 2 Sg: We often have equality here if A is commutative (see Proposition 3.23). If the algebra A is non-commutative, things get more interesting: Example 3.7. Consider the subset fx; yg  M2 .C/ with       0 1 0 0 1 0 x WD ; y WD ; xy D ; 0 0 1 0 0 0



 0 0 yx D : 0 1

Then x 2 D y 2 D 0, so that %.x/ D 0 and %.y/ D 0. Nevertheless, %.fx; yg/ D %.fx; yg  fx; yg/ =2 D %.fx 2 ; y 2 ; xy; yxg/ =2 D %.fxy; yxg/ =2 D 1 1

1

1

because xy and yx are commuting idempotents. If A is a Banach algebra, then we have %.S /  kS k WD sup kxk: x2S

Conversely, spectral radius estimates can be converted into norm estimates for sufficiently high powers:

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3 The spectral radius of bounded subsets and its applications

Lemma 3.8. Let A be a Banach algebra with unit ball D, equipped with the von Neumann bornology. Let S r 2 .0; 1/, and let S 2 S.A/ satisfy %.S / < 1. Then there is n 2 N such that S n  N n S N  r D. S Proof. By hypothesis, .c S /1 is bounded for some c > 1, that is, .c S /n  C D for some C > 0. Thus S n  c n C D, which is contained in r D for sufficiently large n. Lemma 3.9. Let V be a bornological vector space and equip Hom.V; V / with the equibounded bornology. A subset S  Hom.V; V / is power-bounded if and only if the subset of S -invariant subsets is cofinal in S.V /. Proof. If T 2 S.V /, then S 1 .T /  V is the smallest S -invariant subset containing T . Hence S 1 is equibounded if and only if any bounded subset is contained in (or absorbed by) an S-invariant bounded subset.

3.2 Locally multiplicative bornological algebras Theorem 3.10. The following are equivalent for a bornological algebra A: (1) %.S/ < 1 for all S 2 S.A/; (2) any bounded subset of A is absorbed by a bounded submultiplicative disk; (3) A is a direct union of semi-normed subalgebras. Condition .3/ means that there is a reduced inductive system of semi-normed algebras whose direct limit is isomorphic to A. If A is separated or complete, then we can choose this reduced inductive system to consist of normed algebras or Banach algebras, respectively. }

Proof. If %.S/ < r, then ..r 1 S/1 / is a bounded submultiplicative disk that absorbs S . Conversely, if S  r T for a bounded submultiplicative disk T , then .r 1  S /1  T 1 D T is bounded. Hence the first two conditions are equivalent. It is clear that the third condition implies the first two. Suppose conversely that A satisfies the second condition. Let Sm .A/ be the set of all submultiplicative bounded disks in A. If S 2 Sm .A/, then the gauge norm on AS is submultiplicative, so that AS is a semi-normed algebra. By assumption, any bounded subset of A is absorbed by one in Sm .A/. Therefore, Sm .A/ is a cofinal subset of .Sd .A/; /. Now Theorem 1.139 yields A D lim A as desired. !S2Sm .A/ S If A is separated, the algebras AS are normed, not just semi-normed. If A is complete, we may restrict further to the set Scm .A/ of complete submultiplicative bounded disks. This set is still cofinal in Sd .A/ because the complete disked hull of a submultiplicative bounded subset is again submultiplicative.

3.2 Locally multiplicative bornological algebras

109

Definition 3.11. We call a bornological algebra A locally multiplicative if it satisfies the equivalent conditions of Theorem 3.10. Complete locally multiplicative bornological algebras are also called local Banach algebras. Warning 3.12. This should not be confused with the notion of a locally multiplicatively convex topological algebra. The precompact or von Neumann bornologies on such algebras need not be locally multiplicative. An example of a locally multiplicatively convex Fréchet algebra that is not locally multiplicative as a bornological algebra is Q Q n2N C; the problem is that %.x/ D 1 if x 2 n2N C is an unbounded sequence. Example 3.13. Let X  C be a compact subset. For a compact neighbourhood K X, let H.K/ be the algebra of continuous functions K ! C that are holomorphic on the interior of K; this is a Banach algebra because it is a closed subalgebra of C.K/. If K0  K1 are two such compact neighbourhoods, then we have a natural bounded restriction homomorphism H.K1 / ! H.K0 /. There is a fundamental decreasing sequence .Kn / of such compact neighbourhoods, that is, any compact neighbourhood contains Kn for sufficiently large n. Choosing this sequence suitably, we can achieve that the restriction maps H.Kn / ! H.KnC1 / are all injective. We let O.X / WD lim H.Kn /; this is the algebra of germs of holomorphic functions near X . It is a local ! Banach algebra by definition.

3.2.1 Holomorphic functional calculus The usual Banach algebra functional calculus can be extended easily to local Banach algebras. This was one of the historical motivations for studying bornological algebras. The bornological algebras O.X / defined in Example 3.13 play a crucial role for this. Definition 3.14. Let A be a unital algebra over C. The spectrum of x 2 A is the set †.xI A/ of all  2 C for which x    1A is not invertible in A. Theorem 3.15. Let A be a unital local Banach algebra and x 2 A. The spectrum †.xI A/  C is a non-empty compact subset of C. There is a unique bounded homo morphism O †.xI A/ ! A – called holomorphic functional calculus for x – which   sends the identical function in O †.xI A/ to x. We have     f @†.xI A/  †.f .x/I A/  f †.xI A/   for all f 2 O †.xI A/ and %.f .x/I A/ D maxfjf ./j j  2 †.xI A/g. In particular, %.xI A/ D maxfjj j  2 †.xI A/g: S

Proof. Write A D S2S AS as above. First we recall why the spectrum of an element x in a unital Banach algebra AS is non-empty and compact. P n The inverse of 1x may be computed by the geometric series 1 nD0 x if %.x/ < 1. This shows that the inversion in a Banach algebra is a continuous map near 1. Hence

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3 The spectral radius of bounded subsets and its applications

the subset of invertible elements is open and the inversion map on it is continuous. Therefore, the spectrum of an element is closed. Since x   for  ¤ 0 is invertible if and only if 1  1 x is invertible, the spectrum is contained in  the ball of radius %.x/ around 0. If the spectrum were empty, then the functions l .  x/1 would be bounded entire functions for all bounded linear functionals l W AS ! C and hence constant. This implies that  7! .  x/1 is constant, which is impossible. Hence the spectrum cannot be empty. We conclude that the subsets †.xI AS /  C are non-empty and compact for all S 2 Scm .A/. Moreover, the map  7! .x  /1 is a continuous map with values in AS for  … †.xI AS /. Since the indexing set Scm .A/ is directed, the subsets †.xI AS /  C form a directed family of non-empty compact subsets of C. It is known that the intersection of such a family of subsets is again non-empty and compact. This intersection is †.xI A/. Now let K  C be a compact neighbourhood of †.xI A/. We want to construct the functional calculus homomorphism on H.K/. Using a compactness T argument, we can find a finite subset F  S such that K is a neighbourhood of S2F †.xI AS /. Since Scm .A/ is directed, we can even find a single S 2 Scm .A/ such that †.xI AS / is contained in the interior of K. Recall that the functional calculus in the unital Banach algebras AS is constructed using the Cauchy integral formula I 1 f ./ f .x/ WD d 2i    x for f 2 H.K/; here  is any cycle (disjoint union of paths) in K that has winding number 1 around all points of †.xI AS / †.xI A/. This clearly defines a bounded linear map H.K/ ! AS . It maps the identical function to x and is an algebra homomorphism, and it is unique with these properties; this remains so if we enlarge the range to A. Letting K vary, we get the desired unique bounded homomorphism O †.xI A/ !  A.    If  … f †.xI A/ , then f   is invertible in O   †.xI A/ , so that f .x/   is invertible in A. Thus †.f .x/I A/  f †.xI A/ . Conversely, suppose that  is a boundary point of †.xI A/; we claim that f ./ 2 †.f .x/I A/. Consider a sequence .n / in C X†.xI A/ that converges towards . The sequence .x n /1 in A cannot converge because its limit would be an inverse for x  , so that  … †.xI A/. Nevertheless, we have convergence lim

n!1

f  f ./ f  f ./ D z  n z

  in O †.xI A/ , where z denotes the identical function. Thus multiplication by f .x/  f ./ makes a divergent sequence converge. Since multiplication by an invertible  element cannot do this, f .x/  f ./ cannot be invertible. Thus f @†.xI A/  †.f .x/I A/. Let r.x/ be the maximum of jj for  2 †.xI A/. We claim that r.x/ D %.xI A/. It is easy to check that r.x/ is the spectral radius of the identical function in O †.xI A/ .

3.2 Locally multiplicative bornological algebras

111

  Since we have a bounded homomorphism O †.xI A/ ! A, we get r.x/  %.xI A/. The geometric series shows that x   is invertible for jj > %.x/; hence we have r.x/  %.xI A/ as well. Finally, this implies the more general formula for %.f .x/I A/. We have just seen that this of jj for  2 †.f .x/I A/. The maximum  is the maximum  principle for f and f @†.xI A/  †.f .x/I A/  f †.xI A/ show that this is equal to the maximum of f on †.xI A/.

3.2.2 Locally multiplicative Fréchet algebras The following theorem goes back to Michael Puschnigg ([83]). He calls Fréchet algebras with these equivalent properties “nice” or “admissible” ([83], [88]). They play a crucial role in his theory because he works in the category of inductive systems of admissible Fréchet algebras throughout. Theorem 3.16. Let A be a Fréchet algebra .not necessarily locally multiplicatively convex/. Then the following are equivalent: (1) Cpt.A/ is locally multiplicative, that is, for all precompact subsets S  A, there is  > 0 such that . S /1 is precompact; (2) for each null-sequence .xn /n2N in A, there exists an N 2 N such that the tail fxn j n  N g1 is precompact; (3) A is locally multiplicatively convex and there is a submultiplicative absorbing disk U  A such that S 1 is precompact for all precompact subsets S  U . Proof. (1) ) (2). Let .xn /n2N be a null-sequence in A. Since A is a Fréchet space, the topological and bornological notions of null-sequence are equivalent by Theorem 1.36. Hence there is a null-sequence of positive scalars ."n / such that S WD f"1 n xn j n 2 Ng is precompact. By hypothesis,  S } is power-bounded for some  > 0. We can choose N 2 N such that  > "n for all n  N . Then xn 2 ."n =/   S   S } for n  N . Hence fxn j n  N g is power-bounded. (2) ) (3). First we show that A is locally multiplicatively convex. Let .Un /n2N be a decreasing sequence of submultiplicative absorbing disks that form a basis for the neighbourhoods of 0 in A. Let Vn be the disked multiplicative hull of Un ; these are unit balls of submultiplicative norms on A. We have to check that .Vn /n2N is a neighbourhood basis as well. Assume the contrary, then there is an open neighbourhood U that does not contain Vn for any n 2 N. Hence there is a function l W N ! N1 and there are an;1 ; : : : ; an;l.n/ 2 Un with an;1    an;l.k/ … U . The sequence l.n/ cannot be bounded because the multiplication in A is continuous. We may assume limn!1 l.n/ D 1. The elements a::: form a sequence converging to 0. We may find a null-sequence ."n /n2N such that xn;j WD "1 n an;j still converges to 0. By construction, .xn;1    xn;l.n/ /n2N is unbounded. Hence this null-sequence violates (2). This shows that A must be locally multiplicatively convex.

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3 The spectral radius of bounded subsets and its applications

The second part of (3) is also shown by contradiction. We can find a decreasing sequence of submultiplicative absorbing disks .Un /n2N that form a basis for the neighbourhoods of 0 in A. If (3) is false, then there are precompact subsets Sn  Un for all n 2 N such that Sn1 is not precompact. By Theorem 1.18, there is a null-sequence .xn;j /j 2N such that Sn is contained in the complete disked hull of fxn;j j j 2 Ng; we may choose xn;j 2 Un for all n; j . Rearrange the elements .xn;j / in a sequence. We claim that this sequence is a null-sequence. Fix a neighbourhood of the origin U . Since .xn;j /j is a null-sequence for each n 2 N, there is j.n/ 2 N such that xn;j 2 U for all j  j.n/. Since UN  U for some N 2 N, xn;j 2 Un  U for all n  N . Thus all but finitely many of the xn;j are in U . This means that .xn;j /n;j 2N is a null-sequence for any ordering of the indices. Let F  N N be finite and let S WD fxn;j j .n; j / 2 N 2 nF g. There is N 2 N for which .fN g  N/ \ F D ;. Thus SN is contained in the disked hull of S . Therefore, 1 is not precompact. Consequently, the S 1 cannot be precompact because even SN null-sequence .xn;j /n;j 2N 2 violates (2). Therefore, if (2) holds, then (3) must also hold. (3) ) (1) is evident. As we will see later, Condition (3) of Theorem 3.16 means that A is an isoradial subalgebra of the Banach algebra AcU . Thus a Fréchet algebra is locally multiplicative if and only if it is an isoradial subalgebra of a Banach algebra.

3.2.3 Permanence properties Definition 3.17. A subobject of an object B in a category C is usually defined to be a monomorphism A ! B (or the object A involved if the map is obvious). Thus a bornological subalgebra is simply an injective bounded algebra homomorphism. Subalgebras of locally multiplicative algebras in this sense need not remain locally multiplicative because, for instance, CŒt  is a subalgebra of O.f0g/. A bornological subalgebra A ! B is called an embedded bornological subalgebra if the map A ! B is a bornological embedding, that is, A carries the subspace bornology from B. Lemma 3.18. Completions, quotients, direct sums, inductive limits, and embedded bornological subalgebras of locally multiplicative algebras remain locally multiplicative. Proof. These assertions are mostly trivial, using Lemma 3.6. The example in Warning 3.12 shows that infinite products of locally multiplicative algebras need not be locally multiplicative. As in Notation 1.10, we write S } T if S  T } . y A2 . Lemma 3.19. If A1 and A2 are locally multiplicative, so are A1 ˝ A2 and A1 ˝ Let S 2 S.A1 ˝ A2 /. Then %.SI A1 ˝ A2 / < 1 if and only if there are S1 2 S.A1 / and S2 2 S.A2 / with %.S1 / < 1 and %.S2 / < 1 and n 2 N with S n } S1 ˝ S2 .

3.2 Locally multiplicative bornological algebras

113

Proof. It is easy to see that %.S1 ˝ S2 /  %.S1 /  %.S2 /. Hence A1 ˝ A2 is again y A2 by Lemma 3.18. locally multiplicative; so is its completion A1 ˝ It is clear that %.S / D %.S n /1=n < 1 if S n } S1 ˝ S2 as in the statement of the lemma. Conversely, S suppose that we have %.S / < 1. Then r S is power-bounded for some r > 1, so that r n S n is absorbed by .S1 ˝ S2 /} for some S1 2 S.A1 /, S2 2 S.A2 /. Since A1 and A2 are locally multiplicative, both S1 and S2 are absorbed by subsets of spectral radius less than 1; therefore, we may assume that %.S1 /; %.S2 / < 1. By construction, there is a constant C > 0 with r n S n } C  .S1 ˝ S2 / for all n 2 N. Hence S n } S1 ˝ S2 for sufficiently large n as desired (compare also Lemma 3.8). i

p

Theorem 3.20. Let K  E  Q be an extension of bornological algebras. Then E is locally multiplicative if and only if Q and K are locally multiplicative. We reduce the proof to a lemma that will be used again later; its statement uses the following notation: Notation 3.21. If S1 ; S2  A, we write hS1 iS2 WD S1 S2 [ S2 and S1 hS2 i WD S1 S2 [ S1 . Thus S 1 D hS 1 iS D ShS 1 i. This should remind you of the notation for optional arguments in computer manuals. Lemma 3.22. Let A be a bornological algebra and let S 2 S.A/. Suppose that there is T 2 S.A/ such that S 2 } S [ T and hS iT hS i is power-bounded. Then S is power-bounded. Proof. Let U WD hSiT hSi. We have U  S D hSiT S [ hS iT S 2 } hS iT  .S [ T /  U [ U 2 : Hence

U 1  S D hU 1 iU  S } hU 1 i.U [ U 2 / D U 1 :

A similar computation shows that S  U 1  U 1 . Now .U 1 [ S /2  U 1 [ S  U 1 [ U 1  S [ S 2 } U 1 [ T [ S  U 1 [ S shows that U 1 [ S is submultiplicative. Since U is power-bounded by assumption, U 1 [ S is bounded. Therefore, S is power-bounded. Proof of Theorem 3.20. Lemma 3.18 already shows that Q and K are locally multiplicative if E is. It remains to prove, conversely, that E is locally multiplicative if Q and K are. Let S  E be bounded. We must show that   %.S / < 1. Since Q is locally multiplicative, we have % p.S / < 1. Rescaling S if necessary, we can achieve that p.S/ is power-bounded. Since p is a quotient map, we may therefore lift p.S/1 to a bounded subset S 0  E. We choose S 0 with S  S 0 , so that %.S/  %.S 0 /. Thus we lose nothing if we replace S by S 0 . This achieves that p.S / is submultiplicative.

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3 The spectral radius of bounded subsets and its applications

  Therefore, if x; y 2 S , then there is s.x; y/ 2 S with p.xy/ D p s.x; y/ , so that xy  s.x; y/ 2 K. Let 1 > > 0 and ˚  ˇ  T WD .1  /1 2  xy  s.x; y/ ˇ x; y 2 S I this is a bounded subset of K. If x; y 2 S, then   xy D 2 s. 1 x; 1 y/ C 2 1 x 1 y  s. 1 x; 1 y/ 2  S C .1  / T } S [ T: Thus . S/2 } S [ T . We define another bounded subset of K by T 0 WD h S iT h Si: Since K is locally multiplicative, there is  2 .0; 1 such that 2 T 0 is power-bounded. We check that we can apply Lemma 3.22 to the pair of subsets . S; 2 T /: we have . S/2 }  S [ 2 T because . S /2 } S [ T and   1; the set h Si2 T h Si is power-bounded because h S i2 T h S i } 2 T 0 . Hence  S is power-bounded, so that S has finite spectral radius as desired.

3.2.4 Spectral radius for commutative algebras Now we show that, roughly speaking, the joint spectral radius gives nothing new for commutative algebras: Proposition 3.23. Let A be a commutative local Banach algebra with a precompact bornology .Definition 1.46/. Then %.S / D supf%.x/ j x 2 S g for all bounded subsets S  A. Proof. It is clear that %.S /  %.x/ for all x 2 S . Consider finitely many bounded subsets S1 ; : : : ; Sn  A. Since xi  xj D xj  xi for all xi 2 Si , xj 2 Sj , we get [ Y 1  1 D .r 1  Sj /1 : r  .S1 [    [ Sn / I 2f1;:::;ng j 2I

(The union over I is necessary because some products may contain no factors from certain Sj ). Therefore, %.S1 [    [ Sn / D max %.Sj /: j D1;:::;n

This yields the desired formula for %.S / if S is finite. Now let S 2 S.A/ be arbitrary. Since the bornology of A is precompact, S is precompact in the normed space AT for some T 2 Sd .A/. We can replace this T by

3.2 Locally multiplicative bornological algebras

115

a bounded submultiplicative disk because A is locally multiplicative. Let " > 0 be arbitrary. Since S is precompact in AT , we can find a finite subset F  S such that S } F [ "  T . Hence %.S /  %.F [ "  T / D maxf%.x/; %."  T / j x 2 F g because of the above result for finite unions. We have %."  T /  " because T is submultiplicative. Hence maxf%.x/ j x 2 S g  %.S /  maxf%.x/; " j x 2 Sg: We get the assertion by letting " & 0. Notice that we require A to be locally multiplicative. If we drop this assumption and merely require %.x/ < 1 for all x 2 A, then the above proof breaks down. It is unclear whether a commutative algebra with a precompact bornology is locally multiplicative once all elements have finite spectral radius. The following counterexample is inspired by [92, Example 1.13]. It shows that Proposition 3.23 may fail if A is a Banach algebra equipped with the von Neumann bornology. Hence the precompactness assumption in Proposition 3.23 is really necessary. Example 3.24. Let A WD `1 .Z/ and let B D C  .Z/ Š C.T 1 /. By Wiener’s Theorem, the commutative Banach algebra A has spectrum T 1 . Hence an element of A is invertible in A if and only if it is invertible in B. Since the bornology Cpt.A/ is precompact, Proposition 3.23 and Theorem 3.15 yield     % SI Cpt.A/ D sup % f I Cpt.A/ D sup max jf .x/j D sup kf k1 f 2S

f 2S x2T 1

f 2S

for precompact S  A. We claim that this fails for the von Neumann bornology. Let kf k1 be the norm of f in A. The difficult part of the argument is to show that for each n 2 N, there is a self-adjoint element n 2 `1 .Z/ with k n k1 D n and kexp.i n /k1 D exp.n/; this is done in [56, proof of Katznelson’s Theorem, p. 82]. Now consider S WD fexp.i n =n/g. This is a bounded subset of A because   kexp.i n =n/k1  exp k n =nk1  exp.1/ for all n 2 N. Since n is self-adjoint, exp.i n =n/ is unitary, so that its supremum norm is 1. Therefore, each element of S has spectral radius 1. However, S n contains elements exp.i n / with norm exp.n/, so that the spectral radius of S in A is at least e. Thus the bounded subset S n has joint spectral radius en , although each of its elements has spectral radius 1.

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3.3 Analytically nilpotent bornological algebras The following notion is crucial for the construction of analytic and local cyclic homology. Definition 3.25. A bornological algebra A is called analytically nilpotent, briefly a-nilpotent, if %.S / D 0 for all S 2 S.A/. Equivalently, all bounded subsets are power-bounded. Of course, a-nilpotent bornological algebras are locally multiplicative. Lemma 3.26. The class of a-nilpotent bornological algebras is closed under passage to embedded bornological subalgebras and quotients, completions and separated quotients, direct sums, inductive limits, arbitrary inverse limits, and tensor products with locally multiplicative bornological algebras. Proof. All assertions are easy to prove. We only write down the argument for inverse limits because this permanence property fails for locally multiplicative bornological algebras (see Warning 3.12). Since inverse limits are bornological subalgebras of Q products, it suffices to prove that A WD i2I Ai is a-nilpotent if all Qthe factors Ai are a-nilpotent. If S 2 S.A/, then there are Si 2 S.Ai / with S  i2I Si . Since the sets Si are all power-bounded, so is S . We have the following analogue of Theorem 3.20: i

p

Theorem 3.27. Let K  E  Q be an extension of bornological algebras. Then E is a-nilpotent if and only if Q and K are a-nilpotent.   Moreover, if K is a-nilpotent, then %.SI E/ D % p.S /I Q for all S 2 S.E/. Proof. Lemma 3.26 shows that K and Q are a-nilpotent if Eis. The converse impli cation follows from the second assertion because % p.S /I   Q D 0 for all S 2 S.E/ if Q is a-nilpotent. The inequality %.SI E/  % p.S /I Q is trivial (Lemma 3.6). For the converse inequality, it suffices to prove that %.SI E/  1 whenever p.S /  Q is power-bounded. This follows by going through the proof of Theorem 3.20 again. In this proof, we may choose  D 1 because K is a-nilpotent. Hence we get that S is power-bounded for all 2 .0; 1/. That is, %.S/  1. Analytically nilpotent algebras are rather rare. We will meet important examples of them later in connection with analytic tensor algebras. For the time being, we remark that Proposition 3.23 provides some commutative examples. Example 3.28. The kernel of the evaluation homomorphism ev0 W O.f0g/ ! C is analytically nilpotent because %.f / D jf .0/j for all f 2 O.f0g/. We also view O.f0g/ as the algebra of analytic power series, that is, power series with non-zero radius of convergence; therefore, we also denote it by C..t //. The kernel of the evaluation homomorphism corresponds to analytic power series without constant term and is denoted by C..t //0 .

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117

The following class of extensions will play a crucial role for analytic cyclic homology: i

p

Definition 3.29. An extension K  E  Q of bornological algebras is called analytically nilpotent, briefly a-nilpotent, if K is a-nilpotent.

3.4 Isoradial homomorphisms Now we introduce a class of homomorphisms that behave nicely with respect to the spectral radius. Definition 3.30. Let A1 and A2 be locally multiplicative bornological algebras and let f W A1 ! A2 be a bounded homomorphism with uniformly dense range. We call f isoradial if %.SI A1 / D %.f .S /I A2 / for all bounded subsets S  A1 . If f is injective, then we call A1 an isoradial subalgebra of A2 . The notion of an isoradial homomorphism only makes sense if A2 is locally multiplicative because we need sufficiently many subsets with %.f .S /I A2 / < 1. The domain A1 of an isoradial homomorphism is automatically submultiplicative because the isoradiality condition implies %.SI A1 / < 1 for all S 2 S.A1 /. For locally multiplicative Fréchet algebras, our definition of an isoradial subalgebra is equivalent to Michael Puschnigg’s definition of a smooth subalgebra in [88]. We only consider homomorphisms with uniformly dense range because some crucial applications require this condition. For the applications to local cyclic homology, we even need f to have approximably dense range. However, this hypothesis seems artificial for the general theory. One instance where density is needed is the following lemma: Lemma 3.31. Let A1 and A2 be unital local Banach algebras and let f W A1 ! A2 be an isoradial bounded unital algebra homomorphism. Then x 2 A1 is invertible if and only if f .x/ 2 A2 is invertible. Hence f preserves spectra of elements. Proof. It is clear that invertibility of x implies invertibility of f .x/. Conversely, suppose that f .x/ is invertible. Since the range of f is sequentially dense, there is a sequence .yn / in A1 such that f .yn / converges towards f .x/1 . This implies that f .yn /  f .x/  1 and f .x/  f .yn /  1 converge towards 0 in some Banach subalgebra of A2 . Therefore, lim %.f .yn /  f .x/  1I A2 / D 0;

n!1

lim %.f .x/  f .yn /  1I A2 / D 0:

n!1

Since f is isoradial, we get lim %.yn  x  1I A1 / D 0;

n!1

lim %.x  yn  1I A1 / D 0:

n!1

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3 The spectral radius of bounded subsets and its applications

P P1 k k This implies that the geometric series 1 kD0 .1  yn  x/ and kD0 .1  x  yn / converge in A1 for sufficiently large n. Their limits are inverses for yn  x and x  yn . Since x has both a left and a right inverse, it is invertible. Lemma 3.32. Let A1 and A2 be locally multiplicative bornological algebras and let f W A1 ! A2 be a bounded homomorphism with uniformly dense range. Suppose that there are n 2 N1 and C  1 such that %.S n I A1 /  C n for all S 2 S.A1 / with %.f .S/I A2 / < 1. Then f is isoradial. Proof. If %.f .S /I A2 / < 1, then %.f .S k /I A2 / < 1 for all k 2 N1 by Lemma 3.6. Hence we get %.SI A1 /nk D %.S nk I A1 /  C n for all k 2 N1 . Letting k ! 1, we get %.S I A1 /  1. Now use %.c S / D c%.S / to prove that %.SI A1 /  %.f .S /I A2 / for all S 2 S.A1 /.

3.4.1 Isoradiality and precompact bornologies Recall that we usually equip Banach algebras and Fréchet algebras with the precompact bornology, which we get by applying the precompact bornology functor Cpt of Definition 1.47 to the von Neumann bornology. It is easy to see that Cpt maps bornological algebras again to bornological algebras. Lemma 3.33 ([67]). Let A be a locally multiplicative bornological algebra. If S  A is bornologically precompact then %.SI A/ D %.SI Cpt.A//. Thus Cpt.A/ is locally multiplicative. Proof. Suppose that .r 1 S/1 is bounded in A. We are done if we show that .R1 S /1 is bornologically precompact for all R > r. Let T 0  A be a bounded disk such that S is precompact in AT 0 . Let T be a submultiplicative bounded disk in A that absorbs .r 1 S /1 [ T 0 . Then .R1 S/n is a precompact subset of AT for all n 2 N. Since T absorbs .r 1 S /1 and R > r, for any " > 0 there is n0 2 N such that .R1 S /n  " T for all n  n0 . Thus .R1 S/1 is a precompact subset of AT . Theorem 3.34. If A is a Fréchet algebra, then Cpt.A/ is locally multiplicative if and only if it is an isoradial subalgebra of Cpt.B/ for some Banach algebra B. Proof. Let U  A be a disk as in condition (3) of Theorem 3.16. Then AU is a normed algebra, so that AcU is a Banach algebra. We claim that the natural homomorphism  W Cpt.A/ ! Cpt.AcU / is isoradial. It is clear that this homomorphism has dense range in the usual sense; hence it has uniformly dense range (by Example 1.54). Let S  A be such that %..S /I AcU / < 1. Then .S/n  U for sufficiently large n by Lemma 3.8. By assumption, it follows that .S n /1 is precompact, so that %.S n /  1. Now apply Lemma 3.32.

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3.4.2 Permanence properties i

p

Example 3.35. Let K  E  Q be an a-nilpotent extension of locally multiplicative bornological algebras. Then Theorem 3.27 asserts that p is isoradial. Moreover, a bounded homomorphism f W Q ! A is isoradial if and only if f ı p W E ! A is isoradial. If f W A1 ! A2 is isoradial, then ker f must be a-nilpotent. Conversely, let f W A1 ! A2 be any bounded homomorphism with a-nilpotent kernel. By Example 3.35, f is isoradial if and only if the induced homomorphism A1 = ker f ! A2 is isoradial. Therefore, it usually suffices to study injective isoradial homomorphisms, that is, isoradial subalgebras. The following useful theorem is already proved in [67]. The argument below is essentially equivalent but formulated slightly more elegantly. Theorem 3.36. Let A1 , A2 and B be local Banach algebras. Suppose that B is nuclear. If f W A1 ! A2 is an isoradial homomorphism, then so is the induced homomorphism y idB W A1 ˝ y B ! A2 ˝ y B. f ˝ There is a similar theorem for ˝. Q y Proof. We omit the proof that   f WD f ˝ idB has uniformly dense range. Let S 2 Q y S.A1 ˝ B/ satisfy % f .S/ < 1. We have to prove %.S /  1. y B, we can find submultiplicative complete bounded Since S is bounded in A1 ˝ disks S1 2 Scm .A1 / and SB 2 Scm .B/ with %.S1 /; %.SB / < 1 and S ~ S1 ˝ SB . The analogue of Lemma 3.19 for complete tensor products yields T2 2 Sc .A2 / and TB 2 Sc .B/ with %.T2 /; %.TB / < 1 and f .S /n ~ T2 ˝ TB . Enlarging n, we may replace T2 and TB by other complete disks that absorb them. Hence we can achieve, in addition, that T2 absorbs f .S1 /, TB absorbs SB , and that the Banach space spanned by TB is isomorphic to `1 .N/; here we use the nuclearity of B, which implies that B is a direct union of Banach subspaces isomorphic to `1 .N/. Since %.S n / D %.S /n and S1 and SB are submultiplicative, we may replace S by S n . Thus we may assume without loss of generality that already f .S / ~ T2 ˝ ˚  TB . We conclude that x; f .x/ j x 2 S is contained in the closed unit ball of y BTB ˚ .A2 /T2 ˝ y BTB with respect to the norm .A1 /˛ S1 ˝ ˚  k.x1 ; x2 /k WD max kx1 k.˛ S1 ˝TB /~ ; kx2 k.T2 ˝TB /~ for sufficiently large ˛. Furthermore, S is mapped to the kernel of y BTB ˚ .A2 /T2 ˝ y BTB ! .A2 /T2 ˝ y BTB : y idB ; id/ W .A1 /˛ S1 ˝ .f ˝ y V Š Now we use a special property of the Banach space `1 .N/: since `1 .N/ ˝ y `1 .N; V / for any Banach space V , the functor `1 .N/ ˝ preserves kernels even in the metric sense. Thus we have an isometry of Banach spaces   y BTB y idB ; id/ Š ker.f; id/ W .A1 /˛ S1 ˚ .A2 /T2 ! .A2 /T2 ˝ ker.f ˝ y BTB : Š .A1 /˛ S1 \f 1 .T2 / ˝

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3 The spectral radius of bounded subsets and its applications

y TB with T1 WD ˛ S1 \ f 1 .T2 /. We have Hence S ~ T1 ˝ %.T1 I A1 / D %.f .T1 /I A2 /  %.T2 I A2 / < 1: Together with %.TB / < 1, this yields %.S / < 1 as desired. Remark 3.37. The proof still works if B is a direct union of Banach spaces that y BT commutes with kernels up to are metrically flat in the sense that the functor ˝ isometry. It is unclear what happens for arbitrary B. A special case of Theorem 3.36 is that an isoradial homomorphism f W A1 ! A2 induces isoradial homomorphisms Mm .f / W Mm .A1 / ! Mm .A2 /, where Mm denotes the algebra of m  m-matrices. This fact is important for K-theory (see [21]). Similarly, we conclude that f W Cc1 .M; A1 / ! Cc1 .M; A2 / is isoradial for any y A and Cc1 .M / is nuclear smooth manifold M because Cc1 .M; A/ Š Cc1 .M / ˝ by [36]. Theorem 3.38. Consider a morphism of extensions of bornological algebras K0 /

i0

˛

 K /

/ E0

p0



ˇ i

 /E

/ / Q0

p

 / / Q.

Suppose that ˛, ˇ, and are homomorphisms with uniformly dense range and that K, E and Q are locally multiplicative. Then ˇ is isoradial if and only if both ˛ and are isoradial. Proof. First we show that ˇ is isoradial if ˛ and are. By Lemma 3.32, it suffices to prove that S n is power-bounded for some n 2 N if S 2 S.E 0 / satisfies %.ˇ.S /I E/ < 1. Choose r with %.ˇ.S /I E/ < r < 1, and choose " 2 .0; 1/. In a first step, we construct S 0 2 S.E 0 / with the following properties: • r 1 p 0 .S 0 /  Q0 is submultiplicative; • S n  S 0 for some n 2 N1 ; • %.ˇ.S 0 /I E/ < ". Since %.ˇ.S /I E/ < r, Lemma 3.6 yields the same inequality for p ı ˇ.S / D 0 ı p 0 .S/. We get %.p 0 .S /I Q is isoradial. Hence r 1 p 0 .S / is  /1< 0r because 1 power-bounded. We may lift r r p .S / 2 S.Q0 / to T 2 S.E 0 / because p 0 is a bornological quotient map. The subset U WD ˇ.r 1 S/1 [ r 1 ˇ.T / of E is bounded and hence absorbed by some bounded submultiplicative disk V 2 Sm .E/. Hence ˇ.S n / [ r n1 ˇ.T /  r n U  r" V for sufficiently large n 2 N1 . Let ˇ ˚  1 } S 0 WD x 2 S n [ r n1 T ˇ p 0 .S 0 / 2 r n r n p 0 .S n / :

3.4 Isoradial homomorphisms

121

 1 } This is a disk that satisfies S n  S 0 , ˇ.S 0 /  r" V , and p 0 .S 0 / D r n r n p 0 .S n / because  1  1 p 0 .T / D r r 1 p 0 .S /

r r n p 0 .S n / : Thus p 0 .r 1 S 0 /  p 0 .r n S 0 / is submultiplicative. Since V is submultiplicative, we get %.ˇ.S 0 /I E/  r" < ". Thus S 0 has the desired properties. We are going to prove that S 0 is power-bounded (for suitable choice of "). This implies that S is power-bounded, so that ˇ is isoradial. To simplify notation, we write S instead of S 0 in the following. The following argument is similar  to theproof of Theorem 3.20. We can choose .x; y/ 2 r 1 S with p 0 .xy/ D p 0  .x; y/ for all x; y 2 r 1 S because p 0 .r 1 S / is submultiplicative. Now we decompose     xy D r  r .r 1 x; r 1 y/ C .1  r/  .1  r/1 xy  r 2  .r 1 x; r 1 y/ for all x; y 2 S . Letting ˚  ˇ  T WD .1  r/1 xy  r 2  .r 1 x; r 1 y/ ˇ x; y 2 S ; we get S 2 } S [ T and T  .1  r/1 .S 2  r S /. We have T  K 0 by construction of  . Let T 0 WD hSiT hS i. Then T  T 0  .1  r/1 hS i.S 2  r S /hS i and hence   ˛.T 0 /  .1  r/1 hˇ.S /i ˇ.S /2  r ˇ.S / hˇ.S /i: We have ˇ.S /  " V for a bounded submultiplicative disk V  K. Therefore, ˛.T 0 /  .1  r/1 "." C r/V . For sufficiently small ", the right hand side is contained in rV , so that %.˛.T 0 /I K/ < 1. This implies that T 0 is power-bounded because ˛ is isoradial. Finally, Lemma 3.22 yields that S is power-bounded as desired. Thus ˇ is isoradial. Now we assume, conversely, that ˇ is isoradial. Since K  E and K 0  E 0 are bornological embeddings, it follows immediately that ˛ is isoradial. It remains to show that is isoradial. This is where we need ˛ to have uniformly dense range. Let S  Q0 be a bounded subset with %. .S /I Q/ < 1. We must show that %.S nI Q0 / < 1for some n 2 N (Lemma 3.32). By assumption, there is r 2 .0; 1/ such 1 that r 1 .S / is bounded. Since p W E ! Q is a quotient map, there is a bounded  1 subset T1  E with p.T1 / D r 1 .S/ . Let T  E be a submultiplicative disk that absorbs T1 . Then p.T / r 1 .S n / for sufficiently large n. Since we may replace S by S n , we may assume without loss of generality that n D 1. Let  W S ! E 0 be a bounded map with p 0 ı  .x/ D x for all x 2 S . If we knew that %.ˇ.S /I E/ < 1, then we would get the desired conclusion %.S I Q0 / < 1. We must construct  rather carefully to fulfil this estimate. There is a map W S ! r T such that p ı .x/ D .x/ for all x 2 S . Then .ˇ ı   /.S / is a bounded subset of K. Since ˛ has uniformly dense range, we can find a sequence of bounded maps nK W S ! K 0 such that ˇ ı . C nK / converges uniformly on S towards . This uniform convergence happens in EU for some bounded disk U  E that absorbs T . We may assume without loss of generality that U is submultiplicative.

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3 The spectral radius of bounded subsets and its applications

Since T is submultiplicative, we have .r T /m  r m T  r 2 U for sufficiently large m. Since the multiplication in EU is continuous, it follows that .r T [ " U /m  r U for sufficiently small " > 0. The uniform convergence of ˇ ı . C nK / yields ˇ ı . C nK /.x/  .x/ 2 " U for all x 2 S for sufficiently large n. Let SO WD . C nK /.S/ for such an n. This is a bounded subset of E 0 that lifts S , and such that O n  r U . It follows that %.ˇ.SO /I E/ < 1. ˇ.SO / 2 .S/C" U  r T C" U . Hence ˇ.S/ 0 O E / < 1, and hence %.SI Q0 / < 1 as desired. Since ˇ is isoradial, this implies %.SI Example 3.39. If the map ˛ W K 0 ! K does not have dense range, then Theorem 3.38 becomes false, as the following counterexample shows. Let E D C .S1 / and let E 0 D C ! .S1 / be the algebra of real-analytic functions on S1 . We will see below in Corollary 3.43 that the obvious map ˇ W E 0 ! E is isoradial. Let X  S1 be an infinite compact subset and let K Š C0 .S1 n X / be the ideal of functions vanishing on X. Then K 0 WD K \ E 0 D f0g because a non-zero real-analytic function has only finitely many zeros in S1 . The map K 0 ! K preserves spectral radii, but does not have dense range. The quotients Q WD E=K and Q0 WD E 0 =K 0 are isomorphic to C .X/ and C ! .S1 /, respectively. The map W C ! .S1 / ! C.X / has uniformly dense range, but it cannot be isoradial because the spectrum of the identical function is S1 in C ! .S1 / and X in C.X /, contradicting Lemma 3.31.

3.5 Examples of isoradial homomorphisms Proposition 3.40. Let .Ai / be a reduced inductive system in Cborn with inductive limit A, and let  W A ! B be an injective bounded homomorphism with uniformly dense range. Suppose that the resulting maps Ai ! B are bornological embeddings for all i 2 I . Then  is isoradial. These hypotheses hold if B is the inductive limit C  -algebra of a reduced inductive system of C  -algebras .Ai / .all equipped with the precompact bornology/. Proof. The maps Ai ! B preserve spectral radii because they are bornological embeddings. Since any bounded subset of A is already bounded in Ai for some i 2 I , the map  W A ! B is isoradial. For direct limits of C  -algebras, the maps Ai ! B are injective -homomorphisms and hence bornological embeddings for the precompact bornologies. The fact that the map A ! B has uniformly dense range follows from Theorem 1.51. Example 3.41. Consider the inductive system of matrix algebras M1 ! M2 ! M3 !    ; whose direct limit is M1 . Since the fine bornology is the unique separated bornology on a finite-dimensional vector space, the hypothesis of Proposition 3.40 is satisfied

3.5 Examples of isoradial homomorphisms

123

for any locally multiplicative complete bornological algebra B that contains M1 as a uniformly dense subalgebra. For instance, we may take the C  -algebra of compact operators on `2 .N/ or the Banach algebra of nuclear operators on a Banach space with Grothendieck’s approximation property.

3.5.1 Real-analytic, smooth, and continuous functions The following result and its proof are taken from [67]. Let X be a pointed compact space with base point 1 and let A be a complete locally multiplicative bornological algebra. We equip C0 .X; A/ with the bornology of uniform continuity (see Definition 1.35). Proposition 3.42. If S  C0 .X; A/ is bounded and x 2 X , let Sx  A be the image of S under the evaluation homomorphism f 7! f .x/. Then   % S I C0 .X; A/ D sup %.Sx I A/: x2X

Let k 2 N [ f1g, let M be a C k -manifold. If S  Cck .M; A/ is bounded, then   % S I Cck .M; A/ D sup %.Sx I A/: x2M

An analogous assertion holds for the space C ! .M; A/ of real-analytic functions M ! A if M is a compact C ! -manifold and for A.Œ0; 1; A/. In all these cases, the function x 7! %.Sx I A/ is upper semicontinuous and vanishes at 1, so that the supremum is actually a maximum. Proof. Write A as a direct union of Banach algebras AT . Then C0 .X; A/ and Cck .M; A/ are direct unions of the bornological algebras C0 .X; AT / and Cck .M; AT /, respectively. By (3.5) the assertion holds for general A once it holds for Banach algebras. Hence we may assume A to be a Banach algebra with unit ball T . Let x 2 XC and let r2 > r1 > r0 > %.Sx /. Since .r01 Sx /1 is bounded, we have .r11 Sx /n  T for sufficiently large n. Since S is a uniformly continuous set of functions, we have .r21 Sy /n  T for y in some neighbourhood of x. Therefore, the function %.Sx / is upper semicontinuous. (This fails if C0 .X; A/ carries the von Neumann bornology.) Since %.S1 / D 0, the function %.Sx / attains its maximum on X. Let r > %.Sx / for all x 2 X ; we want to show that %.SI C0 .X; A//  r. We have seen above that there are nj 2 N1 and an open covering .Uj / of XC such that f .y/n 2 r n T for all f 2 S , y 2 Uj , n  nj . Since XC is compact, we can find a finite subcovering. Hence there is n 2 N1 such f .y/ 2 r n T for all y 2 XC ,  that n n n f 2 S . This easily implies % S I C0 .X;A/ < r , so that % S I C0 .X; A/ < r as desired. Conversely, Lemma 3.6 yields % S I C0 .X; A/  %.Sx I A/ for all x 2 X . Thus   % SI C0 .X; A/ D max %.Sx I A/: x2X

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3 The spectral radius of bounded subsets and its applications

Now we establish the corresponding formula for Cck .M; A/. It only remains to prove that a bounded subset S  Cck .M; A/ satisfies %.S I Cck .M; A//  1 if %.Sx / < 1 for n all x 2 M . As above, we findr 2 .0; 1/ and n 2 N1 with f .x/ 2  r T kfor all f  2 S . n k We claim that this implies % S I Cc .M; A/  1 and hence % S I Cc .M; A/  1. To begin with, the supports of products do not increase, so that all functions in S 1 are again uniformly compactly supported. It remains to estimate the supremum norm of X1 ı    ı Xm .f1    fl / for vector fields X1 ; : : : ; Xm and f1 ; : : : ; fl 2 S n . Using the Leibniz rule, we rewrite this as a sum of k  m monomials, each of the form Xw1 .f1 /Xw2 .f2 /    Xwl .fl /, where the sets w1 ; : : : ; wl form a partition of f1; : : : ; mg and Xw WD Xi1 ı    ı Xij if w D fi1 ; : : : ; ij g with i1      ij ; by convention, X; D id. Since there are only finitely many possibilities for the differential operators Xw , the factors Xw .f / remain uniformly bounded by some constant C1 > 0. The crucial observation is that since there are at most m possible letters, no more than m of the functions fj are hit by a non-trivial differential operator. Thus we may estimate the supremum norm of the occurring monomials by C1m  r lm . The sum of l  m such monomials is then estimated by m.C1 =r/m  lr l . This remains bounded for l ! 1 because 0 < r < 1. The above argument deals with the cases k  1. If k D !, then we argue as follows. Since M is real-analytic, we can embed M in a complex manifold MC ; any real-analytic function on M extends to a holomorphic function on a neighbourhood of MC . Thus C ! .M; A/ is the direct limit of the spaces H.U; A/ of bounded holomorphic functions U ! A, where U runs through the directed set of compact neighbourhoods of M in MC . The bornology on H.U; A/ is the subspace bornology from C.U; A/, so that the spectral radius of a subset in H.U; A/ is the same as in C.U; A/. Letting U & M , we get       % S I C ! .M; A/ D lim % S I H.U; A/ D lim max % Sx I A/ D max % Sx I A/: U &M

U &M x2U

x2M

We leave it to the reader to check the assertion for A.Œ0; 1; A/. Corollary 3.43. Let A be a complete, locally multiplicative bornological algebra. If M is a C k -manifold, then Cck .M; A/ ! C0 .M; A/ is an isoradial subalgebra; if M is a compact C ! -manifold, then C ! .M; A/ ! C .M; A/ is an isoradial subalgebra; A.Œ0; 1; A/ is an isoradial subalgebra of C .Œ0; 1; A/. Proof. Proposition 3.42 yields that these maps preserve spectral radii. We claim that they have approximably dense range. We can construct a sequence of bounded linear maps C0 .M; A/ ! Cc1 .M; A/, using an approximation of the identity map on C0 .M / by a sequence of smooth integral kernels. This yields a sequence of maps C0 .M; A/ ! Cc1 .M; A/, which converge uniformly on uniformly continuous subsets to the identity map. A similar argument using real-analytic integral kernels works for C ! .M; A/.

3.5 Examples of isoradial homomorphisms

125

3.5.2 Isoradial subalgebras from group actions Let G be a locally compact group. We are going to define continuous, smooth, and real-analytic representations of G on bornological vector spaces, following [68]. First, we restrict attention to a convenient class of locally compact groups. If U  G is an open subgroup, then a representation of G is continuous, smooth, or real-analytic if and only if its restriction to U is so. Since any locally compact group contains an almost connected open subgroup, we may restrict attention to almost connected locally compact groups. Such groups have been classified and shown to be projective limits of Lie groups. More precisely, call a compact normal subgroup N  G smooth if G=N is a Lie group (the Lie group structure is unique if it exists). Then the smooth compact normal subgroups form a directed set, and G Š lim G=N , where N runs through this  set. This structural information is used by François Bruhat to define the space of smooth functions on G ([5]); this is extended in [68], where the space C 1 .G; V / of smooth functions G ! V for a complete bornological vector space V is defined. The idea is that a function is smooth if and only if it is N -invariant for some smooth compact normal subgroup N  G and descends to a smooth function on the Lie group G=N ; more precisely, we only require this condition locally because we want smoothness to be a local property. By a similar recipe, we can define a space C ! .G; V / of realanalytic functions G ! V . Finally, we have the space C.G; V / of continuous functions G ! V . It is important to equip C .G; V / with the bornology of uniform continuity. Definition 3.44. Let  W G V ! V be a representation of G by bounded linear maps. If x 2 V , then we define a function  .x/ W G ! V by  .x/.g/ WD .g; x/. We call  continuous if  defines a bounded map V ! C.G; V /. We call  smooth if  defines a bounded map V ! C 1 .G; V /. We call  analytic if  defines a bounded map V ! C ! .G; V /. Definition 3.45. Let  W G  V ! V be a continuous representation of G. The subspace V 1  V of smooth elements (with respect to ) is defined as the inverse limit of the diagram V ! C .G; V / C 1 .G; V /; explicitly, V 1 WD f.x; y/ 2 V ˚ C 1 .G; V / j  .x/ D yg Š fx 2 V j  .x/ 2 C 1 .G; V /g: The subspace V !  V of analytic elements (with respect to ) is defined as the inverse limit of the diagram V ! C .G; V / C ! .G; V /. The natural bounded maps V ! ! V 1 ! V are injective because we have bounded injective maps C ! .G; V /  C 1 .G; V /  C .G; V /. It is shown in [68] that the above notions of continuous and smooth representation and of smooth subspace are the usual ones if V is a Fréchet space and G is a Lie group or if V is a fine bornological space and G is totally disconnected. In the above definitions, we may replace C ::: .G; V / by C ::: .L; V / for any compact neighbourhood of the identity element L  G: if  .x/ is continuous or smooth or analytic near 1, then it has this property on all of G because  .x/.g1 g2 / D  .g2 x/.g1 /.

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3 The spectral radius of bounded subsets and its applications

Now let A be a complete bornological algebra. We get complete bornological algebras C ! .G; A/  C 1 .G; A/  C .G; A/. The maps  in Definition 3.44 are automatically algebra homomorphisms, so that A! and A1 become complete bornological algebras. Theorem 3.46 ([67, Proposition 6.12]). Let A be a complete, locally multiplicative bornological algebra, and let G be a locally compact group. Let  W G  A ! A be a continuous group representation by algebra automorphisms. Then A1  A is an isoradial subalgebra; so is A!  A provided the map A! ! A has uniformly dense range. Proof. First we check that the map A1 ! A has approximably dense range. We can map A ! A1 by taking convolutions with compactly supported smooth functions on G. Let S 2 S.A/. Since  is a continuous representation,  .S /  C .L; A/ is uniformly continuous. Therefore, we can choose a sequence of smooth compactly supported functions .fn / such that fn  x ! x uniformly for x 2 S . In many cases, one can show similarly that A! ! A has approximably dense range. However, since real-analytic functions cannot be compactly supported, we must assume that .g/ 2 Hom.A; A/ grows sufficiently slowly, so that the convolutions that we need are defined. This is automatic if A is a Banach algebra. To avoid this technical issue here, we assume A! ! A to have uniformly dense range. For the rest of the proof, it makes no difference whether we consider A1 or A! . We will only write down the argument for A1 . Let L  G be a compact neighbourhood of 1. We claim that C 1 .L; A/  C .L; A/ is an isoradial subalgebra. Before we check this claim, we show that it implies the assertion of the theorem. Since A1 is a bornological subalgebra of A ˚ C 1 .L; A/, a subset of A1 is power-bounded if and only if its images in A and C 1 .L; A/ are power-bounded. Let S 2 S.A1 / satisfy %.SI A/  1; then the image of S in C .L; A/ also has spectral radius at most 1 because  W A ! C.L; A/ is a bounded algebra homomorphism; by the claim, the image of S in C 1 .L; A/ has spectral radius at most 1 as well. Therefore, %.SI A1 /  1 as well. Now we verify the claim. Proposition 3.42 shows that C.L; A/ is locally multiplicative, so that the claim makes sense. If G is a Lie group, then the assertion follows from Corollary 3.43. We reduce the general case to this special case. First, we may assume without loss of generality that G is almost connected, so that there exist many smooth compact normal subgroups. Replacing L by N L for some such group, we can achieve that N  L D L for a cofinal set of smooth subgroups. By definition, we have C 1 .L; A/ D lim C 1 .L=N; A/. Since L=N  U=N is a compact subset of a smooth ! manifold, C 1 .L=N; A/ has the usual meaning. Although C .L; A/ is not equal to the direct union of the spaces C .L=N; A/, Proposition 3.40 yields that lim C.L=N; A/ ! is an isoradial subalgebra of C.L; A/. Corollary 3.43 implies that the subalgebras C 1 .L=N; A/  C .L=N; A/ are isoradial. Hence C 1 .L; A/ is isoradial in C.L; A/ as asserted. Example 3.47. Let C  .T 2 / be the rotation C  -algebra with parameter  . This is the universal C  -algebra generated by two unitaries U; V that satisfy the commutation

127

3.5 Examples of isoradial homomorphisms

relation U V D exp.2i/V U .P Let S.T 2 /  C  .T 2 / be the usual smooth subalgebra, consisting of all infinite series m;n2Z amn U m V n with .amn / 2 S.Z2 /. This dense subalgebra is isoradial in C  .T 2 / because it is the subalgebra of smooth vectors for the gauge action of T 2 on C  .T 2 / defined by .; /.U n V m / D n m U n V m : The analytic subspace consists of all infinite series with coefficients in S ! .Z2 /. Example 3.48. Let IsoC  be the Toeplitz C  -algebra, that is, the universal C  -algebra generated by an isometry. Recall that there is a C  -algebra extension K.`2 N/  IsoC   C  .Z/;

(3.49)

where C  .Z/ Š C .T 1 / is the group C  -algebra of Z. Moreover, this extension has a canonical completely positive section, so that we may identify IsoC  Š K.`2 N/ ˚ C  .Z/ as a Banach space. We equip these Banach spaces with the precompact bornology. It follows from Corollary 3.43 that the Schwartz algebra S.Z/ Š C 1 .T 1 / (with the precompact bornology) is an isoradial subalgebra of C  .Z/ Š C.T 1 /. Let KS  K.`2 N/ be the subspace of smooth compact operators; as a bornological vector space, KS Š S.N 2 / with the precompact bornology. We claim that KS is isoradial in K.`2 N/. The crucial point is that the multiplication in K.`2 N/ restricts to a bounded 3-linear map KS  K.`2 N/  KS ! KS : If S  KS is bounded in KS and has power-bounded image SQ in K.`2 N/, then it follows that S 1 D S [ ShSQ 1 iS is bounded in KS . Now put IsoS WD KS ˚ S.Z/ and view this as a subspace of IsoC  . The multiplication of IsoC  restricts to a bounded bilinear map on IsoS , so that IsoS is a complete bornological algebra. By construction, it fits into an extension of bornological algebras KS ! IsoS ! S.Z/, and we have a morphism from this extension to the extension (3.49). Theorem 3.38 implies that IsoS is an isoradial subalgebra of IsoC  . There is also a real-analytic version of this result, where we replace S.Z/ Š C 1 .T 1 / by the space C ! .T 1 / D O.T 1 / of real-analytic functions on T 1 and KS by the space of matrices whose entries decay faster than some exponential. The same arguments as above show that this yields an isoradial subalgebra of IsoC  .

3.5.3 Related notions for topological algebras Next we compare our notion of isoradial homomorphism with related notions due to Joachim Cuntz and Bruce Blackadar ([3]) and Larry B. Schweitzer ([92]); these approaches define when a Fréchet subalgebra is a “smooth” subalgebra of a Banach algebra.

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3 The spectral radius of bounded subsets and its applications

Definition 3.50 ([3]).   A differential norm on an algebra A is an increasing sequence of norms .x/ D s s2N such that s .x  y/ 

s X

k .x/  sk .y/:

kD0

Such a sequence of norms determines a metrisable topology on A. We assume that A with this topology is complete; we let A0 be the Banach algebra completion of A with respect to the norm 0 . Then A  A0 is a “smooth” subalgebra according to [3]. Actually, [3] only considers the case where A0 is a C  -algebra, but the generalisation to Banach algebras is straightforward. It is important to treat C  -algebras separately because in this context one can also study the smooth or even continuous functional calculus for normal elements. But we do not consider such issues here. One shortcoming of differential norms is that they behave badly for quotients. Therefore, the more general notion of derived norm is needed in [3] to construct smooth closures of subalgebras. Another shortcoming is that the inequalities in Definition 3.50 are sensitive to the choice of the sequence .n / and may be destroyed by passing to another sequence of norms that defines the same topology. Therefore, it is quite hard to prove that a given Fréchet subalgebra of a C  -algebra does not admit a differential norm. Definition 3.51 ([92]). Let A0 be a Banach algebra with norm 0 and let A  A0 be a dense Fréchet subalgebra; let .n /n2N be any increasing sequence of norms on A defining its topology, starting with 0 . We call A strongly spectrally invariant in A0 if there is a C > 0 such that for all m 2 N there exist Dm > 0 and pm  m with X m .x1    xn /  Dm C n k1 .x1 /    kn .xn / (3.52) k1 C Ckn pm

for all n 2 N; x1 ; : : : ; xn 2 A. If A is defined by a differential norm .n /, then A is strongly spectrally invariant in A0 : we have (3.52) with C D 1, Dm D 1, and pm D m for all m 2 N. A virtue of strong spectral invariance is that it only depends on the topology of A, that is, Definition 3.51 applies to all sequences that define the topology once it applies to one such sequence. Theorem 3.53. Let A  A0 be strongly spectrally invariant. Then the canonical map Cpt.A/ ! Cpt.A0 / is isoradial. Proof. It is clear that this map has dense range; since Cpt.A0 / is precompact, it has uniformly dense range by Proposition 1.53. Let S  A be precompact and assume that %.S I A0 / < 1. We have to check that S 1 is precompact in A; actually, it suffices to check that S 1 is von Neumann bounded by Lemma 3.33. Equivalently, m .S 1 / is bounded for all m 2 N. Replacing S by S n if necessary, we can achieve that 0 .x/  1=2C for all x 2 S.

3.5 Examples of isoradial homomorphisms

129

We use (3.52) to estimate m .x1    xn / for x1 ; : : : ; xn 2 S by the sum of all terms of the form k1 .x1 /    kn .xn /, where the numbers k1 ; : : : ; kn  0 satisfy k1 C    C kn  pm . The crucial point is that at most pm of the numbers kj may be non-zero. Hence at least n  pm factors in our product are 0 .xj /  1=2C . Since S is von Neumann bounded, there is a constant ˛ with j .x/  ˛=2C for all x 2 S , j  pm . Hence we may estimate each summand k1 .x1 /    kn .xn / in (3.52) by ˛ pm  .2C /n ; since the number of n-tuples .k1 ; : : : ; kn / with k1 C  Ckn  pm grows polynomially in n, we obtain a uniform bound for m .S n / for all n 2 N. Thus all the examples in [3], [92], [93] become isoradial subalgebras when we apply the precompact bornology functor. Our notion of an isoradial homomorphism is closely related to the notion of an analytic semi-norm in [3, Definition 3.11]. In our notation, this definition amounts to the following. Let A0 be a normed algebra with norm 0 and let ˛ be another semi-norm on A, which is not continuous with respect to the given norm on A; let A denote A0 with the new semi-norm ˛ C 0 . Then 0 is analytic in the sense of [3, Definition 3.11] if and only if %.F I A/ D %.F I A0 / for all finite subsets F  A; here it makes no difference whether we take the spectral radius with respect to the precompact or von Neumann bornologies on A and A0 .     We get potentially different notions if we require % SI vN.A/ D % S I vN.A0 / for all precompact subsets or for all von Neumann bounded subsets S of A. This condition for all precompact subsets is equivalent to Cpt.A/ ! Cpt.A0 / being isoradial; the condition for all von Neumann bounded subsets is stronger: Example 3.54. We consider once again the situation of Example 3.24, that is, the map of commutative Banach algebras `1 .Z/ ! C  .Z/ Š C.T 1 /. We have already seen that this is isoradial for the precompact bornologies. Therefore, the norm that defines `1 .Z/ is analytic with respect to the usual C  -norm in the sense of [3]. But there is a von Neumann bounded subset S  `1 .Z/ with       % SI vN `1 .Z/ ¤ % SI vN C.T 1 / D max %.Sx I C/: x2T 1

If A  A0 is strongly spectrally invariant, then necessarily %.S I A/ D %.S I A0 / for all von Neumann bounded subsets S  A. Therefore, Example 3.54 exhibits a dense Banach subalgebra of a commutative Banach algebra that is not strongly spectrally invariant but nevertheless isoradial with respect to the precompact bornologies. Question 3.55. Is there a concrete criterion when a dense Fréchet subalgebra of a Banach algebra is isoradial?

3.5.4 Some examples involving group algebras Next we consider some examples involving group convolution algebras. The following bornological algebras also play a role in [72]. Let G be a finitely generated discrete

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3 The spectral radius of bounded subsets and its applications

group with word-length function L W G ! RC . For k 2 N, let S k .G/ be the Banach space of all functions f W G ! C for which the norm X jf .g/j.L.g/ C 1/k < 1 (3.56) kf k.k/ WD g2G

remains bounded. This is a Banach algebra with respect to convolution. In particular, S 0 .G/ is the usual `1 -group algebra. We let S 1 .G/ WD lim S k .G/ be the space of  all functions that satisfy (3.56) for all k 2 N; this is a locally multiplicatively convex Fréchet algebra. We equip S k .G/ with the precompact bornology; for k D 1, this is equal to the von Neumann bornology on S 1 .G/. ! be the space of all functions f W G ! C for which the norm ˛ .f / WD P Let S .G/ L.g/ jf .g/j˛ remains finite for some ˛ > 1. Each of the norms ˛ is submultig2G plicative, so that S ! .G/ is a Silva algebra. If G D Z, then the Fourier transform identifies S k .Z/ with an algebra of functions on T 1 . For k D 1 and k D !, we get the algebras C 1 .T 1 / and C ! .T 1 /, respectively. For finite k, these function algebras are hard to describe. Theorem 3.57. The subalgebras S k .G/  S 0 .G/ D `1 .G/ are isoradial for all k 2 N [ f1; !g. Proof. Using L.gh/ C 1  L.g/ C 1 C L.h/ C 1, we compute X kf1  f2 k.k/  jf1 .g/j  jf2 .h/j  .L.gh/ C 1/k g;h2G



X

jf1 .g/j  jf2 .h/j 

D

j

.L.g/ C 1/j .L.h/ C 1/kj

j D0

g;h2G k X

k X k 

k  j

kf1 k.j / kf2 k.kj / :

j D0

This means that the sequence of norms kf k.j / =j Š for j 2 N is a differential seminorm. Hence Theorem 3.53 yields the assertion for k 2 N [ f1g. (For finite k, we truncate this sequence of norms at k.) Now consider the case k D !. Let S  S ! .G/ be bounded and have spectral radius strictly less than 1 in `1 .G/. We claim that its spectral radius in S ! .G/ is also strictly less than 1. This condition implies that S ! .G/ ! `1 .G/ is isoradial. Replacing S by some power S n if necessary, we can achieve that kf k.0/  r 3 for all f 2 S , for some 0 < r < 1. Moreover, S is bounded with respect to ˛ .f / for some ˛ > 1. Thus X jf .g/j  .r 1 C "  ˛ L.g/ /  r r 1 kf k.0/ C "  ˛ .f / D g2G 1 L.g/  for sufficiently small " > 0. For sufficiently small  ˇ >! 1, we  have r C "  ˛ L.g/ ˇ for all g 2 G. Hence ˇ .S /  r, so that % S I S .G/ < 1.

3.6 Isoradial hulls of subalgebras

131

3.6 Isoradial hulls of subalgebras Let B be a local Banach algebra. Notation 3.58. We write A B and call A a dense bornological subalgebra if A ! B is an injective bounded algebra homomorphism between complete bornological algebras with uniformly dense range (compare Definition 3.17). The set of dense bornological subalgebras is partially ordered by A  A0 if A  A0 and the inclusion map A ! A0 is bounded; we also say that A0 dominates A in this case. Our goal is to complete a dense bornological subalgebra with respect to functional calculus. There are two possible approaches to define this closure. We may either look for a minimal isoradial subalgebra containing A. Or we may consider all sums of convergent power series with entries in A that converge in B. Our main result is that both approaches succeed and yield the same result. Let A B as above and let .Ai /i2I be a non-emptyTset of dense bornological subalgebras of B that dominate A for all i 2 I . We equip i2I Ai with the canonical bornology: a subset is bounded if and only if it is bounded in Ai T for all i 2 I . This turns T A into a complete bornological algebra. We still have i2I Ai B because Ti2I i i2I Ai dominates A. Definition 3.59. Let I be the set of all A0 B that are isoradial in B and dominate A. This is a non-empty set of subalgebras because it contains B itself. Let \ hull% .AI B/ WD A0 : A0 2I

This intersection is called the isoradial hull of A in B. This formalises what the minimal isoradial subalgebra containing A should be. It is clear that hull% .AI B/ is again a dense bornological subalgebra that dominates A0 ; we will show soon that it is isoradial in B as well. The disadvantage of this approach is that it does not tell us how the elements of hull% .AI B/ look like. In order to construct them, we let J be the set of all bounded subsets S  A with %.SI B/ < 1. If S 2 J , then its complete submultiplicative disked necessarily of A; explicitly, it consists hull .S 1 /~ is still a bounded subset of B but not P1 of all sums of non-commutative power series iD1 i xi;1    xi;l.i/ with xi;j 2 S and P1 j j  1. These infinite sums necessarily converge in B (even uniformly for all i iD1 possible entries) because %.SI B/ < 1. Theorem 3.60. The dense subalgebra hull% .AI B/ B is the minimal isoradial subalgebra of B that dominates A. We have [ hull% .AI B/ D R  .S 1 /~ ; S2J

and a subset of hull% .AI B/ is bounded if and only if it is absorbed by .S 1 /~ for some S 2 J.

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3 The spectral radius of bounded subsets and its applications

Proof. Let S  hull% .AI B/ be bounded with %.SI B/ < 1. Then S 1 is bounded in A0 for all A0 2 I because A0 dominates hull% .AI B/ and A0 is isoradial in B. Hence S is power-bounded in hull% .AI B/. This means that hull% .AI B/ is isoradial. By construction, it is the minimal isoradial subalgebra of B that dominates A. It is clear that .S 1 /~ for S 2 J is bounded in any isoradial subalgebra of B that dominates A and hence in hull% .AI B/. By construction, .S 1 /~ is a complete submultiplicative disk and hence the unit ball of a Banach subalgebra of B. We are going to prove that the resulting set of subalgebras is directed, that is, for any S1S ; S2 2 J , there is T 2 J for which .T 1 /~ absorbs .S11 /~ [ .S21 /~ . This means that S2J B.S 1 /~ is a bornological subalgebra of B. To construct T , we observe that there is r > 1 such that %.r Sj / < 1 for j D 1; 2, so that .r S1 /1 [ .r S2 /1 is still bounded in B. Let U  B be a bounded submultiplicative disk that absorbs .r S1 /1 [ .r S2 /1 . Then S1n [ S2n  r 1 U for sufficiently large n because r > 1. For sufficiently small " > 0, S k k 1 we also have "  n1 U . Let kD1 .S1 [ S2 /  r T WD S1n [ S2n [ " 

n1 [

.S1k [ S2k /:

kD1

This is a bounded subset of A because A is a bornological algebra, and its spectral radius in B is at most r 1 < 1. Hence T 2 J . By construction, we have Sj1 D

n [

Sjk  .Sjn /1  "1  T 1

kD1 1 1 1 ~ 1 ~ 1 ~ for j D 1; 2, that is, T absorbs S S1 [ S2 . Hence .T / absorbs .S1 / [ .S2 / . So far, we have seen that S2J B.S 1 /~ is a complete bornological algebra. Since B is locally multiplicative, any bounded subset of A is absorbed by S an element of J . S Therefore, S 2J B.S 1 /~ dominates A. It remains to prove that S2J B.S 1 /~ is isoradial in B. S Let T be a bounded subset of S2J B.S 1 /~ with %.T I B/ < 1. We have to show S that T is power-bounded in S2J B.S 1 /~ . For this, we are going to construct SQ 2 J and n 2 N such that T n ~ SQ 1 . This implies that T is power-bounded. Since T is bounded, there is S 2 J such that T is absorbed by .S 1 /~ . Since %.T I B/ < 1 and %.SI B/ < 1, there is r > 1 such that .r T /1 and .r S /1 are bounded in B. Let U  B be a submultiplicative complete disk that absorbs .r S /1 [ .r T /1 . Hence 1 [ n S k  1=8 U; T n  1=8 U S WD 1

kDn

for some n 2P N. Since T  S 1 , we can write P any x 2 T n as a non-commutative power series i xi;1    xi;m.i/ with xi;j 2 S and ji j  1. We split off monomials S k with m.i/  n and write x D x1 C x2 with x1 2~ S n and x2 2~ n1 kD1 S . Then we have ~ x2 2 T n  .S n /  1=8 U  1=8 U  1=4 U: n

~

3.6 Isoradial hulls of subalgebras

133

Let SQ WD 2

 2n1 [

Sk

~

\ 1=2 U:

kD1

Q B/ < 1. Therefore, This is a bounded subset of A with SQ  1=2 U and hence %.SI 1 n ~ Q1 Q Q S 2 J . Since S contains 2 S , we have 2 x1 ; 2 x2 2 S and thus x 2~ SQ 1 for all x 2 T n . Thus T n ~ SQ 1 as desired, so that T n is indeed power-bounded. Example 3.61. Consider the subalgebra of Laurent polynomials CŒz; z 1  in the C  -algebra C .T 1 / of continuous functions on the circle. Then   hull% CŒz; z 1 I C .T 1 / D C ! .T 1 / D O.T 1 /: The proof is rather easy. On the one hand, it follows from Corollary 3.43 that C ! .T 1 / is an isoradial subalgebra of C .T 1 /. On the other hand, any bounded subset of C ! .T 1 / is ~ contained in .frz; rz 1 g1 / for some r 2 .0; 1/ and therefore bounded in the isoradial 1 1 hull of CŒz; z  in C .T /. More generally, we may consider an affine algebraic variety V over C and let VR be its set of real points. Let O.V / be the coordinate ring of V . We assume that VR is compact and that V has enough real points in the sense that the restriction map O.V / ! C.VR / is injective. We get example 3.61 if we let V WD f.x; y/ 2 C 2 j x 2 C y 2  1 D 0g: Notice that O.V / is also generated by the functions z D x C iy and z D x  iy.   Question 3.62. When do we have hull% O.V /I C.VR / D C ! .VR /? Corollary 3.43 yields that C ! .VR / is isoradial in C .VR /, so that   hull% O.V /I C .VR / C ! .VR /: Example 3.63. Let G be a discrete group and define S ! .G/ as in §3.5.4. We claim  that S ! .G/ D hull% CŒGI `1 .G/ . We have already checked in Theorem 3.57 that S ! .G/ is isoradial in `1 .G/. Let S be a symmetric set of generators of G with 1 2 S . We view  S as a subset of CŒG, sending g 2 G to the characteristic function ıg . Clearly, % SI `1 .G// D 1. If r < 1, then the disked hull of .r S /1 is equal to the disked hull of fr L.g/ ıg j g 2 Gg. This is exactly the unit ball of the norm 1=r in the definition of S ! .G/. Hence S ! .G/ is the minimal isoradial subalgebra of `1 .G/ containing CŒG. Question 3.64. Let G be a discrete group. What is the isoradial hull of CŒG in the reduced group C  -algebra? This question is almost certainly intractable for general G. We should restrict attention to some class of groups like, say, groups with the rapid decay property of Paul Jolissaint ([54], [55]). In this case, we may guess an answer, namely, the real-analytic analogue of the Jolissaint algebra. But even for free groups, it is unclear whether this answer is correct.

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3 The spectral radius of bounded subsets and its applications

3.6.1 Isoradial Silva subalgebras Our next goal is to choose particularly small isoradial subalgebras, where small means Silva algebra. Any locally multiplicative Silva algebra is isomorphic to lim.An /n2N ! for a reduced countable inductive system of Banach algebras with compact maps An ! AnC1 for all n 2 N. Isoradial Silva subalgebras are easy to get: Proposition 3.65. Let B be a locally multiplicative Fréchet algebra and let A be a dense subalgebra of B with a countable basis, equipped with the fine bornology. Then its isoradial hull hull% .AI B/ is a locally multiplicative Silva algebra. Proof. It follows from Theorem 3.16 that any locally multiplicative Fréchet algebra is an isoradial subalgebra of a Banach algebra. Hence we may assume without loss of generality that B be a Banach algebra. Our explicit description of the bornology on hull% .AI B/ in Theorem 3.60 now shows that it is countably generated and precompact, hence Silva. If U  B is the closed unit ball and .xn /n2N is a basis for A, then the desired cofinal sequence can be taken to be .Sn1 /~ with Sn WD spanfx1 ; : : : ; xn g \ .1  1=n/ U . It is easy to see that Sn1 becomes relatively compact in A.S 1 /~ . nC1

We are particularly interested in Silva algebras with the approximation property because for such algebras both Theorems 2.13 and 2.44 apply, so that HA .A/ Š HL .A/;

HA .A/ Š HL .A/:

The approximation property for a Silva algebra means that for each n 2 N there is N  N such that the map An ! AN is a norm limit of finite rank maps. We can achieve that the maps An ! AnC1 are norm limits of finite rank maps for all n 2 N by re-indexing. Theorem 3.66. Let B be a separable, locally multiplicative Fréchet algebra with the approximation property, equipped with the precompact bornology. Then there is an isoradial Silva subalgebra A  B with the approximation property. If V  B is a dense Silva subspace, then we can achieve V  A. Proof. Since B is separable, we can always find a dense vector subspace with a countable basis; this becomes a Silva space for the fine bornology. We choose such a dense subspace if no dense Silva subspace is handed to us. We write V D lim Vn with Banach ! spaces Vn and compact injective maps Vn ! VnC1 . If it were not for the approximation property, we could get an isoradial Silva subalgebra that dominates V already from Proposition 3.65. We have to work harder now. We need an increasing sequence of Banach subalgebras .An / such that the maps An ! AnC1 are norm limits of finite rank operators and A WD lim An  B is an ! isoradial subalgebra of B containing V . Density is automatic if V  A. We construct An by induction, starting with A0 D f0g. We fix r 2 .0; 1/ and let U be the closed unit ball of B. Suppose that A0 ; : : : ; An have been constructed. Then AnC1 should have the following properties:

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3.7 Passage to inductive systems

1. the map nC1 W AnC1 ! B is a compact operator; 2. the map An ! AnC1 is a norm limit of finite rank operators; 3. VnC1  AnC1 ; 4. .rU \ An /1 is bounded in AnC1 ; 5. An is a Banach subalgebra of B. By hypothesis, B has the approximation property. Since n W An ! B is compact by the induction hypothesis, we can approximate it uniformly by finite rank maps. That is, there is a sequence of bounded finite rank maps .fj / W An ! B that converges bornologically towards n . This means that they converge uniformly in Hom.An ; BT / for some T 2 Sd .B/. Enlarging T , we can achieve VnC1  BT . Since rU \ An is a compact disk in B whose spectral radius is less than 1, the set .rU \ An /1 is precompact. Enlarging T further, we can achieve that BT absorbs this set as well. Finally, replacing T by "T for some " > 0, we can achieve that %.T / < 1. Thus T 1 is again precompact. We let AnC1 be the Banach subalgebra whose closed unit ball is .T 1 /~ . SinceSBT  AnC1 , this has the required properties. Let A WD An . This is a locally multiplicative Silva algebra with the approximation property by construction. The map A ! B has uniformly dense range because already V  B is dense and B is a Fréchet space (see Example 1.54). If S  A is bounded and contained in rU , then S  An \ rU for some n 2 N and hence S 1 is bounded in AnC1  A. Thus A is isoradial as well.

3.7 Passage to inductive systems Now we extend the notions of locally multiplicative bornological algebra, analytically nilpotent bornological algebra, and isoradial homomorphism to the setting of inductive systems. This is notationally more difficult because we lack elements and bounded subsets. The following definition provides our substitute: ! Definition 3.67. Let S be a set and let A D .Ai ; ˛ij /i2I be an algebra in Norm1=2. The set of bounded maps S ! A is defined as the inductive limit of the sets of bounded maps S ! Ai . Thus two bounded maps f1 W S ! Ai , f2 W S ! Aj are considered equal if there is k 2 I with k  i; j such that ˛ik ı f1 D ˛jk ı f2 . ! ! More formally, we have a forgetful functor Forget W Norm1=2 ! Sets, where Sets is the category of sets, and a bounded map S ! A is a morphism S ! Forget.A/ in ! Sets. We also need substitutes for familiar constructions with bounded subsets like rescaling and powers. Let f W S ! A be a bounded map. For a scalar , we get a bounded map   f W S ! A by composing f with the operator of multiplication by  on A. fn

mult

We let f n W S n ! A for n 2 N1 be the composite map S n ! An ! A˝n ! A.

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3 The spectral radius of bounded subsets and its applications

3.7.1 Spectral radius and power-bounded subsets ! Definition 3.68. Let S be a set and let A 22 Alg.Norm1=2/. A bounded map f W S ! A is called power-bounded if there is a semi-normed algebra B with closed unit ball D, a bounded homomorphism f1 W B ! A, and a map f2 W S ! D  B such that f D f1 ı f2 . The spectral radius of a bounded map f W S ! A is the infimum of the set of all r > 0 for which r 1 f W S ! A is power-bounded, or 1 if no such r exists. There is a canonical choice for the factorisation S ! D  B ! A of a powerbounded map f W S ! A. Namely, let F.S / be the free algebra generated by the set S; the set of all words formed by letters in S is a basis for F.S /. We equip F.S / with the `1 -norm with respect to this basis; hence the unit ball of F.S / is .S 1 /} . Any map S ! D  B to the unit ball of a semi-normed algebra can be extended to a bounded homomorphism F.S / ! B. Therefore, f W S ! A is power-bounded if and only if we can extend f to a bounded homomorphism F.S / ! A. If A 22 Alg.Born1=2/ is a bornological algebra, then diss.A/ is an algebra in ! Norm1=2. A bounded map f W S ! diss.A/ is nothing but a bounded map f W S ! A in the usual sense. We may replace such a map by its image f .S /, which is a bounded subset of A: the map f W S ! diss.A/ is power-bounded if and only if the subset f .S / is so, and the map f and the subset f .S/ have the same spectral radius. Equation (3.5) and Lemma 3.6 continue to hold in this situation. The only nontrivial point is the following: Lemma 3.69. If f n W S n ! A is power-bounded, then so is f W S ! A. Proof. By assumption, f n extends to a bounded homomorphism fOn W F.S n / ! A. Inspecting the bases, we see that F.S / D

n1 M kD1

CŒS k  ˚ F.S n / ˚

n1 M

CŒS k  ˝ F.S n /:

kD1

! We want to define a morphism fO W F.S / ! A in Norm1=2 extending f . Of course, we take the given map fOn on F.S n / and the linearisation of f k on CŒS k . On CŒS k  ˝ F.S n /, we first use these maps to get a map to A˝A and compose with the multiplication map A ˝ A ! A. We claim that this morphism F.S / ! A is an algebra homomorphism. This follows from the associativity of the multiplication in A by an annoying computation. To reduce notational overhead, we only check a typical special case. Assume that n D 2 and restrict attention to the direct summand CŒS  ˝ F.S 2 / ˝ CŒS  ˝ F.S 2 / of F.S/ ˝ F.S/. If we first map to A ˝ A and then multiply, then we get a map that may be described symbolically by x1 .x2 x3 /.x4 x5 /    .x2k x2kC1 /x2kC2 .x2kC3 x2kC4 /    .x2.kCl/C1 x2.kCl/C2 /:

137

3.7 Passage to inductive systems

If we first multiply in F.S /, we map the relevant summand into F.S 2 /. Symbolically, the resulting map to A is described by .x1 x2 /.x3 x4 /    .x2.kCl/C1 x2.kCl/C2 /: To move between these two formulas, we have to shift around lots of brackets. This is possible because A is associative, but it may force us to move up in the inductive system. We must check that, even for k; l ! 1, we have to move up only finitely many steps in the inductive system to transform the first product into the second one. If we move the brackets one at a time, then in each step we are transforming a product in F.S 2 / ˝ CŒS  ˝ CŒS 2  ˝ F.S 2 / ˝ CŒS  ˝ F.S 2 / to a product in F.S 2 / ˝ CŒS 2  ˝ CŒS  ˝ F.S 2 / ˝ CŒS  ˝ F.S 2 /: Regardless of k; l, we can control these products in a uniform way. Hence we get the required associativity statement.

3.7.2 Locally multiplicative algebras in inductive systems ! Theorem 3.70. The following assertions are equivalent for A 22 Alg.Norm1=2/: (1) %.S/ < 1 for all bounded maps f W S ! A; (2) A is isomorphic to an inductive system of semi-normed algebras; if A belongs ! ! to Ban or Norm, then A is even isomorphic to an inductive system of Banach algebras or normed algebras, respectively. ! Moreover, the subcategory of Alg.Norm1=2/ described by these equivalent conditions ! is equivalent to the category Alg.Norm1=2/ of inductive systems of semi-normed algebras, and similarly for Banach and normed algebras. We call A locally multiplicative if it satisfies these equivalent conditions. Proof. It is clear that (1) holds if A is an inductive limit of semi-normed algebras. Conversely, assume that %.S / < 1 for any bounded map f W S ! A. Write A D .Ai ; ˛ij /i2I . For each Ai , let Si  Ai be the closed unit ball and consider the bounded gi

hi

map Si  Ai ! A. Since %.Si I A/ < 1, we get a factorisation Si ! Bi ! A of fi , where Bi is a semi-normed algebra, hi W Bi ! A is an algebra homomorphism, and gi W Si ! Bi is bounded; the image of Si need not be contained in the closed unit ball of Bi . We extend gi to a bounded homogeneous map gi W Ai ! Bi by gi .x/ WD kxk  gi .x=kxk/ for x ¤ 0 and gi .0/ WD 0.

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3 The spectral radius of bounded subsets and its applications

For each i 2 I , there is j.i / 2 I such that Bi ! A factors through Aj.i/ . Let W Bi ! Aj for j  j.i / be the resulting map. Let Kij  Bi be the closed ideal generated by the kernel of hji . Since hi is an algebra homomorphism whose restriction to ker hji vanishes, there must exist a factorisation of hi through the semi-normed algebra Bij WD Bi =Kij . Thus we get factorisations Bij ! Ak ! A for suitable k 2 I . There is no reason to expect the composite map ˇijk W Bij ! Ak ! Bk to be an algebra homomorphism. However, when we map further to A, we get the algebra homomorphism hi . Therefore, there must be some l 2 I such that the map hlk W Bk ! Al annihilates hji

ˇijk .x/  ˇijk .x/;

ˇijk .x C y/  ˇijk .x/  ˇijk .y/;

ˇijk .xy/  ˇijk .x/ˇijk .y/

for all x; y 2 Bij . Therefore, we do get an algebra homomorphism Bij ! Bkl for sufficiently large l. Now we can define the required inductive system of semi-normed algebras. Its indexing set is the set J of all pairs .i; j / 2 I 2 with j  j.i /. We declare that .i; j /  .k; l/ if the map Bij ! Bkl defined above is an algebra homomorphism. Our argument shows that this partially ordered set is directed. By design, .Bij / is an inductive system of semi-normed algebras indexed by J . The maps Ai ! Bij ! Ak constructed above yield morphisms of inductive systems A ! .Bij / ! A, which are algebra homomorphisms and inverse to each other. Therefore, A Š lim Bij is an ! inductive limit of the required sort. ! It is clear that algebras in Norm1=2 are finitely presented in Alg.Norm1=2/. We have already observed after Definition 2.16 that this implies that the functor ! ! lim W Alg.Norm1=2/ ! Alg.Norm1=2/ ! is fully faithful. This yields the desired equivalence of categories. ! ! Finally, it is easy to adapt the above argument to the categories Norm and Ban and see that we only need normed algebras and Banach algebras in either case. Clearly, a bornological algebra A is locally multiplicative if and only if diss.A/ 22 ! Alg.Norm1=2/ is. The following permanence properties carry over easily: Lemma 3.71. Tensor products, quotients, direct sums, inductive limits, and completions of locally multiplicative algebras remain locally multiplicative. But there is no general result for embedded subalgebras: Example 3.72. Let B WD C..t// be the algebra of analytic power series and let A WD CŒt with the fine bornology. Then A is a dense bornological subalgebra of B. But S ! the map diss A  diss B is part of an extension in Ban because A D An is a direct union of finite-dimensional vector spaces; the cokernel is lim .diss B=An /. !n2N Therefore, if we carry over the notion of an embedded subalgebra in the obvious fashion, then we no longer rule out this simple counterexample.

3.7 Passage to inductive systems

139

This example explains the curiously weak statements in the following lemma: Lemma 3.73. The category of locally multiplicative algebras is closed under finite inverse limits. Closed left or right ideals in locally multiplicative algebras are again locally multiplicative. Proof. To get arbitrary finite limits, we only need finite products and equalisers, that is, inverse limits of diagrams of the special form A  B. Finite products are already treated by Lemma 3.71. We identify the category of locally multiplicative algebras with ! ! Alg.Norm1=2/. A morphism in Alg.Norm1=2/ is represented by a family of algebra homomorphisms between semi-normed algebras. The equaliser of a pair of maps .f1 ; f2 / W A  B is defined as the kernel of f1  f2 , equipped with its natural algebra structure. When we perform this construction as in the proof of Proposition 1.135, we find that the outcome is again an inductive system of semi-normed algebras, as asserted. We may view a two-sided ideal A  B as the equaliser of the pair of maps .0; p/ W B  B=A, where p is the canonical projection; hence A is again locally multiplicative in this case. We check the assertion about one-sided ideals by hand. Let A  B be a closed right ideal and suppose that B D .Bi ; ˇij / is an inductive jl system of semi-normed algebras. Then A D .Aik ; ˛ik / with closed subspaces Aik  jl Bi and ˛ik D ˇi jAi k because A is a closed subspace of B. Since A is even a closed right ideal, we get an isomorphic inductive system if we replace Aik by the closed right ideal in Bi that it generates, that is, by the closed linear span of Aik  Bi . This is automatically a closed subalgebra, so that we get an inductive system of semi-normed algebras as desired. The same argument works for left ideals. Lemma 3.22 also extends to our new setting: ! Lemma 3.74. Let A 22 Alg.Norm1=2/ and let fS W S ! A be a bounded map. Suppose that there is a bounded map fT W T ! A such that fS2 W S 2 ! A factors through the map fS ˚fT

CŒT [ S D CŒT  ˚ CŒS  ! A; and suppose that hfS ifT hfS i W hSiT hS i ! A is power-bounded. Then fS W S ! A is power-bounded. Proof. The argument is very similar to the proof of Lemma 3.22. First, we extend hfS ifT hfS i to an algebra homomorphism F.hSi  T  hSi/ ! A. Then we extend the multiplication in F.hSi  T  hSi/ to V WD F.hSi  T  hSi/ ˚ CŒS , using the factorisation of fS2 through CŒS [T . This multiplication on V need not be associative. Let I  V be the closed two-sided ideal that is generated by the obstructions .x1  x2 /  x3  x1  .x2  x3 / to associativity. Then the map V ! A annihilates I because A is associative and the map is compatible with the multiplication. Replacing V by V =I , we obtain the desired factorisation of fS through the closed unit ball of an associative semi-normed algebra.

140

3 The spectral radius of bounded subsets and its applications

Now the proof of Theorem 3.20 carries over easily, so that the category of locally ! multiplicative algebras in Norm1=2 is closed under extensions. The only change in the proof is that we have to replace subsets by bounded maps everywhere. For example, the subset T should be replaced by the bounded map   S 2 ! K; .x; y/ 7! .1  /1 xy  2 s.x; y/ : The following theorem sums up the permanence properties of local multiplicativity: ! Theorem 3.75. The class of locally multiplicative algebras in Norm1=2 is hereditary for the following constructions: • quotients; • completions; • direct sums and inductive limits; • finite inverse limits; • left and right ideals; • tensor products; • extensions. ! Lemma 3.76. Let A 2 Alg.Ban/, let S be a set, and let hn W S ! A, n 2 N, be a sequence of bounded maps that converges uniformly towards some map h1 W S ! A. Then %.h1 /  lim sup %.hn /: It is important here that A is locally multiplicative, otherwise the result fails. A bornological version of this lemma is used implicitly in the proof of Theorem 3.38. Proof. Let R > %.h1 /, we have to show that %.hn / < R for almost all n 2 N. Write A D .Ai ; ˛ij /i2I for an inductive system of Banach algebras. Both uniform convergence and the spectral radius of a bounded map are local notions; that is, there must be i 2 I and factorisations hn D ˛i ı h0n with maps h0n W S ! Ai such that limn!1 h0n D h01 uniformly and %.h01 / < R. Hence we may assume without loss of generality that A be a Banach algebra. Since R > %.h01 /, there is r 2 .0; R/ such that r 1 h01 .S / is power-bounded. Hence kRk h01 .S /k kA  1=2 for sufficiently large k. Since h0n converges uniformly towards h01 , the maps .h0n /k W S k ! A;

.x1 ; : : : ; xk / 7! h0n .x1 /    h0n .xk /

converge uniformly towards .h01 /k . Therefore, kRk h0n .S /k k < 1 for sufficiently large n. This implies %.h0n / < R as desired.

3.7 Passage to inductive systems

141

3.7.3 Analytically nilpotent algebras and isoradial homomorphisms Analytically nilpotent algebras and isoradial homomorphisms only make sense in the subcategory of locally multiplicative algebras. The difficulties with associativity that we have met above disappear in this subcategory. Therefore, only local multiplicativity creates such problems, and the other notions are much easier to carry over. ! Definition 3.77. An algebra in Norm1=2 is called analytically nilpotent or a-nilpotent if any bounded map into it is power-bounded or, equivalently, has spectral radius 0. It is obvious that a bornological algebra is analytically nilpotent if and only if its ! dissection is an analytically nilpotent algebra in Norm1=2. The permanence properties of analytically nilpotent bornological algebras formulated in Lemma 3.26 and Theorem 3.27 mostly carry over. The only exception is that we get problems for closed subalgebras as in Example 3.72. In summary, we get the following permanence properties: ! Theorem 3.78. The class of analytically nilpotent algebras in Norm1=2 is hereditary for the following constructions: • quotients; • completions; • direct sums and inductive limits; • arbitrary inverse limits; • tensor products with locally multiplicative algebras; • extensions. A closed subalgebra of an a-nilpotent algebra is again a-nilpotent provided it is locally multiplicative. Proof. The proof is a straightforward translation of the corresponding arguments in the setting of bornological algebras. Therefore, we leave it to the reader. Finally, we carry over the definition of an isoradial homomorphism. As a preparation, we need maps with uniformly dense range. ! Definition 3.79. A morphism f W V ! W in Norm1=2 has uniformly dense range if for any bounded map h W S ! W there is a sequence of bounded maps n W S ! V such that f ı n W S ! W converges uniformly to h; this means that there is a morphism W 0 ! W with W 0 22 Norm1=2 such that the maps f ı n and h factor through W 0 ! W and such that we have the desired uniform convergence for the resulting maps S ! W 0 .

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3 The spectral radius of bounded subsets and its applications

We say that f has approximably dense range if for any map h W AS ! W with AS 22 Norm1=2 there is a sequence of maps n W AS ! V such that f ı n W AS ! W converges uniformly to h; this means that there is a morphism W 0 ! W with W 0 22 Norm1=2 such that the maps f ı n and h factor through W 0 ! W and such that the maps S ! W 0 corresponding to f ı n converge in operator norm to the map corresponding to h. Roughly speaking, this is the linearised version of uniformly dense range. Definition 3.80. Let f W A ! B be an algebra homomorphism between two locally ! multiplicative algebras in Norm1=2 with uniformly dense range. We call f W A ! B isoradial if %.SI A/ D %.f .S /I B/ for all bounded maps S ! A. This definition agrees with our previous definition for bornological algebras, that is, if f W A ! B is an algebra homomorphism between bornological algebras, then f is isoradial if and only if diss.f / is isoradial. We still have the same permanence properties as in the case of bornological algebras: ! Theorem 3.81. Let A1 , A2 and B be locally multiplicative algebras in Ban. Suppose that B is nuclear or, more generally, that B is isomorphic to an inductive system of metrically flat Banach spaces. If f W A1 ! A2 is an isoradial homomorphism, then y B ! A2 ˝ y B. y idB W A1 ˝ so is the induced homomorphism f ˝ ! Theorem 3.82. Consider a morphism of extensions of algebras in Norm1=2 K0 /

i0

˛

 K /

/ E0

p0



ˇ i

 /E

/ / Q0

p

 / / Q.

Suppose that ˛, ˇ, and have uniformly dense range and that K, E and Q are locally multiplicative. Then ˇ is isoradial if and only if both ˛ and are isoradial. In each case, the proof is a tedious but routine translation of the corresponding argument for bornological algebras.

Chapter 4

Periodic cyclic homology via pro-nilpotent extensions

A good conceptual understanding of analytic cyclic homology should also have an analogue for periodic cyclic homology. In fact, the periodic case should be considerably simpler. It turns out that the Cuntz–Quillen approach to periodic cyclic homology carries over to the analytic theory. This approach has three ingredients. The first ingredient is the category of algebras for which we define the theory. For the analytic theory, we will use the category of complete bornological algebras. Here we work with pro-algebras, that is, algebras in categories of projective systems, in order to exhibit similarities between the periodic and analytic cyclic theories. More precisely, we fix a Q-linear symmetric monoidal category C and study alge bras in the category C of projective systems over C. If C is the category of Banach  spaces, then Alg. C / contains the category of complete locally convex topological algebras with jointly continuous multiplication as a full subcategory; we are careful to   treat Alg. C / and not just Alg.C/ below because otherwise we would only get the locally multiplicatively convex topological algebras. The second ingredient is the X-complex. This is a very small quotient of the cyclic bicomplex. But if the underlying algebra A is quasi-free, then the canonical projection HP.A/  X.A/ is a chain homotopy equivalence. Hoping that most readers are already familiar with this construction, we relegate its discussion to §A.6.3. The construction of the X-complex is the same for all cyclic theories we consider. The third and most important ingredient is a completed tensor algebra. For the analytic theory, we use the analytic tensor algebra T A, for the periodic theory, we use  even  the pro-tensor algebra TA. This is simply the algebra .  .A/; ˇ/, where 1 Y  even  .A/ WD 2n .A/;

! ˇ  WD !    d.!/  d./:

nD0

This definition is simple enough to work with it directly, as in [25], [26]. But we introduce some more machinery which allows us to formalise several important arguments; this will facilitate the transition to the analytic theory.  The completed tensor algebras TA and T A can be characterised by a universal prop  erty. For the pro-tensor algebra TA, this says that algebra homomorphisms TA ! B are equivalent to maps A ! B with pro-nilpotent curvature. Recall that the curvature of f W A ! B is the map !f WD f ı mA˝2  mB ı f W A ˝ A ! B that measures the failure of multiplicativity of f . The map f has pro-nilpotent curvature if and only if !f factors through a pro-nilpotent algebra, that is, a projective limit of nilpotent pro-algebras.

144

4 Periodic cyclic homology via pro-nilpotent extensions

The class of pro-nilpotent algebras, the class of pronilcurs, and the pro-tensor algebra functor determine each other uniquely; that is, if we know one of them, we can reconstruct the other two. The analytic theory is based instead on the class of analytically nilpotent algebras introduced in Chapter 3 and the corresponding linear maps with analytically nilpotent curvature and analytic tensor algebras. We deal with the relationship between pro-nilpotent algebras, pronilcurs, and pro tensor algebras and then define the periodic cyclic chain complex HP.A/ as X. TA/ in §4.1. Since our main focus is on the analytic theory, we omit some arguments that are identical to the corresponding ones in Chapter 5. The next main step is the homotopy invariance of HP, which we prove in §4.2. This amounts to proving the homotopy invariance of the X-complex for quasi-free algebras; we can use this result again in the analytic case because the analytic tensor algebra is quasi-free as well. Once we have homotopy invariance, we get several important results almost for free. This includes Goodwillie’s Theorem, which asserts that HP is invariant under  semi-split pro-nilpotent extensions. This tells us that we have X.E/ X. TA/ if N  E  A is a semi-split algebra extension with pro-nilpotent N and quasifree E. This is a crucial ingredient in the proof of the Excision Theorem. It also allows us to compute some simple examples. We get matrix stability and additivity for finite sums or infinite products. Since all these results are almost identical to the corresponding ones for the analytic theory, we omit most proofs. The deepest result in this chapter is the Excision Theorem for periodic cyclic homology, which we prove in §4.3. It is a crucial tool for the computation of periodic cyclic (co)homology. Let i

p

KEQ be a semi-split algebra extension. Excision amounts to the statement that the canonical chain map   HP.i / W HP.K/ ! ker HP.p/ W HP.E/ ! HP.Q/ is a chain homotopy equivalence. The first excision result for cyclic homology theories is due to Mariusz Wodzicki ([109], [110]; see also [37] for slightly simpler proofs). He showed that Hochschild homology satisfies excision for semi-split extensions with H-unital kernel; this implies excision for cyclic and periodic cyclic (co)homology for such extensions. We do not treat these excision results here. Counterexamples show that some additional hypothesis is necessary for excision in Hochschild and cyclic homology. Once we stabilise and pass to the periodic cyclic theory, we can remove this hypothesis, as shown in a series of papers by Joachim Cuntz and Daniel Quillen ([18], [22], [23], [27]) in increasing generality. The extension to Q-linear symmetric monoidal categories is due to Guillermo Cortiñas and Christian Valqui ([13]). Michael Puschnigg has carried over these arguments to analytic and local cyclic homology in [86]. All these proofs of excision still rely on an idea of Wodzicki in one important step. This is where they require a good amount of careful book-keeping and computation.

4.1 Pro-algebras

145

Our proof is more conceptual. It dates back to my thesis ([65]) and appeared   previously in [66]. The crucial ingredient is the left ideal L in TE generated by K.     We show that L is quasi-free and that the diagram L ! TE ! TQ satisfies excision in Hochschild homology, although it is not an extension. Both assertions are proved at  the same time by studying a certain projective bimodule resolution of L C of length 1,  which is based on properties of TE. They immediately imply excision when combined with Goodwillie’s Theorem.  Our projective bimodule resolution for L is quite explicit, so that our proof is, in principle, constructive. But this construction is impossible to carry out in practice.  Already the connection on 1 . L /, whose existence we get from our proof, seems too complicated to work with. The beauty of our method is that we get the Excision Theorem without any serious computations. Finally, in §4.4, we compute HP for tensor products, following [26], [84]. We prove that there is a canonical chain homotopy equivalence HP.A ˝ B/ HP.A/ ˝ HP.B/ for all pro-algebras A and B. For unital algebras, this result is due to Cuntz and Quillen [26, §14]. We use the Excision Theorem 4.42 to remove the unitality hypothesis. There are other reasonable classes of nilpotent algebras that can be used to define variants of periodic cyclic homology. But since I am not yet convinced that the resulting cyclic theories are worthwhile, I only discuss the two cases of pro-nilpotent and analytically nilpotent algebras here. An amusing trivial case occurs if we let all algebras be nilpotent; then any linear map has nilpotent curvature, so that the completed tensor algebra is just the usual tensor algebra. Goodwillie’s Theorem asserts that the resulting cyclic theory is zero.

4.1 Pro-algebras Let C be an additive symmetric monoidal category with tensor product ˝ and unit object 1. We assume that morphisms in C have kernels and cokernels and that ˝ preserves cokernels. Several important results need C to be Q-linear, but the more basic theory in this section does not.  We embed C in the category C of projective systems in C. Projective systems are  ! dual to inductive systems (see §1.4), that is, . C /op Š Cop . A projective system X determines a covariant functor Hom.X; / W C ! Ab, and morphisms of projective systems are, by definition, natural transformations between these functors. There is also a more concrete description of morphisms similar to (1.132) for inductive systems.  The category C is again additive symmetric monoidal by .Ai /i2I ˝ .Bj /j 2J WD .Ai ˝ Bj /i;j 2I J

 for all .Ai /i2I ; .Bj /j 2J 22 C :

 Definition 4.1. A pro-algebra in C is an algebra in C .

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4 Periodic cyclic homology via pro-nilpotent extensions

Example 4.2. Let Ban be the additive category of Banach spaces with tensor prod y  . The category Ban contains the category of complete locally convex topological uct ˝ ! vector spaces as a full subcategory. This is dual to the embedding Cborn ! Ban. Thus the category of pro-algebras in Ban contains the category of complete locally convex topological algebras with jointly continuous multiplication as a full subcategory. Our definition of periodic cyclic homology for such topological algebras agrees with the usual one. There is no need to restrict to locally multiplicatively convex algebras because everything goes through for general pro-algebras.

4.1.1 Pro-nilpotent pro-algebras Let A be a pro-algebra. We denote the n-fold multiplication map by mAn W A˝n ! A. Recall that A is nilpotent if mAn D 0 for some n 2 N. We use the following more general notion:  Definition 4.3. A pro-algebra A is pro-nilpotent if for any morphism f W A ! X in C n mA

f

with X 22 C there is n 2 N such that the composite map A˝n ! A  ! X vanishes.   More generally, a morphism ˛ W V ! A in C for some V 22 C is pro-nilpotent  if for any morphism f W A ! X in C with X 22 C there is n 2 N such that the n mA

˛ ˝n

f

composite map V ˝n ! A˝n ! A  ! X vanishes. In [65], pro-nilpotent algebras are called locally nilpotent. Theorem 4.4. A pro-algebra is pro-nilpotent if and only if it is isomorphic to a projective limit A D lim An of nilpotent pro-algebras An .  Let f W A ! B be a pro-algebra homomorphism. If f is monic and B is pronilpotent, then A is pro-nilpotent; if f is epic and A is pro-nilpotent, then B is pronilpotent. Thus the class of pro-nilpotent algebras is closed under subalgebras and quotients in a very strong sense. In addition, it is closed under extensions, arbitrary inverse limits, and tensor products with arbitrary pro-algebras. Proof. Let B be pro-nilpotent and let f W A ! B be monic. Disregarding the algebra structures, we write f as a projective limit of maps fi W Ai ! Bi in C (compare Remark 1.133). We can achieve that the maps fi are all monic because f is monic (compare Lemma 1.146). The following argument takes place in the diagram A˝n f ˝n



B ˝n

n mA

/A _ f

n mB

 /B

/ Ai _ fi

 / Bi .

/X

4.1 Pro-algebras

147

Any map A ! X with X 22 C factors through Ai for some i 2 I ; hence it suffices to find, for each i 2 I , a number n 2 N such that the map A˝n ! A ! Ai vanishes. Since B is pro-nilpotent and Bi 22 C, for each i 2 I , there is n 2 N such that the map B ˝n ! B ! Bi vanishes. Since the above diagram commutes, the map A˝n ! A ! Ai ! Bi vanishes as well. So does A˝n ! A ! Ai because fi is monic. Thus A is pro-nilpotent. Now suppose that A is pro-nilpotent and let f W A  B be an epimorphism. This time, we work in the diagram ˝n

A f ˝n



B ˝n

n mA

/A f

n mB

 /B

/ X.

The maps f ˝n are epimorphisms for all n 2 N because we require ˝ to preserve cokernels. Take any map B ! X with X 22 C. The composite map A˝n ! A ! B ! X vanishes for some n 2 N because A is pro-nilpotent. Since f ˝n is epic and the above diagram commutes, the map B ˝n ! B ! X vanishes as well. Thus B is pro-nilpotent. Next we deal with inverse limits. Recall that inverse limits of arbitrary diagrams are subobjects of products. Since we already know that pro-nilpotence is hereditary for subalgebras, it suffices to prove that products of pro-nilpotent algebras are again pro-nilpotent. This is easy for finite products, and follows in general because any map  from an infinite product in C to an object of C factors through some finite sub-product. In particular, we get that projective limits of nilpotent algebras are pro-nilpotent. Conversely, let A be pro-nilpotent; we claim that A is a projective limit of nilpotent pro-algebras. Since ˝ preserves cokernels, the quotients An WD coker.mAn W A˝n ! A/  in C inherit algebra structures; they are nilpotent (mAn n D 0). The canonical map A ! An descends to maps AN ! An for N  n, turning .An /n2N into a projective system of nilpotent pro-algebras. The maps A ! An induce an algebra homomorphism A ! lim An . The definition of being pro-nilpotent means that this map induces  isomorphisms Hom.A; X / Š Hom.lim An ; X / for all X 22 C. This is equivalent   to A ! lim An being an isomorphism in C . Thus A is pro-nilpotent if and only if  A ! lim An is an isomorphism, if and only if A is a projective limit of nilpotent  pro-algebras. It is clear that A ˝ B is nilpotent if A is. Therefore, if A is a projective limit of nilpotent pro-algebras, so is A ˝ B. This shows that being pro-nilpotent is hereditary for tensor products with arbitrary pro-algebras. Let K  E  Q be an extension of pro-algebras and suppose that K and Q are pro-nilpotent. We claim that E is pro-nilpotent. The analogue of Lemma 1.146

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4 Periodic cyclic homology via pro-nilpotent extensions

for projective systems shows that K  E  Q is a projective limit of a projective system of extensions Ki  Ei  Qi in C; its structure maps are morphisms of extensions j j / Ej / / Qj Kj / 

˛ji

Ki /

ˇji

 / Ei

i

ji

i

 / / Qi

for all i; j 2 I with j  i . Fix i 2 I . We must show that the map E ˝n ! E ! Ei vanishes for some n. We can describe the n-fold multiplication mnE D mn in E by maps mn;i W Ej˝n ! n .i/

Ei . We let mjn;i WD mn;i ı ˇjj.i/ W Ej˝n ! Ei for j  jn .i /. Since K is pro-nilpotent, the commuting diagram / E ˝n

K ˝n n mK

/ E ˝n jn .i/

mn E

 K

mn;i

 / Ei

 /E

shows that the composite map K ˝n ! Ei vanishes for some n. This means that the map ˝n

j

mjn;i

Kj˝n ! Ej˝n ! Ei vanishes for some n; j . We fix such n; j . Since Q is pro-nilpotent, the map Q˝N ! Q ! Qj vanishes for some N 2 N. Equivalently, the map E ˝N ! E ! Ej factors through Kj . Hence the N  n-fold multiplication map E ˝N n ! E ! Ei factors through the n-fold multiplication map Kj˝n ! Ej , which is already known to vanish. Thus E is pro-nilpotent.

4.1.2 Maps with pro-nilpotent curvature and the pro-tensor algebra  Definition 4.5. Let A and B be pro-algebras and let f W A ! B be a morphism in C . Its curvature is the morphism !f W A ˝ A ! B;

!f D f ı mA  mB ı f ˝2 :

We say that f has pro-nilpotent curvature if !f is pro-nilpotent as in Definition 4.3. Maps with pro-nilpotent curvature are briefly called pronilcurs, and we denote by pronilcur.A; B/ the set of pronilcurs A ! B. We will see eventually that pronilcurs form a category. To get started, we observe that pronilcurs are closed under composition with algebra homomorphisms. That is, pronilcur is a bifunctor on the category of pro-algebras.

4.1 Pro-algebras

149

 Definition 4.6. The pro-tensor algebra TA of a pro-algebra A is a co-representing object for pronilcur.A; /, that is, we require natural isomorphisms  Alg. TA; B/ Š pronilcur.A; B/ for all pro-algebras B. These isomorphisms are of the form f 7! f ıA for a canonical  pronilcur A W A ! TA.  We are going to construct TA explicitly, using the description of tensor algebras by Joachim Cuntz and Daniel Quillen in [25, §1]; we recall the latter construction in §A.4. Notice that our construction works regardless of whether the tensor algebra TA exists;   the projective limits that we need for TA automatically exist in C . Let n .A/ Š AC ˝ A˝n for n 2 N be the space of non-commutative n-forms and 1 Y  n .A/; .A/ WD nD0

1 Y  even  .A/ WD 2n .A/; nD0

1 Y  odd  .A/ WD 2nC1 .A/: nD0

 The Fedosov product on .A/ is defined by ! ˇ  WD !    .1/k l d!  d;

! 2 k A;  2 l A:

 even This yields an associative algebra, and  .A/ is a subalgebra for ˇ. The finite products n Y Tn A WD 2j .A/ j D0

 even are quotients of  .A/, and the projection map is an algebra homomorphism if we equip Tn A with the truncated Fedosov product, where we drop all terms in >2n .A/. Then .Tn A; ˇ/ is a projective system of algebras. Its projective limit T1 A WD lim Tn A  even  is canonically isomorphic to .  .A/; ˇ/. We will soon see that T1 A has the  universal property of TA; for the time being, we distinguish the two pro-algebras in our notation. Notation 4.7. We have a canonical algebra homomorphism A W T1 A  T0 A Š A   and a canonical linear map A W A ! T1 A. Let JA WD ker A  TA. Since A is a   linear section for A , we get a semi-split algebra extension JA  TA  A called the pro-tensor algebra extension of A.  Lemma 4.8. The algebra JA is pro-nilpotent and A W A ! T1 A is a pronilcur.

150

4 Periodic cyclic homology via pro-nilpotent extensions

Q  Proof. We have JA D lim Jn A with Jn A D jnD1 2j .A/ with the truncated Fedosov  product. The (truncated) Fedosov product maps k .A/ ˝ l .A/ ! kCl .A/ for all k; l 2 N. Hence the n C 1-fold truncated Fedosov product on Jn A vanishes.  Thus Jn A is nilpotent and JA D lim Jn A is pro-nilpotent by Theorem 4.4. Since   A W TA ! A is an algebra homomorphism, we have A ı !A D !A ıA D 0, that is,  the curvature of A W A ! T1 A factors through JA; since the latter is pro-nilpotent, A is a pronilcur.  Theorem 4.9. There is a canonical isomorphism T1 A Š TA that intertwines the  canonical maps A W A ! T1 A; TA. The algebra homomorphism hf i W T1 A ! B induced by a pronilcur f W A ! B is given by hf i.a0 da1 da2 : : : da2n / WD fC .a0 /  !f .a1 ; a2 /    !f .a2n1 ; a2n / nC1 ı .fC ˝ !f˝n / on 2n .A/ Š in terms of elements, that is, hf i restricts to mB ˝2n AC ˝ A .

Proof. We have seen in Lemma 4.8 that A W A ! T1 A is a pronilcur. Therefore, fQ ı A is a pronilcur for any algebra homomorphism fQ W T1 A ! B. Conversely, let f W A ! B be a pronilcur. For any map B ! X with X 22 C, there is n 2 N ˝N

N C1 mB ı.fC ˝!f

/

such that the maps 2N .A/ ! B ! X vanish for all N  n (use associativity). This yields a natural transformation Hom.B; X / ! Hom.T1 A; X /, that is, a morphism of projective systems hf i W T1 A ! B. As in Theorem A.63, we see that this is the unique algebra homomorphism fQ W T1 A ! B with fQ ı A D f .  This theorem allows us to identify T1 A Š TA. We do this from now on. Corollary 4.10. A map f W A ! B is a pronilcur if and only if its curvature factors as g ! A˝A  ! B for an algebra homomorphism B and a pro-nilpotent algebra N . !N  More generally, a linear map f W V ! B is pro-nilpotent if and only if it factors through an algebra homomorphism N ! B with pro-nilpotent N . Proof. The second assertion certainly implies the first one. If N ! B is an algebra homomorphism with pro-nilpotent N , then any composite map V ! N ! B is pronilpotent. Conversely, suppose that f is pro-nilpotent. View V as an algebra with zero multiplication. Then f is a pronilcur and hence factors as f D hf i ı V for an algebra   homomorphism hf i W TV ! B. By Lemma 4.8, JV is pro-nilpotent; so is V and  hence TV by Theorem 4.4. Hence f D hf i ı A is a factorisation with the required properties. Two algebra homomorphisms A  B for pro-algebras A and B are called close if their difference is pro-nilpotent. The same arguments as in §5.1.3 show that such

4.1 Pro-algebras

151

  pairs correspond to algebra homomorphisms .A/; ˇ ! B. Since close homomorphisms behave exactly like their counterparts in §5.1.3, we do not discuss them further here. Theorem 4.11. Pronilcurs form a category, that is, they are closed under composition. In particular, the composite map A   TA   A2 W A ! TA ! T. TA/

is a pronilcur for any pro-algebra A. Proof. The proof of the analogous Theorem 5.23 for lanilcurs carries over literally. The forgetful map Alg.A; B/ ! pronilcur.A; B/ is a faithful functor from the  category of algebra homomorphisms to the category of pronilcurs, and T is its left  adjoint. As such, Tmust be functorial for pronilcurs. This functoriality can be described as follows: if f W A ! B is a pronilcur then there is a unique algebra homomorphism    Tf W TA ! TB    with Tf ı A D B ı f , namely, Tf WD hB ı f i. It satisfies B ı Tf D hf i.  The formula for Tf becomes rather complicated in general; if f is multiplicative, we simply have  Tf .x0 dx1 : : : dx2n / D fC .x0 / df .x1 / : : : df .x2n /: Theorem 4.12. If f1 W A1 ! B1 and f2 W A2 ! B2 are pronilcurs, so is f1 ˝ f2 W A1 ˝ A2 ! B1 ˝ B2 : Proof. It suffices to prove that f1 ˝ idB and idA ˝ f2 are pronilcurs because f1 ˝ f2 is a composite of two such maps and we can use Theorem 4.11. Of course, the arguments for f1 ˝ idB and idA ˝ f2 are identical. If f W A1 ! A2 is a pronilcur, we can factor its curvature !f through an algebra homomorphism N ! A2 with pro-nilpotent N by Corollary 4.10. The curvature of f ˝ idB is simply !f ˝ mB W A1 ˝ A1 ˝ B ˝ B ! A2 ˝ B and factors through the algebra homomorphism N ˝ B ! A ˝ B. The pro-algebra N ˝ B is pro-nilpotent by Theorem 4.4. Hence f ˝ idB is a pronilcur by the other direction of Corollary 4.10. This theorem is the analogue for pronilcurs of Theorem 5.28. The two theories differ here because analytically nilpotent algebras are not closed under tensor products with arbitrary algebras; therefore, tensor products of lanilcurs only remain lanilcurs if the algebras involved are locally multiplicative. Definition 4.13. Two pronilcurs (or two algebra homomorphisms) f0 ; f1 W A ! B are called .polynomially/ homotopic if there is a pronilcur (or an algebra homomorphism) F W A ! BŒx with ev t ı f D f t for t D 0; 1.

152

4 Periodic cyclic homology via pro-nilpotent extensions

L n This definition presupposes that the countably infinite direct sum 1Œx WD 1 nD0 1 x exists, so that we can put BŒx WD B ˝ 1Œx. If C is, say, Ban or Cborn, then we have smooth or continuous homotopies and homotopies with bounded variation instead. The following proposition also holds for these notions of homotopy.  Proposition 4.14. The functor T from the pronilcur category to the category of algebra homomorphisms is a homotopy functor, that is, if f0 and f1 are homotopic pronilcurs,   then Tf0 and Tf1 are homotopic algebra homomorphisms.  Proof. Theorem 4.12 yields a canonical pronilcur B ˝ 1Œx ! . TB/ ˝ 1Œx, which   induces a canonical pro-algebra homomorphism T.B ˝ 1Œx/ ! . TB/ ˝ 1Œx/. Thus a pronilcur F W A ! BŒx induces an algebra homomorphism      TF TA ! T.B ˝ 1Œx/ ! . TB/ ˝ 1Œx D . TB/Œx:

  This is a homotopy between Tf0 and Tf1 if F is a homotopy between f0 and f1 .

4.1.3 Quasi-free pro-algebras Definition 4.15. An algebra extension N  E  A is called a square-zero extension of A if the multiplication map N ˝ N ! N vanishes. Theorem 4.16. The following are equivalent for a pro-algebra A:   (1) the pro-tensor algebra extension JA  TA  A splits; (2) any semi-split pro-nilpotent extension N  E  A of pro-algebras splits; (3) if f W A ! B is a homomorphism and N  E  B is a semi-split pronilpotent extension of pro-algebras, then f lifts to a homomorphism fO W A ! E; (4) the square-zero extension J1 A  T1 A  A splits; (5) any semi-split square-zero extension N  E  A splits; (6) if f W A ! B is a homomorphism and N  E  B is a semi-split square-zero extension, then f lifts to a homomorphism fO W A ! E; (7) 1 .A/ is a projective A-bimodule. The analogous result for analytically nilpotent extensions (Proposition 5.24) is much weaker because the analogues of (1)–(3) and (4)–(7) are no longer equivalent; this leads to the stronger notion of analytic quasi-freeness (Definition 5.25).

4.1 Pro-algebras

153

Proof. Recall that J1 A D 2 .A/; the Fedosov product on J1 vanishes. Hence the implications (3) ) (2) ) (1) and (6) ) (5) ) (4) are obvious. To prove (1) ) (3),  let  W A ! TA be an algebra homomorphism with A ı  D idA . If s W B ! E is a linear section, then !s factors through the kernel N , which is pro-nilpotent; hence  s ı f is a pronilcur and induces hs ı f i W TA ! E. The map hs ı f i ı  W A ! E is the required lifting homomorphism for f . This shows that (1)–(3) are equivalent. The implication (4) ) (6) is proved similarly, so that (4)–(6) are equivalent.  It is also clear that (3) ) (6). Conversely, we prove (6) ) (1). Let J n A be the    ideal in TA spanned by k .A/ with k  n, so that TA= J n A D Tn1 A. We have semi-split square-zero extensions 2n 2nC1 J A= J A  T2nC1 1 A  T2n 1 A for all n 2 N. Hence (6) allows us to lift algebra homomorphisms A ! T2n 1 A to algebra homomorphisms A ! T2nC1 1 A. Starting with the identity map A ! A D T0 A, we recursively construct a compatible family of algebra homomorphisms  A ! T2n 1 A, which combine to an algebra homomorphism A ! TA. Thus (1)–(6) are equivalent. Finally, the equivalence of (3) and (7) is a well-known fact of homological algebra, for which we refer to [26, §3]. These arguments are, in principle, constructive and therefore work in any additive symmetric monoidal category. Definition 4.17. A pro-algebra is called quasi-free if it has these equivalent properties. The following results are analogous to Theorems 5.26 and 5.32 and Corollary 5.33, and they are proved in exactly the same fashion.  Theorem 4.18. The pro-tensor algebra TA is quasi-free. Theorem 4.19. Let N1  E1  A1 and N2  E2  A2 be semi-split extensions of pro-algebras and let f W A1 ! A2 be an algebra homomorphism. Suppose that E1 is quasi-free and N2 is pro-nilpotent. Then we can lift f to a morphism of extensions N1 /

/ E1

/ / A1

 N2 /

 / E2

 / / A2 ;

f

the lifting is unique up to a polynomial homotopy that is constant on A1 . Corollary 4.20. Any semi-split algebra extension N  E  A with pro-nilpotent N  and quasi-free E is homotopy equivalent to the pro-tensor algebra extension JA    TA  A. In particular, E and TA are homotopy equivalent.

154

4 Periodic cyclic homology via pro-nilpotent extensions

4.1.4 Differential 1-forms Like the analogous results for T A and TA in §5.2.1 and §A.4.5, the results in this  section follow from the universal property of TA. Hence we omit the proofs here.   Notation 4.21. To avoid confusion between the differentials on . A; ˇ/ TA and   . TA/, we denote the latter by D.   Theorem 4.22. The bimodule 1 . TA/ is isomorphic to the free TA-bimodule on A via   Š    ! TA; ! ˝ x ˝  7! ! ˇ DA .x/ ˇ : TC A ˝ A ˝ TC A    Corollary 4.23. As a left or right TA-module, TA is isomorphic to the free module on A via Š   ! TA; ! ˝ x 7! ! ˇ A .x/; TC A ˝ A  Š   A ˝ TC A  ! TA; x ˝ ! 7! A .x/ ˇ !:  odd Š 1  !  . TA/=Œ ;  induced by the map There is a natural isomorphism  A   odd    A Š TC A ˝ A ! 1 . TA/; ! dx 7! ! ˇ DA .x/:

4.1.5 Periodic cyclic homology  From now on, we assume that the underlying category C is Q-linear. Hence so is C . Otherwise, homotopy invariance and Theorem 4.24 fail, which invalidates all our results. Periodic cyclic homology and cohomology are often defined using the chain  complex . .A/; B C b/; we briefly recall this approach in Definition A.119. The X-complex is defined in Definition A.122. Theorem 4.24. For any pro-algebra A, there is a natural chain homotopy equivalence   between X. TA/ and . .A/; B C b/. Proof. The proof is similar to (and slightly easier than) the proof of the corresponding Theorem 5.38 for the analytic theory. Hence the following definition is legitimate:  Definition 4.25. Let A be a pro-algebra. We let HP.A/ WD X. TA/. Periodic cyclic homology HP .A/, periodic cyclic cohomology HP .A/, and bivariant periodic cyclic homology HP .A; B/ for two pro-algebras A; B are defined by      HP .A/ WD H HP.A/ WD H Hom 1; HP.A/ ;     HP .A/ WD H HP.A/ WD H Hom.HP.A/; 1/ ;    HP .A; B/ WD H Hom HP.A/; HP.B/ :

4.2 Homotopy invariance of periodic cyclic homology

155

An algebra homomorphism is an HP-equivalence if it becomes invertible in HP0 .A; B/. Such homomorphisms induce invertible maps on periodic cyclic homology and cohomology and on bivariant periodic cyclic homology in both variables. Remark 4.26. Since projective systems and inductive systems are dual to each other, the theory of local chain homotopy equivalences in §2.3 applies to chain complexes  in C as well and yields a local homotopy category of chain complexes. It is not a bad idea to replace homology, cohomology, and bivariant homology in Definition 4.25 by local versions; for projective systems indexed by N, this theory is developed in [13]. It has slightly better properties, but the effect is not as striking as for local cyclic homology. We will not treat this variant explicitly here, but we remark that statements about the chain complex valued functor HP hold regardless of the localisations that we incorporate into the homology of chain complexes.

4.2 Homotopy invariance of periodic cyclic homology ! We prove that periodic cyclic homology for pro-algebras over Cborn or Ban is invariant under homotopies of bounded variation. Our argument works for the X-complex of any quasi-free algebra, so that it carries over literally to analytic and local cyclic homology. A similar argument yields invariance under polynomial homotopy for algebras in Q-linear symmetric monoidal categories; Q-linearity is needed to integrate polynomials. Theorem 4.27. Let F W A ! A.Œ0; 1; B/ be a homotopy of bounded variation between two algebra homomorphisms f0 ; f1 W A ! B. If A is quasi-free, then the induced chain maps X.f0 /; X.f1 / W X.A/ ! X.B/ are chain homotopic. Proof. In the first part of the proof, we do not yet use the quasi-freeness of A. We abbreviate D WD A.Œ0; 1; B/ and define Z n W  .D/ !  n

n1

1

.B/; n .'0 d'1 : : : d'n / WD 0

'0 .t /'10 .t / d'2 .t / : : : d'n .t / dt;

using the Stieltjes integral of 1.3.7. More precisely, we consider '0 d'2 : : : d'n as y a continuous function Œ0; 1 ! BC ˝ B ˝n and '1 as a bounded variation function  Œ0; 1 ! B to apply Lemma 1.100; this construction carries over to algebras in Cborn   ! or Ban and similar categories as well.

156

4 Periodic cyclic homology via pro-nilpotent extensions

We also let 0 D 0 on 0 .D/. Lemma 1.101 shows that  ı b.'0 d'1 : : : d'n / is Z 0

1

 dt '0 '1 '20 d'3 : : : d'n  '0 .'1 '2 /0 d'3 : : : d'n

C .1/n1 '0 '10  d.'2 d'3 : : : d'n1  'n / C .1/n 'n '0 '10  d'2 : : : d'n1 Z 1 n D .1/ dt Œ'n ; '0 '10 d'2 : : : d'n1  D b.'0 d'1 : : : d'n /;



0

that is, Œ; b D b C b D 0: (4.28)   Hence  maps b 3 .D/ into b 2 .B/ and therefore descends to a map X .2/ .D/ ! X.B/, which we again denote by ; here we use the chain complex X .2/ .D/ defined in Definition A.122. Let 2 W X .2/ .D/ ! X.D/ be the canonical projection. We claim that   (4.29) Œ; @ D X.ev1 /  X.ev0 / ı 2 W X .2/ .D/ ! X.B/: 



We use (4.28) to check this on j .D/ for j D 0; 1: Z 1 Œ; @.'/ D . d'/ D ' 0 dt D ev1 .'/  ev0 .'/; 0

Œ; @.'0 d'1 / D .d ı  C  ı B/.'0 d'1 / Z 1   D d.'0 '10 / C '00 d'1  '10 d'0 dt 0 Z 1 Z 1   D Œ d'0 ; '10  dt '0 d'10 C '00 d'1 dt C 0 0 Z 1  2  @

'0 d'1 dt mod b  .B/ 0 @t D .ev1  ev0 /.'0 d'1 /: Finally, we have Œ@;  D  ı b C b ı  D 0 on 2 .D/=Œ ; . This finishes the proof of (4.29). Now we pull back  to a map  ı F W X .2/ .A/ ! X.B/, which satisfies   Œ; @ D X.f1 /  X.f0 / ı 2 W X .2/ .A/ ! X.B/; where 2 W X .2/ .A/ ! X.A/ is the canonical projection. Since A is quasi-free, Theorem A.123 yields that 2 is a chain homotopy equivalence. There is a chain homotopy inverse ˛ W X.A/ ! X .2/ .A/ with 2 ı ˛ D idX.A/ . Then  ı ˛ W X.A/ ! X.B/ provides a chain homotopy between X.f0 / and X.f1 / because   Œ@;  ı ˛ D Œ@;  ı ˛ D X.f1 /  X.f0 / ı 2 ı ˛ D X.f1 /  X.f0 /:

4.2 Homotopy invariance of periodic cyclic homology

157

Let r W 1 .A/ ! 2 .A/ be a connection. Then (A.125) provides an explicit formula for the chain map ˛ W X.A/ ! X .2/ .A/ and hence for the chain homotopy  ı ˛ induced by a homotopy of algebra homomorphisms; since  vanishes on 0 .A/,  ı ˛ is  ı r ı d on 0 .A/ and  ı .id  b ı r/ on 1 .A/=Œ ; . An equivalent formula appeared already in [26, §7].  Theorem 4.30. Let A and B be pro-algebras in Cborn and let F W A ! A.Œ0; 1; B/ be a homotopy of bounded variation between two algebra homomorphisms f0 ; f1 W A ! B. Then the induced chain maps HP.f0 / and HP.f1 / are chain homotopic, that is, HP.f0 / D HP.f1 / in HP0 .A; B/. Proof. Let F W A ! A.Œ0; 1; B/ be a homotopy between f0 and f1 . By Propo  sition 4.14, it induces an algebra homomorphism TA ! A.Œ0; 1; TB/ that pro   vides a homotopy between Tf0 and Tf1 . Since TA is quasi-free (Theorem 4.18), the Homotopy Invariance Theorem 4.27 for the X-complex yields a chain homotopy   X. Tf0 / X. Tf1 /. The same argument yields invariance under polynomial homotopies for pro-algebras over an arbitrary Q-linear symmetric monoidal category. We need Q-linearity here in order to integrate polynomials.

4.2.1 Invariance under pro-nilpotent extensions p

Theorem 4.31 (Goodwillie’s Theorem). Let N  E  A be a pro-nilpotent semisplit extension of pro-algebras. Then p is an HP-equivalence and HP.N / 0. Proof. It is easy to see that N is contractible and p is a homotopy equivalence in the pronilcur category. This yields the assertions because HP is a homotopy functor on the pronilcur category. We give a more detailed proof for the corresponding statement (Theorem 5.46) about analytic and local cyclic homology. Theorem 4.31 is a crucial step in the proof of the Excision Theorem. Theorem 4.32. Let N  E  A be a semi-split extension of pro-algebras with   pro-nilpotent N and quasi-free E. Then the canonical maps E TE ! TA induce   chain homotopy equivalences X.E/ X. TE/ X. TA/.   Proof. Corollary 4.20 implies that the maps E TE ! TA are homotopy equivalences. Since all algebras involved are quasi-free, the Homotopy Invariance Theorem 4.27 for the X-complex yields the assertion.  Corollary 4.33. If A is quasi-free, then X.A/ X. TA/.

158

4 Periodic cyclic homology via pro-nilpotent extensions

  This is equivalent to Theorem A.123 with X. TA/ instead of . .A/; B C b/. The advantage of Corollary 4.33 is that literally the same proof works for analytic and local cyclic homology (Corollary 5.48). Quasi-freeness is a rare property. For instance, the algebra of polynomial functions on an affine variety is quasi-free if and only if the variety is smooth and at most 1-dimensional. Nevertheless, there are some interesting examples of quasi-free algebras. In particular, the unit object 1 is quasi-free. Hence we get HP.1/ X.1/ 1. Thus HP .1; A/ Š HP .A/ and HP .A; 1/ Š HP .A/ for all pro-algebras A. §5.3.1 contains some more examples of (analytically) quasi-free algebras. The computations in Theorem 5.63 also apply to periodic cyclic homology because HP.A/ X.A/ HA.A/ if A is analytically quasi-free.

4.2.2 Some consequences of homotopy invariance We briefly mention that homotopy invariance implies stability and finite additivity of HP, and we discuss the behaviour for countably infinite products of pro-algebras. We omit the proofs because the arguments in §5.3.2 and §5.3.3 carry over almost literally. Stability. Let A be a pro-algebra in C. The algebra of finite matrices Mn .A/ is defined in the obvious way, and comes with a canonical embedding A ! Mn .A/ in one corner. More generally, we can treat certain biprojective algebras (see §A.7.1).  Let V; W 22 C and let b W W ˝ V ! 1 be a linear map. Then b defines a multiplication on ` WD V ˝ W that is analogous to the composition of finite-rank operators (see §A.7.1). Assume that there are maps W 1 ! V and  W 1 ! W with f ı . ˝ / D id1 ; then ` is a biprojective algebra. The map WD  ˝ W 1 ! ` is called the corner embedding associated to ; . It is an algebra homomorphism. Theorem 4.34. The algebra homomorphism A WD idA ˝ W A ! A ˝ ` is an HPequivalence for all pro-algebras A. The proof is identical to the proof of Theorem 5.65. Another proof uses the exterior product constructed in §4.4. First, the biprojectivity of ` allows us to check by hand that the corner embedding W 1 ! ` is an HPequivalence (see Proposition A.131). Secondly, A is the exterior product of idA and . Remark 4.35. We can generalise Theorem 4.34 considerably using techniques of bivariant K-theory (see [17]). When we work in the category of topological algebras, this yields stability with respect to the Schatten ideals `p .H / on a Hilbert space H for all p 2 Œ1; 1/; only `1 .H / is covered by Theorem 4.34. Using Theorem 4.34, we get finite additivity of HP: Proposition 4.36. The functor HP is additive in the following two senses: first, if A and B are pro-algebras, then the coordinate embeddings and projections induce a chain

4.3 Excision in periodic cyclic homology

159

homotopy equivalence HP.A ˚ B/ HP.A/ ˚ HP.B/; secondly, if f; g W A ! B are algebra homomorphisms for which f C g is an algebra homomorphism as well, then HP.f C g/ and HP.f / C HP.g/ are chain homotopic, that is, HP.f C g/ D HP.f / C HP.g/ in HP0 .A; B/. Lemma 5.67 shows that both notions of additivity are equivalent. The proof of additivity on objects is identical to the proof of Proposition 5.68. Whereas analytic and local cyclic homology are compatible with direct sums, periodic cyclic homology is compatible with products of pro-algebras: Theorem 4.37. Let .Ai /i2I be a set of pro-algebras. If I is countable, then the canonical chain map Y Y  HP Ai ! HP.Ai / i2I

i2I

is a chain homotopy equivalence; for uncountable I , it is a local homotopy equivalence. The main point of the proof is that we have HP.lim.Ai /i2I / Š lim HP.Ai / for any    projective system .Ai /i2I of pro-algebras; this follows because ˝ on C commutes with projective limits. When we combine this with finite additivity (Proposition 4.36), we get Theorem 4.37 as in the proofs of Theorem 5.70 and Proposition 5.72. Warning 4.38. Periodic cyclic homology is not compatible with infinite direct sums, even if the tensor product in C commutes with direct sums. The problem is that the construction of HP involves projective limits; but these do not commute with direct sums without additional assumptions, compare Proposition 1.155.

4.3 Excision in periodic cyclic homology i

p

Let K  E  Q be a semi-split pro-algebra extension with linear section s W Q ! E. The Excision Theorem for bivariant periodic cyclic homology yields long exact sequences HP0 .D; K/ O

i

/ HP0 .D; E/

p

@

HP1 .D; Q/ o HP0 .Q; D/ O

@

p p

HP1 .D; E/ o / HP0 .E; D/

i i

@

HP1 .K; D/ o

/ HP0 .D; Q/  HP1 .D; K/, / HP0 .K; D/ @

i

HP1 .E; D/ o

p

(4.39)

 HP1 .Q; D/,

(4.40)

160

4 Periodic cyclic homology via pro-nilpotent extensions

and similarly without D. We prefer statements about chain complexes because the passage to homology forgets information in non-Abelian categories. For conceptual reasons, we formulate excision using a relative version of periodic cyclic homology.  Definition 4.41. Let A1 ; A2 22 Alg. C / and let f W A1 ! A2 be homo an algebra  morphism. Let HPrel .f / be the desuspended mapping cone cone HP.f / Œ1 of the induced chain map HP.f / W HP.A1 / ! HP.A2 / (see Definition A.11). Thus we get an exact triangle HP.f /

HPrel .f / ! HP.A1 / ! HP.A2 / ! HPrel .f /Œ1

 in the homotopy category of Z=2-graded chain complexes over C . The (co)homology of HPrel .f / is the .relative/ periodic cyclic .co/homology of f and denoted by HPrel  .f / and HPrel .f /, respectively. We may also define relative bivariant groups:     HPrel HPrel  .D; f / WD H HP.D/; HPrel .f / ;  .f; D/ WD H HPrel .f /; HP.D/ : This relative theory has built-in exact sequences like

HP1 .D; A2 / o

f

/ HP0 .D; A1 /

HPrel f/ 0 .D; O

HP1 .D; A1 / o

f

/ HP0 .D; A2 /  HPrel 1 .D; f /

and similarly in the first variable; this follows from the Puppe exact sequence (Theorem A.18). Since HP.p/ ı HP.i / D 0, we get a canonical chain map .HP.i /; 0/ W HP.K/ ! HPrel .p/: p  i Theorem 4.42. Let K  E  Q be a semi-split algebra extension in Alg.C/. The canonical chain map HP.K/ ! HPrel .p/ is a chain homotopy equivalence. This implies HP .K/ Š HPrel  .p/ and a natural long exact sequence

HP0 .K/ O

i

/ HP0 .E/

p

@

HP1 .Q/ o

p

HP1 .E/ o

/ HP0 .Q/ 

i

@

HP1 .K/,

and similar statements for periodic cyclic cohomology and for bivariant periodic cyclic homology in both variables, see (4.39) and (4.40). Remark 4.43. The excision theorem of [13] still applies to certain algebra extensions that are not semi-split, namely, to locally semi-split extensions; this means that any

4.3 Excision in periodic cyclic homology

161

map A ! Q with A 22 C lifts to a map A ! E. Then the map HP.K/ ! HPrel .p/ is a local chain homotopy equivalence; we have met similar notions for inductive systems in §2.3. We can deduce the version of Theorem 4.42 for locally semi-split pro-algebra extensions from the statement above by a formal argument as in §5.4.4. This generalisation has important applications because extensions of nuclear topological vector spaces are always locally semi-split, but need not be semi-split. A simple example is the Taylor series projection C 1 .Œ0; 1/  CŒŒx, where CŒŒx is the algebra of formal power series in one variable. Similarly, the projection from the algebra of classical pseudodifferential operators of order 0 to the corresponding symbol algebra is not semi-split. Remark 4.44. We can describe the boundary maps in the long exact sequences that we get from Theorem 4.42 using a single natural class in HP1 .Q; K/ associated to the extension. We discuss this in greater detail for the analytic and local theory in §5.4.2. Recall that HP .A/ D lim.HCC2n .A/; S / for  D 0; 1, where S is Connes’ ! periodicity operator. A similar description is available for the relative theories. The Excision Theorem yields an isomorphism HP .K/ Š HPrel .p/. The canonical maps HCrel .p/ ! HC .K/ need not be isomorphisms unless K is H-unital. But Theorem 4.42 still implies the following weaker result: for each n 2 N, there is some ƒ.n/ 2 N and a commuting diagram ƒ.n/

S / HCnC2ƒ.n/ .K/ HCn .K/ RRR O O RRR RRR RRR R( S ƒ.n/ n / HCrel .p/ HCnC2ƒ.n/ .p/. rel

(4.45)

Using the boundary maps HCnrel .p/ ! HCnC1 .Q/ in the Puppe exact sequence, we may also lift the boundary map HPn .K/ ! HPnC1 .Q/ to maps HCn .K/ ! HCnC1C2ƒ.n/ .Q/. Given an element x 2 HP .A/, the first n 2 N such that x belongs to the range of n HC .A/ often plays the role of a dimension of x. The function ƒ tells us how much information about the dimension we lose in the Excision Theorem. We state some results about this function in §4.3.5.

4.3.1 Preparations i

p

We fix an algebra extension K  E  Q with a linear section s W Q ! E throughout this section. Sometimes we decompose E Š K ˚ Q as an object of C using s. The diagram HP.K/ ! HP.E/ ! HP.Q/ is (almost) never an extension of chain complexes because HP.E/ contains mixed tensor powers like K ˝ Q. Lemma 4.46. The map HP.p/ W HP.E/ ! HP.Q/ is a split epimorphism and the map HP.i/ W HP.K/ ! HP.E/ is a split monomorphism.

162

4 Periodic cyclic homology via pro-nilpotent extensions

Proof. The section s W Q ! E induces sections s ˝n W Q˝n ! E ˝n for p ˝n . Using the standard trivialisations n .A/ Š AC ˝ A˝n and n .A/ Š A˝n ˝ AC as left or right A-modules, we get two sections sL ; sR W n .Q/ ! n .E/I these are defined in terms of elements by sL .q0 dq1 : : : dqn / WD s.q0 / ds.q1 / : : : ds.qn /; sR . dq1 : : : dqn  qnC1 / WD ds.q1 / : : : ds.qn /  s.qnC1 /:

(4.47)

Combining these maps for all n 2 N, we get sections sL ; sR W HP.Q/ D

1 Y

n .Q/ !

nD0

1 Y

n .E/ D HP.E/

nD0

for HP.p/. Similarly, a section E ! K for i yields a section for HP.i /. The maps sL and sR defined in (4.47) will play a role in the following arguments. The canonical map C1 ! cone.p/Œ1 is a chain homotopy equivalence for any p semi-split extension of chain complexes C1  C2 ! C3 (Theorem A.19). Hence Lemma 4.46 yields a canonical chain homotopy equivalence   ker HP.p/ W HP.E/ ! HP.Q/ HPrel .p/: We view HP.K/ as a subcomplex of HP.E/ using Lemma4.46. Since the composition HP.p/ ı HP.i / vanishes, we have HP.K/  ker HP.p/ . The Excision Theorem asserts that this embedding is a chain homotopy equivalence. This is equivalent to the contractibility of the quotient complex   C WD ker HP.p/ =HP.K/ because we have a semi-split extension of chain complexes 



HP.K/  ker HP.p/  C

(4.48)

by Lemma 4.46 (use a long exact sequence argument). This observation does not simplify the proof of the Excision Theorem, but it yields a strengthening of it. If C   is contractible, then (4.48) splits by a chain map, that is, ker HP.p/ Š HP.K/ ˚ C is a direct sum of HP.K/ and a contractible chain complex. Therefore, HP.K/ is a  is a chain deformation retract of ker HP.p/ : there   map retraction   f W ker HP.p/ ! HP.K/ and a chain homotopy h W ker HP.p/ ! ker HP.p/ between f and the identity map.

4.3 Excision in periodic cyclic homology

163

4.3.2 A left ideal in the tensor algebra  even  Notation 4.49. For all algebras A, we identify TA Š .  A; ˇ/ where ˇ denotes  the Fedosov product; let A W A ! TA be the canonical embedding of A Š 0 A into  even  .  A; ˇ/; we view A as a subspace of TA and often drop A from our notation. The following computations in terms of elements work in any additive symmetric monoidal category, as explained in §A.2.1.  Notation 4.50. We reserve the letters k, e, q, !, and  for elements of K, E, Q, TE,  and TQ, respectively.    Corollary 4.23 yields that TE is the free left TE-module generated by E  TE; that is, the multiplication map restricts to an isomorphism Š   TC E ˝ E  ! TE;

! ˝ e 7! ! ˇ e:

(4.51)

Hence the decomposition E Š K ˚ Q in C yields a decomposition    TE Š .TC E ˝ K/ ˚ .TC E ˝ Q/

(4.52)

 of TE into left modules or, equivalently, left ideals. These left ideals inherit an algebra  structure from TE.   Notation 4.53. Let L be the left ideal in TE generated by K.   Equation (4.52) identifies L with the free TE-module over K via Š   TC E ˝ K  ! L;

! ˝ k 7! ! ˇ k:

(4.54)

 The next lemma gives a simple criterion when a differential form belongs to L ; there is no similar criterion for the complementary left ideal generated by s.Q/.  Lemma 4.55. A form e0 de1 : : : de2n belongs to L if and only if e2n 2 K, that is, 1 Y  L D  .K/  2n1 .E/ dK: nD1

Proof. If ! 2 2n .E/, k 2 K, then ! ˇ k D !  k  d.!/ dk. Since K is an ideal, the last entries of all summands of !  k and d! dk belong to K. Conversely, the inverse   map TE ! TC E ˝ E of the isomorphism in (4.51) maps e0 de1 : : : de2n 7! e0 de1 : : : de2n2 ˝ e2n1  e2n  .e0 de1 : : : de2n2 ˇ e2n1 / ˝ e2n :  This maps 2n1 dK to TE ˝ K because K is a left ideal in E.

164

4 Periodic cyclic homology via pro-nilpotent extensions

  We have already observed that L is a free TE-module. Conversely, it is quite   important that TC E is a free left L -module. This is a consequence of the following lemma: Lemma 4.56. The map    mult ı .id ˝ sL / W L C ˝ TC Q ! TC E;

l ˝  7! l ˇ sL ./

    is an isomorphism. Thus TC E is the free left L -module generated by sL .TC Q/  TC E.  The proof uses the two-sided ideal generated by K  TE,     I WD ker. Tp W TE  TQ/:      Since sL is a section for Tp, we have TC E Š sL TC Q ˚ I .         Proof. We have L C ˝ TC Q Š TC Q ˚ . L ˝ TC Q/ and TC E Š sL .TC Q/ ˚ I . Therefore, the assertion of the lemma is equivalent to an isomorphism    L ˝ TC Q ! I ;

l ˝  7! l ˇ sL ./:

(4.57)

 The decomposition E Š K ˚ s.Q/ induces a direct product decomposition of TE into the subspaces V0 dV1 dV2 : : : dV2n and dV1 dV2 : : : dV2n with Vi 2 fK; s.Q/g.  Such a subspace is contained in I if and only if K occurs at least once among the Vj .   We let F . I /  I be the product of those subspaces where the last occurrence of K  is V2i for an even number 2i. In terms of elements F . I / is generated by e0 de1 : : : de2i1 dk2i ds.q2iC1 / : : : ds.q2n / D e0 de1 : : : de2i1 dk2i ˇ sL . dq2iC1 : : : dq2n /; k0 ds.q1 / : : : ds.q2n / D k0 ˇ sL . dq1 : : : dq2n /:

(4.58)

We also consider the subspace 1  Y     F . L ˝ TC Q/ WD L ˝ 1 ˚ .dQ/2n : nD1

   Using Lemma 4.55 and (4.58), we see that the multiplication map L ˝ TC Q ! I restricts to an isomorphism Š    ! F . I /: F . L ˝ TC Q/ 

  The subspace F . L ˝ TC Q/ has an obvious complement, namely, 1    Y Q.dQ/2n : F ? . L ˝ TC Q/ WD L ˝ nD0

4.3 Excision in periodic cyclic homology

165

It remains to check that the composite map        F ? . L ˝ TC Q/  L ˝ TC Q ! I  I =F . I / is an isomorphism as well.    We use the product decomposition L Š TC E ˝K in (4.54). Let ! 2 TC E, k 2 K, and qi 2 Q, then we have   ! ˇ k ˇ sL .q0 dq1 : : : dq2n / D ! ˇ k ˇ s.q0 / ˇ ds.q1 / : : : ds.q2n /   D ! ˇ k  s.q0 /  dk ds.q0 / ˇ ds.q1 / : : : ds.q2n / 

! dk ds.q0 / ds.q1 / : : : ds.q2n / mod F . I /  because k  s.q0 / 2 K. That is, if we disregard F . I /, then the multiplication map Q1  becomes concatenation of tensors on TC E ˝K ˝ nD0 s.Q/˝2nC1 . Now it is manifest     that the map F ? . L ˝ TC Q/ ! I =F . I / is an isomorphism as well.   Next we show that L is free as a left or right L -module. By the way, projectivity as a one-sided module is necessary for quasi-freeness.    Lemma 4.56 and (4.54) show that L is free as a left L -module over TC Q ˝ K:     L C ˝ TC Q ˝ K Š TC E ˝ K Š L ;

l ˝  ˝ k 7! l ˇ sL ./ ˇ k:

(4.59)

Lemma 4.60. The map    mult ı .sR ˝ id ˝ id/ W TC Q ˝ K ˝ L C ! L ;

 ˝ k ˝ l 7! sR ./ ˇ k ˇ l;

  is an isomorphism. Thus L is isomorphic to the free right L -module on   G WD TC Q ˝ K: Proof. We have seen that the map    TC E ˝ K ˝ TC Q ! I ;

! ˝ k ˝  7! ! ˇ k ˇ sL ./;

is an isomorphism. The map x0 dx1 : : : dx2n 7! dx2n : : : dx1 x0 yields an algebra iso  morphism . TE/op Š T.E op /. Our previous reasoning applied to the algebra extension K op  E op  Qop yields an isomorphism Š    ! I; TC Q ˝ K ˝ TC E 

 ˝ k ˝ ! 7! sR ./ ˇ k ˇ !:

(4.61)

   Recall that TC E D L C ˚ TC E ˇ s.Q/ by (4.52). The isomorphism (4.61) maps         TC Q ˝ K ˝ L C into L  I and TC Q ˝ K ˝ TC E ˇ s.Q/ into I ˇ s.Q/, which   intersects trivially with L . Therefore, (4.61) restricts to an isomorphism TC Q ˝ K ˝   LC Š L.

166

4 Periodic cyclic homology via pro-nilpotent extensions

  In the following, we will view G as a subspace of L . Computations as in the proof   of Lemma 4.55 show that G  L agrees with  G D ds.Q/even  K  ds.Q/odd dK; that is, it is generated by elements of the form ds.q1 / ds.q1 / : : : ds.q2n /  k;

ds.q1 / ds.q1 / : : : ds.q2n1 / dk:

Lemma 4.60 asserts that the multiplication map    G ˝ LC ! L;

g ˝ l 7! g ˇ l;

(4.62)

is an isomorphism. Warning 4.63. It is important in the above isomorphisms to use sL and sR in the right places. These two maps are only equal if s is an algebra homomorphism. As a  consequence, (4.59) and (4.62) use isomorphic but different subspaces sL .TC Q/ ˇ K     and sR .TC Q/ˇK of L to generate L freely as a left and right L -module, respectively.  This does not happen for algebras of the form TA, which are generated by the same subspace .A/ as a left and right module. The corresponding left ideal L  TE turns   out to be isomorphic to TG, but this seems to fail for L  TE.

4.3.3 A projective bimodule resolution    We use the embedding L ! TE and the known quasi-freeness of A WD TE to  construct a projective bimodule resolution for L C of length 1. Since A is quasi-free, m

1 .A/  AC ˝ AC  AC

(4.64)

is a projective A-bimodule resolution; here m is the multiplication map and the map 1 .A/ ! AC ˝ AC is induced by the derivation x 7! 1 ˝ x  x ˝ 1 and maps x0 Dx1 7! x0 ˝ x1  .x0  x1 / ˝ 1:  As in Notation 4.21, we write differential forms in 1 . TE/ as x0 Dx1 . Definition 4.65. We let       P0 WD L C ˝ L C C TC E ˝ L  TC E ˝ TC E;    P1 WD TC E D L  1 . TE/:

4.3 Excision in periodic cyclic homology

167

 Theorem 4.66. Both P0 and P1 are free L -bimodules, and we have a semi-split   extension of L -bimodules P1  P0  L C and a commuting diagram of extensions  1 . L / /

  / L C ˝ LC



 P1 / 

 1   . TE/ /

 // L C



 / P0

 // L C



   /T C E ˝ TC E





 //T C E.

The induced maps   even       P0 =Œ L ; P0  ! TC E ˝ TC E=Œ TE; TC E ˝ TC E Š TC E Š 1 ˚  E;   odd     P1 =Œ L ; P1  ! 1 . TE/=Œ TE; 1 . TE/ Š TE DE Š  E    are isomorphisms onto 1 ˚ ker. p W E ! Q/.  Proof. Let A WD TE. The maps 1 .A/ ! AC ˝ AC and AC ˝ AC ! AC in (4.64)   map P1 to P0 and P0 to L C because L is a left ideal in AC . The extension (4.64) has an explicit contracting homotopy id ˝ 1 W AC ! AC ˝ AC ;  W AC ˝ AC !  .A/; 1

x 7! x ˝ 1; x ˝ y 7! x Dy;

 where we interpret x Dy D 0 if y 2 1  AC . This restricts to maps L C ! P0 ! P1  and hence contracts the chain complex P1 ! P0 ! L C . It is easy to see that  P0  AC ˝ AC and P1  1 .A/ are sub- L -bimodules and the maps between them     L -bimodule maps. Hence P1  P0  L C is an L -bimodule resolution of L C . It  remains to check that P1 and P0 are free L -bimodules and compute their commutator quotients.    Concerning P0 , we use that TC E and L are free as a left and right L -module,  respectively, by Lemmas 4.56 and 4.60. It follows that P0 is the free L -bimodule   generated by .1 ˝ 1/ ˚ sL . TQ/ ˝ G . Hence the commutator quotient of P0 is isomorphic   P0 =Œ L ; P0  Š L C ˚  Š LC ˚

       L C ˝ sL . TQ/ ˝ G Š L C ˚ G ˝ L C ˝ sL . TQ/       L ˝ sL . TQ/ Š 1 ˚ L ˝ sL .TC Q/ Š 1 ˚ I Š I C ;

168

4 Periodic cyclic homology via pro-nilpotent extensions

   where we use Lemma 4.60 and (4.57). Recall that I D ker. TE ! TQ/. It is easy Š   ! I C constructed above is exactly the map to see that the isomorphism P0 =Œ L ; P0     induced by the embedding P0 ! TC E ˝ TC E. Concerning P1 , we first claim that    P1 D 1 . TE/ ˇ K C TC E DK  1 . TE/:  It is clear that TC E DK is contained in P1 . Equation (4.54) and !0 D.!1 ˇ k/ D .!0 D!1 / ˇ k  !0 ˇ !1 Dk     show that TC E D L and 1 . TE/ ˇ K agree modulo TC E DK. This proves the claim. Š    Theorem 4.22 provides a bimodule isomorphism 1 . TE/  ! TC E ˝ E ˝ TC E.       It maps 1 . TE/ ˇ K onto TC E ˝ E ˝ .TC E ˇ K/ Š TC E ˝ E ˝ L and TC E DK  onto TC E ˝ K ˝ 1. Hence we get isomorphisms        P1 Š .TC E ˝E ˝ L /C.TC E ˝K ˝1/ Š .TC E ˝s.Q/˝ L /˚.TC E ˝K ˝ L C /:    Since TC E and L are free as a left and right L -module, respectively, by Lemmas 4.56  and 4.60, this shows that P1 is a free L -bimodule. More precisely, P1 is free on the subspace     sL .TC Q/ Ds.Q/ G ˚ sL .TC Q/ DK: As a consequence, we have     P1 =Œ L ; P1  Š .sL .TC Q/ Ds.Q// ˇ L ˚ TC EDK      Š L ˇ sL .TC Q/ Ds.Q/ ˚ TC E DK Š I Ds.Q/ ˚ TC E DK  odd  odd   Š ker.TC E DE ! TC Q DQ/ Š ker.  E !  Q/:  This isomorphism agrees with the map induced by the embedding P1 ! 1 . TE/.

4.3.4 Proof of the Excision Theorem    Other proofs of the Excision Theorem use the two-sided ideal I  TE instead of L .  They have to overcome the difficulty that I is neither quasi-free nor H-unital and hence  does not satisfy excision in Hochschild homology. In contrast, the algebra L is quasi   free and the diagram L ! TE ! TQ satisfies excision in Hochschild homology and hence in cyclic and periodic cyclic homology, although it is not an extension:

4.3 Excision in periodic cyclic homology

169

 Theorem 4.67. The algebra L is quasi-free, and the canonical map      HH. L / ! ker HH. Tp/ HHrel . Tp/ is a chain homotopy equivalence. So are the maps      HC. L / ! ker HC. Tp/ HCrel . Tp/;      X. L / ! ker X. Tp/ Xrel . Tp/:   Proof. Theorem 4.66 asserts that P1  P0  L C is a projective L -bimodule   resolution of L C of length 1. Hence L is quasi-free by Theorem 4.16. Recall that we can compute the Hochschild homology of an algebra A using any   projective bimodule resolution of AC . For quasi-free algebras such as L , TE, and  TQ, we can use the short projective bimodule resolution (4.64). This yields 

b



HH.A/ 1 .A/=Π;  ! A

because we remove the subspace 1  AC ˝ AC =Œ ; . We abbreviate   b Xˇ .A/ WD 1 .A/=ŒA; 1 .A/ ! A I this differs from the X-complex because there is no boundary map from degree 0 to    degree 1. Lemma 4.46 shows that Xˇ . Tp/ W Xˇ . TE/ ! Xˇ . TQ/ is a split epimorphism, so that the canonical map     ker Xˇ . Tp/ ! Xˇ;rel . Tp/ is a chain homotopy equivalence.  We may compute HH. L / from the commutator quotient complex of the projective     bimodule resolution P1 ! P0 ! L C . This yields 1˚ker Xˇ . Tp/ by Theorem 4.66.     Hence the canonical map Xˇ . L / ! ker Xˇ . Tp/ is a chain homotopy equivalence, so that we get excision in Hochschild homology. This implies excision in cyclic and periodic cyclic homology by standard methods. Since the three algebras involved are quasi-free, we can see this directly. We only write down the proof for the X-complexes.   The obvious map L C ˝ L C ! P0 is split injective as a bimodule map; the com   plementary bimodule is L C ˝ sL . TQ/ ˝ L . Since we are comparing two resolutions,  it follows that the map 1 . L / ! P1 is split injective as well, with an isomorphic complement. Passing to commutator quotients, we get     ker Xˇ . Tp/ Š Xˇ . L / ˚ C for the contractible chain complex  id  !C ; C D C 

  C Š L ˝ sL . TQ/:

170

4 Periodic cyclic homology via pro-nilpotent extensions

    The map Xˇ . L / ! ker Xˇ . Tp/ is compatible with the additional boundary map d as well; but b is already invertible on C , so that d must vanish on C . Hence we get      ker X. Tp/ Š X. L / ˚ C X. L /:    Proof of Theorem 4.42. The semi-split pro-algebra extension . L \ JE/  L    K is pro-nilpotent because L \ JE is a subalgebra (even a left ideal) in the pro  nilpotent algebra JE (use Theorem 4.4 and Lemma 4.8). The algebra L is quasi-free by Theorem 4.67. Theorem 4.32 yields 



HP.K/ D X. TK/ X. L /:

  More precisely, any algebra homomorphism TK ! L that lifts the identity map on K induces such a chain homotopy equivalence. Of course, the standard embedding      TK ! L will do here. Since HPrel .p/ ker X. Tp/ , Theorem 4.67 yields that the standard embedding HP.K/ ! HPrel .p/ is a chain homotopy equivalence. If we drop the Q-linearity assumption, then much of the above argument still goes through and yields the following weaker result, which appears to be new. i

p

Theorem 4.68. Let K  E  Q be a semi-split algebra extension in an additive symmetric monoidal category that is not necessarily Q-linear. Then the canonical map   j W X. TK/ ! Xrel . Tp/ is a split monomorphism in Kom.CI Z=2/, that is,   Xrel . Tp/ Š X. TK/ ˚ C for some chain complex C .which need not be contractible/. Proof. Theorems 4.66 and 4.67 never use Q-linearity and still show   X. L / Xrel . Tp/I   as above, this yields Xrel . Tp/ Š X. L / ˚ C1 for a contractible chain complex C1 .   Since L is quasi-free, we can lift the projection L ! K to an algebra homomor  phism f W L ! TK; we will construct f more explicitly later. It turns out that we can    arrange f to restrict to the identity map on TK. Thus the embedding X. TK/ ! X. L /   is a split monomorphism. Hence so is the composite map X. TK/ ! Xrel . Tp/. Since Kom.CI Z=2/ is additive, this yields the asserted direct sum decomposition. As a result, we get a decomposition HPrel  .p/ Š HP .K/˚‹, where ‹ is the obstruction to excision for the particular extension.

171

4.3 Excision in periodic cyclic homology

4.3.5 Application to non-periodic cyclic homology There are relative versions HCrel .p/ and HHrel .p/ of cyclic and Hochschild homology, which are defined as in Definition 4.41. As before, let i

p

KEQ be a semi-split algebra extension in a Q-linear symmetric monoidal category C. The chain  map HC  .p/ W HC.E/ ! HC.Q/ is a split epimorphism by Lemma 4.46.  Hence  ker HC.p/ HCrel .p/. The map i induces a chain map HC.K/ ! ker HC.p/ . This map is a split monomorphism in C. The chain complex   C WD ker HC.p/ W HC.E/ ! HC.Q/ = HC.K/ measures the obstruction to excision in cyclic homology: there is an exact triangle HC.K/ ! HCrel .p/ ! C ! HC.K/Œ1

in HoKom.C/, which induces a long exact homology sequence    ! HCn .K/ ! HCrel n .p/ ! Hn .C / ! HCn1 .K/ ! HCrel n1 .p/ ! Hn1 .C / !    and a dual long exact sequence in cohomology. Thus excision in cyclic homology is equivalent to exactness of C . The Excision Theorem for cyclic homology by Mariusz Wodzicki asserts that C is exact if the ideal K is H-unital. Although cyclic homology does not satisfy excision in general, the Excision Theorem 4.42 for HP still yields some information. Recall that the periodicity operator S on cyclic homology comes from a canonical chain map SA W HC.A/ ! HC.A/Œ2. These maps for A D K; E; Q induce a canonical chain map S W C ! C Œ2 on the subquotient complex C . Proposition 4.69. The projective system of chain complexes SŒ4

SŒ2

SŒ6

SŒ8

C  C Œ2  C Œ4  C Œ6  C Œ8



(4.70)

  is contractible .it is an object of Kom.C/ Š Kom. C //. Proof. The projective system of chain complexes HC.A/

SA Œ2

 HC.A/Œ2

SA Œ4

 HC.A/Œ4

SA Œ6

 HC.A/Œ6



is chain homotopy equivalent to HP.A/ viewed as a Z-graded chain complex (see §A.6.2). Hence the system (4.70) is chain homotopy equivalent to the com projective  plex CHP WD ker HP.p/ =HP.K/, which is contractible by the Excision Theorem 4.42 for periodic cyclic homology.

172

4 Periodic cyclic homology via pro-nilpotent extensions

The homology of a projective system of chain complexes is itself a projective system, and its cohomology is an inductive system. If a pro-chain complex is contractible, then its homology and cohomology are isomorphic to 0 as projective or inductive systems, respectively. Together with a long exact sequence argument, Proposition 4.69 implies that the canonical maps HCn .K/ ! HCrel n .p/ form an isomorphism of projective systems     HCnC2k .K/; S k2Z Š HCrel nC2k .p/; S k2Z for n D 0; 1 mod 2; that is, for each n 2 N there is some ƒ.n/ 2 N and a commuting diagram S ƒ.n/

HCnC2ƒ.n/ .K/ HCn .K/ o hRRR RRR RRR RRR R   rel o HC HCrel .p/ nC2ƒ.n/ .p/. n

(4.71)

S ƒ.n/

Similar diagrams exist in cohomology, see (4.45). The growth of the function ƒ depends on the algebra extension in question, of course. Theorem 4.72. Let K  E  Q be a semi-split algebra extension in a Q-linear symmetric monoidal category C. Then diagrams as in (4.71) exist with ƒ.n/ D n C 1, so that n C 2ƒ.n/ D 3n C 2. This estimate is optimal. The optimal estimate for split extensions is ƒ.n/ D bn=2c C 1, so that n 7! n C 2ƒ.n/ maps 2n 7! 4n C 2 and 2n C 1 7! 4n C 3. The upper bounds ƒ.n/  n C 1 for general extensions and ƒ.n/  bn=2c C 1 for split extensions were obtained independently by myself in [65] and by Michael Puschnigg in [86], [89]. Puschnigg also gets better estimates for invertible extensions, that is, retracts of split extensions in [89]; and he shows that the reverse inequalities ƒ.n/  n C 1 and ƒ.n/  bn=2c C 1 hold for the algebra extensions JA  TA  A and qA  QA  A, respectively; this shows that they are optimal. Actually, Puschnigg only proves a slightly weaker statement: in our notation, he constructs commuting diagrams of the form S ƒ.n/

HCnC2ƒ.n/ .K/ o HCn .K/ o hRRR RRR RRR RRR R  o HCrel nC2ƒ.n/ .p/

HPn .K/ 

Š

(4.73)

HPrel n .p/

  because he only controls the retraction ker HP.p/ ! HP.K/. Following Alain Connes, he defines the dimension of a class ˛ in HP .A/ to be the minimal k 2  C 2Z for which the canonical map HP .A/ ! HCk .A/ does not annihilate ˛; his main theorems deal with this dimension. But his more technical results also yield (4.73). With a bit more work, the methods of [86], [89] should also yield (4.71), but I have not checked this thoroughly. When comparing our results to

4.4 Exterior products

173

those of Puschnigg, you should beware that the Hodge filtration is indexed differently in [86], [89]. We postpone the proof of the upper bounds ƒ.n/     in Theorem 4.72 to §5.4.7 because the same techniques are needed to prove the analogous theorems for analytic and local cyclic homology. We do not establish here that these estimates are optimal.

4.4 Exterior products Now we consider the behaviour of periodic cyclic homology for tensor products. There is, in general, no Künneth Formula for HP because there is no Künneth Formula for  tensor products of chain complexes in C . We avoid this problem by working directly with the underlying chain complexes. Theorem 4.74. There are natural chain homotopy equivalences HP.A1 / ˝ HP.A2 / HP.A1 ˝ A2 /

for all pro-algebras A1 and A2 ; these are associative, symmetric, and monoidal up to chain homotopy, that is, the following diagrams commute up to chain homotopy:

HP.A1 / ˝ HP.A2 / ˝ HP.A3 /

 HP.A1 / ˝ HP.A2 ˝ A3 / HP.A1 / ˝ HP.A2 /



HP.A1 ˝ A2 /

Š

Š





/ HP.A2 / ˝ HP.A1 / 

/ HP.A1 ˝ A2 / ˝ HP.A3 /



 / HP.A1 ˝ A2 ˝ A3 /; HP.A/ o O Š

/ HP.A2 ˝ A1 /;

Š

/ 1 ˝ HP.A/



HP.1 ˝ A/



 / HP.1/ ˝ HP.A/I

here we use the canonical isomorphism A1 ˝ A2 Š A2 ˝ A1 and the standard chain homotopy equivalence 1 ! HP.1/ .see Theorem 5.63). Let K  E  Q be a semi-split pro-algebra extension and let A be a proalgebra; then A˝K  A˝E  A˝Q is another semi-split pro-algebra extension. Let @ W HP.Q/ ! HP.K/Œ1 and @A W HP.A ˝ Q/ ! HP.A ˝ K/Œ1 be the classes of their boundary maps. Then the following diagram commutes up to chain homotopy: HP.A/ ˝ HP.Q/ idHP.A/ ˝@



HP.A/ ˝ HP.K/Œ1



/ HP.A ˝ Q/



 / HP.A ˝ K/Œ1.

@A

174

4 Periodic cyclic homology via pro-nilpotent extensions

Theorem 4.74 yields an exterior product HP .A1 ; B1 / ˝ HP .A2 ; B2 / ! HP .A1 ˝ A2 ; B1 ˝ B2 /; which is associative, graded commutative, and compatible with the boundary maps in the Excision Theorem 4.42; it sends f1 ˝ f2 to the composite chain map f1 ˝f2

HP.A1 ˝ A2 / HP.A1 / ˝ HP.A2 / ! HP.B1 / ˝ HP.B2 / HP.B1 ˝ B2 /:

Our proof of Theorem 4.74 follows Joachim Cuntz and Daniel Quillen [26, §14]. This argument is made more explicit by Michael Puschnigg in [84].

4.4.1 Tensor products of quasi-free unital algebras First we prove the following special case of Theorem 4.74; we will see later that it implies the general case. We use the chain complex X .2/ .A ˝ B/ defined in A.122. Theorem 4.75. Let A and B be quasi-free unital algebras. Then there is a natural chain homotopy equivalence X.A/ ˝ X.B/ X .2/ .A ˝ B/. We denote the boundary maps of degree ˙1 in the chain complexes X.A/ ˝ X.B/ and X .2/ .A ˝ B/ by b and B, respectively. The algebra A ˝ B is again unital, but usually not quasi-free. Its bidimension is x 1 .A/  at most 2 because the tensor product of the projective bimodule resolutions  1 x .B/  B ˝ B  B is a projective bimodule resolution of A ˝ A  A and  length 2: x 1 .A/ ˝ B ˝ B 0  @1 @00 ˚ 0 !  .A/ ˝  .B/ ! ! A ˝ A ˝ B ˝ B ! A ˝ B; x 1B A˝A˝   a da ˝ b0 ˝ b1  a0 da1 ˝ b0 b1 ˝ 1 @02 .a0 da1 ˝ b0 db1 / WD 0 1 ; a0 a1 ˝ 1 ˝ b0 db1  a0 ˝ a1 ˝ b0 db1   a0 da1 ˝ b0 ˝ b1 0 WD a0 a1 ˝ 1 ˝ b0 ˝ b1  a0 ˝ a1 ˝ b0 ˝ b1 @1 a2 ˝ a3 ˝ b2 db3 x1

x1

@02

C a2 ˝ a3 ˝ b0 b1 ˝ 1  a2 ˝ a3 ˝ b0 ˝ b1 ; ˝ a1 ˝ b0 ˝ b1 / WD a0 a1 ˝ b0 b1 : (4.76) The commutator quotient complex of the resolution in (4.76) is @00 .a0



  1  b b x 1 .A/=Œ ;  ! x .B/=Œ ;  !   A ˝   B HH.A/ ˝ HH.B/:

We abbreviate D WD A ˝ B. Since all projective bimodule resolutions of D are equivalent, we get a chain homotopy equivalence HH.D/ HH.A/ ˝ HH.B/:

175

4.4 Exterior products

To carry this over to the periodic cyclic theory, we write down a chain homotopy equivalence and check that it is compatible with B. We do this carefully because tensor products in non-periodic cyclic homology are rather subtle (see [52]). x 1 .A/ for a unital algebra A have the same First we observe that 1 .A/ and  commutator quotient. Let e 2 A be the unit element, then b.de dx/ D e dx  dx C x de;

b.e de dx/ D e dx  e dx C x de D x de;

so that the relations x de 0 and e dx dx for x 2 A hold modulo commutators. x 1 .A/ or 1 .A/ to define X.A/. Therefore, it makes no difference whether we use  By the way, this computation also shows that X.AC / Š X.A/ ˚ 1

(4.77)

x 1 .AC /. for any algebra A in a symmetric monoidal category because 1 .A/ WD  .2/ The chain complex X .D/ is different from x .2/ .D/ WD .D ˚  x 1 .D/ ˚  x 2 .D/=Œ ; ; B C b/; X x .2/ .D/ is a chain homotopy equivalence. Hence but the canonical map X .2/ .D/ ! X it is allowed to use reduced differential forms to prove Theorem 4.75. Since D has bidimension 2, we have a chain homotopy equivalence 

b



b

x 2 .D/=Œ ;  ! x 1 .D/ ! HH.D/     D : This is the commutator quotient of the projective bimodule resolution @2 @1 @0 x 1 .D/ ˝ D  x 2 .D/  ! ! D ˝ D ! D; 0!

@0 .x ˝ y/ WD x  y; @1 .x0 dx1 ˝ x2 / WD x0 x1 ˝ x2  x0 ˝ x1 x2 ; @2 .x0 dx1 dx2 / WD x0 dx1 ˝ x2  x0 . dx1 /x2 ˝ 1:

(4.78)

The following is a chain map of D-bimodule homomorphisms: @2

x 2 .D/ /  ˆ2

 x 1 .B/ / x 1 .A/ ˝  

/ x 1 .D/ ˝ D

@1

/D˝D

 / x 1 .B/ x 1 .A/ ˝ B ˝2 ˚ A˝2 ˝ 

//D

Š ˆ0

ˆ1

@02

@0

@01

 / A˝2 ˝ B ˝2

@00

/ / D,

ˆ0 .a0 b0 ˝ a1 b1 / WD a0 ˝ a1 ˝ b0 ˝ b1 ;   1 a0 .da1 /a2 ˝ b0 b1 ˝ b2 C a0 .da1 /a2 ˝ b0 ˝ b1 b2 ˆ1 .a0 b0 d.a1 b1 / ˝ a2 b2 / WD ; 2 a0 ˝ a1 a2 ˝ b0 .db1 /b2 C a0 a1 ˝ a2 ˝ b0 .db1 /b2    1 ˆ2 a0 b0 d.a1 b1 / d.a2 b2 / WD a0 .da1 /a2 ˝ b0 b1 db2  a0 a1 da2 ˝ b0 .db1 /b2 : 2

176

4 Periodic cyclic homology via pro-nilpotent extensions

Here we abbreviate ab WD a ˝ b for elements of D. When we pass to commutator x 1 .D/ ˝ D/=Œ ;  Š  x 1 .D/ and .D ˝ D/=Œ ;  Š D via quotients, we identify . ! ˝ x 7! x  !. Similarly, we identify x 1 .A/ ˝ B ˝2 /=Œ ;  Š  x 1 .A/=Œ ;  ˝ B; .

x 1 .B//=Œ ;  Š A ˝  x 1 .B/=Œ ; : .A˝2 ˝ 

This yields a chain map     1   2 b b b b x 1 .D/ ! x 1 .A/=Œ ;  ! x .B/=Œ ;  ! PW  x .D/=Œ ;  !   D !   A ˝   B ; ˆ P 0 .ab/ D a ˝ b; ˆ     1 a0 da1 ˝ .b0 b1 C b1 b0 / P ˆ1 a0 b0 d.a1 b1 / D ; 2 .a0 a1 C a1 a0 / ˝ b0 db1     P 2 a0 b0 d.a1 b1 / d.a2 b2 / D 1 a2 a0 da1 ˝ b0 b1 db2  a0 a1 da2 ˝ b2 b0 db1 : ˆ 2 The same chain map appears in [84, 2.1]. P is a chain map for the boundary b. Now we check that ˆ P is By construction, ˆ compatible with B as well:   da ˝ b P ˆ1 ı d.ab/ D D .d ˝ 1 C 1 ˝ d /.ab/; a ˝ db   P 2 ı B a1 b1 d.a2 b2 / D a2 da1 ˝ b1 db2  a1 da2 ˝ b2 db1 ; ˆ   P 1 a1 b1 d.a2 b2 / a1 da2 ˝ b1 db2  a1 da2 ˝ b2 db1 .d ˝ 1  1 ˝ d / ı ˆ C a2 da1 ˝ b1 db2 C a1 da2 ˝ b1 db2   P D ˆ2 ı B a1 b1 d.a2 b2 / : x .2/ .A ˝ B/ ! X.A/ ˝ X.B/. It is a chain homoPWX Hence we get a chain map ˆ topy equivalence for the b-boundary maps because it compares the commutator quotient complexes for two projective bimodule resolutions of D. By Kassel’s Perturbation x .2/ .A ˝ B/ ! X.A/ ˝ X.B/ P is a chain homotopy equivalence X Lemma ([58]), ˆ as well. This finishes the proof of Theorem 4.75. P The map ˆ P is natural in an obvious Now we discuss additional properties of ˆ. sense: if f W A1 ! A2 and g W B1 ! B2 are unital algebra homomorphisms between quasi-free unital algebras, then the diagram x .2/ .A1 ˝ B1 / X x .2/ .f ˝g/ X



x .2/ .A2 ˝ B2 / X

P ˆ

/ X.A1 / ˝ X.A2 / X.f /˝X.g/

P ˆ

 / X.B1 / ˝ X.B2 /

commutes exactly. We have canonical isomorphisms X.1/ ˝ X.B/ Š 1 ˝ X.B/ Š X.B/;

x .2/ .1 ˝ B/ Š X x .2/ .B/ X

177

4.4 Exterior products

x .2/ .B/ ! X.B/ in this P becomes the canonical projection B W X for A D 1, and ˆ P is exactly symmetric, that case. Similar things happen for B D 1. The chain map ˆ is, compatible with exchanging A and B; the symmetry constraint for tensor products of chain complexes such as X.A/ ˝ X.B/ is recalled in §A.1.4 and involves a sign on X1 .A/ ˝ X1 .B/. Associativity makes no sense in the situation of Theorem 4.75 because it involves a triple tensor product A ˝ B ˝ C , which has bidimension 3. Therefore, we omit a discussion of associativity here. Proposition 4.79. Let A and B be quasi-free, but not necessarily unital. Then there is a canonical chain homotopy equivalence HP.A ˝ B/ X.A/ ˝ X.B/. Proof. Equation (4.77) yields a canonical isomorphism of chain complexes   X.AC / ˝ X.BC / Š X.A/ ˝ X.B/ ˚ X.A/ ˚ X.B/ ˚ 1: x .2/ .AC ˝ BC / Š X .2/ .A  B/ ˚ 1, where A  B is the augmentation ideal We have X of AC ˝ BC (see also §A.2.4). This algebra has bidimension 2 because its unitalisation P restricts to a chain homotopy equivalence AC ˝ BC has. Naturality implies that ˆ 

 ! X.A/ ˝ X.B/ ˚ X.A/ ˚ X.B/I X .2/ .A  B/ 

(4.80)

that is, we can drop the summands 1. Proposition 4.36 and Theorem A.123 show that the canonical projections HP.A ˚ B/  HP.A/ ˚ HP.B/  X.A/ ˚ X.B/;

HP.A  B/  X .2/ .A  B/

P implies that the chain map are chain homotopy equivalences. The naturality of ˆ P ˆ





! X.A/ ˝ X.B/ ˚ X.A/ ˚ X.B/ HP.A  B/  X .2/ .A  B/  restricts to a canonical chain map HP.A ˝ B/ ! X.A/ ˝ X.B/. To check that it is a chain homotopy equivalence, we apply the Excision Theorem 4.42 to the canonical semi-split algebra extension A ˝ B  A  B  A ˚ B; it shows that we have an exact triangle HP.A ˝ B/ ! HP.A  B/ ! HP.A ˚ B/ ! HP.A ˝ B/Œ1

 in HoKom. C I Z=2/. Since the extension has sections on A and B separately, this triangle splits, that is, HP.A  B/ HP.A ˝ B/ ˚ HP.A/ ˚ HP.B/:

(4.81)

P is compatible with the decompositions in (4.80) The chain homotopy equivalence ˆ and (4.81); hence its restriction HP.A ˝ B/ ! X.A/ ˝ X.B/ is a chain homotopy equivalence as well.

178

4 Periodic cyclic homology via pro-nilpotent extensions

4.4.2 Reduction to the quasi-free case Now let A and B be arbitrary pro-algebras. Proposition 4.82. The canonical projections are HP-equivalences  TA  A;

 TB  B;

  . TA/ ˝ . TB/  A ˝ B:

    Proof. We have semi-split algebra extensions JA  TA  A, JB  TB  B,   N  . TA/ ˝ . TB/  A ˝ B;

and

    JA ˝ JB  N  A ˝ JB ˚ JA ˝ B:

  Since JA and JB are pro-nilpotent, Theorem 4.4 implies that N is pro-nilpotent. Hence the assertions follow from Goodwillie’s Theorem 4.31 for HP. Propositions 4.79 and 4.82 yield the desired natural chain homotopy equivalence: 









HP.A ˝ B/ HP T.A/ ˝ T.B/ X. TA/ ˝ X. TB/ HP.A/ ˝ HP.B/:

It remains to check that these maps have the additional properties listed in Theorem 4.74. Naturality is rather obvious, and symmetry and compatibility with the unit object follow P easily from the corresponding properties of ˆ. p

i

If K  E  Q is an extension, then the resulting diagram HP.A ˝ K/

HP.idA ˝i/



 HP.A/ ˝ HP.K/

/ HP.A ˝ E/

HP.idA ˝p/



id˝HP.i/

 / HP.A/ ˝ HP.E/

/ HP.A ˝ Q/

id˝HP.p/

 / HP.A/ ˝ HP.Q/

P restricts to a chain commutes Hence ˆ  homotopy.   exactly andnot just up to chain map ker HP.idA ˝ p/ ! HP.A/ ˝ ker HP.p/ , which is again a chain homotopy equivalence. This implies compatibility with the boundary map. This finishes the proof of Theorem 4.74, except for the proof of associativity which we omit.

Chapter 5

Analytic cyclic homology and analytically nilpotent extensions

We have seen in Chapter 4 that periodic cyclic homology is closely related to pronilpotent extensions. There is a similar relationship between analytic or local cyclic homology and analytically nilpotent extensions. We mainly state results and definitions for the analytic cyclic theory and complete ! bornological algebras. The local cyclic theory for algebras in Ban usually behaves similarly; we comment on differences between the two categories as we go along. First we study bounded l inear maps with analytically nilpotent curvature, which we briefly call lanilcurs. The analytic tensor algebra T A is defined by a universal property that is similar to that of the pro-tensor algebra: algebra homomorphisms A ! B correspond to lanilcurs A ! B. In §5.1, we construct T A and check that the kernel JA  T A of the natural homomorphism T A ! A is analytically nilpotent. Using this link to analytically nilpotent algebras, we show that lanilcurs form a category, that is, are closed under composition. By the universal property, this implies that any semi-split analytically nilpotent extension of T A splits. Algebras with this property are called analytically quasi-free; this is a stronger property than quasi-freeness. Since T A is (analytically) quasi-free for any A, its X-complex computes the periodic cyclic homology of T A. We show in §5.2 that X.T A/ is chain homotopy equivalent to the chain complex HA.A/ that we have used in §2.1.2. More generally, Q for any semi-split, analytiwe get a chain homotopy equivalence HA.A/ X.A/ Q cally quasi-free, analytically nilpotent extension A  A. This allows to compute HA for some other simple examples like C and CŒz; z 1 ; this is used to construct the Chern–Connes character K ! HA in §7.3. Using X.T A/ as a new realisation of HA, we prove invariance of analytic cyclic homology under homotopies of bounded variation and under analytically nilpotent extensions, and the Excision Theorem for HA; the arguments are mostly the same as for the periodic theory, with some additional steps. To get the Excision Theorem, we show that the left ideal L  T E that makes our argument work is analytically quasi-free, which cannot be detected by the homological algebra of §4.3.2. The advantage of our new definition is that the X-complex is rather small, so that the study of analytic cyclic homology becomes mainly the study of the analytic tensor algebra. The category of algebras is much more rigid than the category of chain complexes. Since there are fewer algebra homomorphisms than chain maps, the universal property of the analytic tensor algebra is a powerful method to construct interesting maps.

180

5 Analytic cyclic homology and analytically nilpotent extensions

5.1 Lanilcurs and the analytic tensor algebra 5.1.1 Definition and universal property The analytic tensor algebra T A of a bornological algebra A is a completion of the usual tensor algebra TA with respect to a certain bornology. Thus Alg.T A; B/ is a subset of Alg.TA; B/ Š Hom.A; B/. Describing T A is equivalent to describing the set of linear maps A ! B that induce algebra homomorphisms T A ! B. Definition 5.1. A bounded linear map f W V ! A from a complete bornological vector space V to a complete bornological algebra A is analytically nilpotent if %.f .S /I A/ D 0 ! ! for all S 2 S.V /. If V 22 Ban and A 2 Alg.Ban/, we require %.f .S /I A/ D 0 for all bounded maps S ! V for sets S instead. Definition 5.2. A bounded linear map f W A ! B between two algebras in Cborn or ! Ban has analytically nilpotent curvature if its curvature y A ! B; !f W A ˝

a1 ˝ a2 7! f .a1  a2 /  f .a1 /  f .a2 /

is analytically nilpotent (Definition 5.1). We briefly call such maps lanilcurs and let lanilcur.A; B/ be the set of lanilcurs A ! B. Since !f is quadratic, we have !f . S;  S/ D 2 !f .S; S /. Hence f is a lanilcur if and only if !f .S; S / is power-bounded for all S 2 S.A/. Algebra homomorphisms are lanilcurs. We will see later that lanilcurs form a category. It is clear that lanilcurs are closed under composition with algebra homoy h/ if g and h are algebra morphisms on both sides because !gıf ıh D g ı !f ı .h ˝ homomorphisms. Thus lanilcur is a functor lanilcur W Alg.Cborn/op  Alg.Cborn/ ! Sets: Example 5.3. Let A be a bornological algebra and let A0 be A with the zero multiplication. The curvature of the identity map A0 ! A is the multiplication map A ˝ A ! A up to a sign. Hence the identity map A0 ! A is a lanilcur if and only if A is analytically nilpotent. Even more, if A is analytically nilpotent, then any bounded linear map into A is a lanilcur. In this way, the lanilcurs determine the analytically nilpotent algebras. Example 5.4. For B D C0 .X /, any lanilcur A ! B is already an algebra homomorphism because %.SI B/ D 0 implies S  f0g. ! Definition 5.5. The analytic tensor algebra T A of an algebra in Ban or Cborn is a (co)representing object for the covariant functor lanilcur.A; /, that is, we require a natural bijection Alg.T A; B/ Š lanilcur.A; B/. Definition 5.6. The natural bijection Alg.T A; B/ ! lanilcur.A; B/ must be of the form f 7! f ı A for a canonical lanilcur A W A ! T A. The identity map on A

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is a lanilcur, which yields a canonical algebra homomorphism A W T A ! A with A ı A D idA . Let JA WD ker A . The resulting semi-split algebra extension JA /

/T Aj

A

/ /A

A

is called the analytic tensor algebra extension of A. We construct T A explicitly using differential forms. TheoL non-commutative 2n  .A/ with the Fedosov product rem A.63 shows that even .A/ D 1 nD0 ! ˇ  WD !    d!  d is isomorphic to the tensor algebra of A; the canonical map A W A ! TA is simply the embedding of A D 0 .A/, and its curvature is ! .a1 ; a2 / D a1  a2  a1 ˇ a2 D da1 da2 : To turn A into a lanilcur, we must ensure that dS dS becomes power-bounded for all S 2 S.A/. This leads to the following definition: Definition 5.7. The analytic bornology San .A/ is the bornology on A generated by subsets of the form hhSii WD hSi.dS/1 D S.dS /1 [ .dS /1 with S 2 S.A/. We let an A be the completion of A with this bornology. Warning 5.8. This differs from the bornology S" .A/ in Notation 2.7 by factors bn=2cŠ. 

0 Lemma 5.9. The algebra .even ! even an A is an A; ˇ/ with the map A W A D  A  isomorphic to T A with the lanilcur A W A ! T A. If f W A ! B is a lanilcur, then

hf i W .even an A; ˇ/ ! B; a0 da1 da2 : : : da2n 7! f .a0 /  !f .a1 ; a2 /    !f .a2n1 ; a2n / is the unique algebra homomorphism with hf i ı A D f . Proof. First we claim that hhS ii  hhS ii } hhS 2 [ 2 S ii, so that the multiplication of differential forms is bounded for the analytic bornology. Let a0 ; : : : ; a2n 2 S , then a0 da1 : : : da2m  a2mC1 da2mC2 : : : da2n D

2m X

.1/i a0 da1 : : : d.ai aiC1 / : : : da2n

iD0

is a sum of 2mC1 monomials with entries in S [S 2 and at least 2m entries in S . Since 2m C 1  22m , this is a convex combination of elements of hhS 2 [ 2 S ii. Dropping a0 or a2mC1 does not invalidate these estimates.

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5 Analytic cyclic homology and analytically nilpotent extensions

Since the differential d is bounded as well, .A; ˇ/ is a bornological algebra; hence so is its completion an A and the closed subalgebra even an A. The analytic bornology is constructed so that dS and hence dS dS are power-bounded for all S 2 S.A/. Therefore, A W A ! .an A; ˇ/ is a lanilcur. The formula for hf i yields the unique extension of f to an algebra homomorphism even A ! B by Theorem A.63. The map hf i is bounded for the analytic bornology if and only if f is a lanilcur. The universal property of completions yields Alg..even an A; ˇ/; B/ Š lanilcur.A; B/ and hence .even an A; ˇ/ Š T A. ! There is a similar concrete description of T A if A is an algebra in Ban. First we ignore the multiplication and write A D lim Ai for an inductive system of Banach ! spaces .Ai /i2I . Equip each Ai with some closed unit ball Si and let IQ WD I  N and Si;n WD n  Si for .i; n/ 2 IQ. We get subsets hhSi;n ii of Ai and let Bi;n be the completion of Ai for the gauge norm of hhSi;n ii} . This defines an inductive system of Banach spaces, which we denote by an A. The multiplication and the differential ! on A extend to an A, turning it into a differential Z=2-graded algebra in Ban. The ! Fedosov product makes sense as well, so that .even an A; ˇ/ becomes an algebra in Ban. The same ideas as above show that this provides a concrete realisation of the analytic tensor algebra. It is clear from the universal property that T is a functor. In our concrete description, an algebra homomorphism f W A ! B induces the algebra homomorphism T f W T A ! T B;

f .a0 da1 : : : da2n / WD f .a0 / df .a1 / : : : df .a2n /:

Example 5.10. Suppose that we work over C. We want to describe T C as an algebra of analytic functions, extending the isomorphism TC Š t CŒt . This is surprisingly subtle and requires several results that we will establish later. Let C..t// be the algebra of analytic power series in the variable t , and let C..t //0 be the ideal of analytic power series without constant term. We want to show that the isomorphism TC Š t CŒt  extends to an isomorphism T C Š C..t//0  C..1  t //; where we map t CŒt  diagonally into the right hand side. Let e 2 C be the identity element. First we claim that the map f W C ! C..t //0  C..1  t //,  e 7! . t;  t / is a lanilcur and hence induces an algebra homomorphism T C ! C..t //0  C..1  t //.  The curvature !f is given by !f .e; e/ D t .1  t /; t .1  t / 2 C..t //0  C..1  t //0 ; since the latter algebra is analytically nilpotent (Example 3.28), Proposition 5.13 shows that f is a lanilcur. To show that the map hf i W T C ! C..t//0  C..1  t // is an isomorphism, we describe its inverse. The main point is that we can lift e to an idempotent eO 2 T C (Proposition 5.50). Since C..t//0  C..1  t // only contains the two idempotents .0; 0/ and .0; 1/ and since e O C .e/ ¤ 0, we must have hf i.e/ O D .0; 1/. Since T C

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183

is commutative, we get a direct product decomposition T C Š eO ? T C  eT O C with O eO ? WD 1  e. It follows from Theorem 5.12 that T C is locally multiplicative and hence has holomorphic functional calculus (Theorem 3.15). The element eO ? e 2 JC has spectrum f0g because JC is analytically nilpotent and hence generates an algebra homomorphism C..t//0 ! eO ? T C with t 7! eO ? e. The element ee O 2 T C maps to e 2 C and has spectrum f1g because the projection T C  C is isoradial and eT O C is unital with unit e. O Hence we get a unital algebra homomorphism C..1  t // ! eT O C with t 7! ee. O As a result, we get an algebra homomorphism C..t //0  C..1  t // ! T C; it is inverse to the homomorphism in the opposite direction constructed above.

5.1.2 Relationship to analytically nilpotent algebras Theorem 5.11. The algebras JA and .1 an A; ˇ/ are analytically nilpotent for any ! algebra A in Cborn or Ban. Proof. We only check the assertion for .1 an A; ˇ/, the subalgebra JA can be treated similarly. Since analytic nilpotence is hereditary for completions, it suffices to work in A. Any bounded subset of 1 A is contained in (and not just absorbed by) hSi.dS /1 dS for some S 2 S.A/. Since the latter is contained in .hSi dS /1 , it suffices to check that hSi dS is power-bounded for the analytic bornology. The idea of the proof is similar to Lemma 3.22. We claim that there is T 2 San .A/ with hS i dS ˇ T } T and hSi dS  T . This implies .hS i dS /n ˇ T } T for all n 2 N and hence .hSi dS/nC1  .hS i dS/n ˇ T  T , so that hSi dS is powerbounded in San .A/. It remains to describe T . Let S .n/ WD S [ S 2 [    [ S n and 1 [ hS .2/ i  .4 dS .3/ /n : T WD nD1

We clearly have hS i dS  T  hh4 S .3/ ii, so that T is bounded in San .A/. To check hS i dS ˇ T } T , let a0 da1 2 hSi dS, a2 da3 : : : da2n 2 T . Then a0 da1 ˇ a2 da3 : : : da2n D a0 d.a1  a2 / da3 : : : da2n  a0  a1 da2 da3 : : : da2n  da0 da1 da2 da3 : : : da2n 2 hS i dS .3/ .4 dS .3/ /1  S .2/ dhS .2/ i.4 dS .3/ /1  dhS i dS.4 dS .3/ /1 } T:  Theorem 5.12. The algebra T A is locally multiplicative if and only if A is. Proof. This follows from Theorem 5.11 because being locally multiplicative is hereditary for extensions and quotients by Theorem 3.75. Proposition 5.13. A bounded linear map f W A ! B is a lanilcur if and only if its g ! y A ! B factors as A ˝ yA curvature !f W A ˝ !N  ! B for a bounded linear map !, an analytically nilpotent algebra N , and an algebra homomorphism g.

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5 Analytic cyclic homology and analytically nilpotent extensions

  Proof. Suppose that !f factors as above. We have !f .S; S /1 D g !.S; S /1 because g is an algebra homomorphism. This set is bounded because N is analytically nilpotent. Hence f is a lanilcur. Conversely, suppose that f is a lanilcur. Let N WD JA, !.a1 ; a2 / WD da1 da2 , and g WD hf ijJA . This factorisation of !f does the job by Theorem 5.11. Proposition 5.14. A bounded linear map V ! A is analytically nilpotent if and only if it factors through an algebra homomorphism N ! A with analytically nilpotent N . Proof. It is clear that maps that factor through an analytically nilpotent algebra are analytically nilpotent. Conversely, let f W V ! A be analytically nilpotent. Equip V with the zero multiplication, then f is a lanilcur. We can factor f as hf i ı A . The algebra T V is analytically nilpotent because V and JV are (Theorem 5.11) and analytic nilpotence is hereditary for extensions by Theorem 3.27. Proposition 5.15. If N  E  B is a semi-split analytically nilpotent extension, then we can lift an algebra homomorphism f W A ! B to a morphism of extensions JA /

/ TA

 N /

 /E

A

//A f

 / / B.

This morphism is not unique, so that we only have a weak universal property. We will improve this in Theorem 5.32. Proof. Choose a section s W B ! E. We have p ı !sıf D !f D 0 because f and p are algebra homomorphisms. That is, !sıf factors through N . Hence s ı f is a lanilcur by Proposition 5.13 and induces a homomorphism hs ı f i W T A ! E. This lifts f W A ! B and therefore restricts to a homomorphism JA ! N as well. We get a square of equivalent notions: analytic tensor algebra o O

/ lanilcurs O gggg3 g g g g g g g ggggg  sggggg  / analytically nilpotent maps. analytically nilpotent algebras o

Analytic tensor algebras and lanilcurs determine each other by the universal property of T A; lanilcurs and analytically nilpotent algebras determine each other by Proposition 5.13 and Example 5.3. As a result, the analytic tensor algebra functor and the class of analytically nilpotent algebras determine each other as well. This explains several fundamental results about lanilcurs and analytic tensor algebras.

5.1 Lanilcurs and the analytic tensor algebra

185

5.1.3 Other analytic completions Here we briefly discuss two variants of the analytic tensor algebra that are related to pairs of close homomorphisms and almost commuting homomorphisms. Definition 5.16. Two homomorphisms f0 ; f1 W A  B are .analytically/ close if the map f0  f1 W A ! B is analytically nilpotent (Definition 5.1). Two lanilcurs A ! B are close if the associated homomorphisms T A ! B are close. Notation 5.17. Motivated by Theorem A.67, we let Qan .A/ WD .an A; ˇ/. The projection Qan .A/ ! 0 .A/ D A is an algebra homomorphism, which we denote by A ; let qan .A/ be its kernel. The algebra homomorphisms ˙ W A  Qan .A/;

x 7! x ˙ dx

are sections for the resulting algebra extension qan .A/  Qan .A/  A. Theorem 5.18. The homomorphisms ˙ W A  Qan .A/ are close. If f˙ W A  B are close homomorphisms then there is an algebra homomorphism fC f W Qan .A/ ! B with .fC  f / ı ˙ D f˙ . As a consequence, two homomorphisms .or lanilcurs/ are close if and only if there is an analytically nilpotent extension N  E  A with two multiplicative .or lanilcur/ sections s˙ W A  E and an algebra homomorphism g W E ! B with f˙ D g ı s˙ . Proof. It is clear from the definition that C and  are close. Recall that .A; ˇ/ is isomorphic to the free product Q.A/ D A  A by Theorem A.67. The algebra homomorphism F W .A; ˇ/ ! B induced homomorphisms  Q  by a pair of algebra f˙ W A  B maps dx1 : : : dxn 7! 2n jnD1 fC .xj /  f .xj / . Hence it extends to an A if and only if fC and f are close. Thus algebra homomorphisms Qan .A/ ! B correspond to pairs of close algebra homomorphisms A ! B. It is clear that a factorisation f˙ D g ı s˙ implies that fC and f are close. Conversely, if fC and f are close homomorphisms, we get such a factorisation with E D Qan .A/, N D qan .A/, s˙ D ˙ , and g D fC  f . Theorem 5.11 asserts that qan .A/ is analytically nilpotent. If fC and f are close lanilcurs, then we use E WD Qan .T A/ instead. Theorem 3.27 yields that the kernel N of the canonical projection E  A is analytically nilpotent because it is an extension of analytically nilpotent algebras qan .T A/  N  JA by Theorem 5.11. Our close lanilcurs A  B first yield close homomorphisms T A  B, then a homomorphism E ! B; this provides the desired factorisation. The relation of being close is reflexive and symmetric, but not transitive. ! Definition 5.19. Let A, B, and D be algebras in Cborn or Ban. Two algebra homomorphisms fA W A ! D and fB W B ! D are .analytically/ almost commuting if the y B ! D, a ˝ b 7! ŒfA .a/; fB .b/, is analytically nilpotent. commutator map A ˝ Two lanilcurs A ! D and B ! D are .analytically/ almost commuting if the associated algebra homomorphisms T A ! B are.

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5 Analytic cyclic homology and analytically nilpotent extensions

Now we use the description of the free product A  B in §A.4.4 as 1   M n .A/ ˝ n .B/; ~ : A  B Š .A; B/ WD A  B ˚ nD1

Recall that we have canonical semi-split extensions .A; B/  .A; B/  A  B;

A ˝ B  A  B  A ˚ B: A

B

There are two canonical algebra homomorphisms A ! .A; B/  B. Definition 5.20. A subset of S .A; B/ belongs to the analytic bornology if it is contained in the disked hull of hSA i.dSA /n ˝ hSB i.dSB /n for some SA 2 S.A/, SB 2 S.B/. We let an .A; B/ be the completion of .A; B/ with respect to the analytic bornology. We let an .A; B/ G an .A; B/ be the kernel of the canonical projection an .A; B/  A  B. Theorem 5.21. Two homomorphisms fA W A ! D and fB W B ! D almost commute if and only if there is an algebra homomorphism fA  fB W an .A; B/ ! D with .fA  fB / ı A D fA and .fA  fB / ı B D fB . The algebra an .A; B/ is analytically nilpotent. Two homomorphisms .or lanilcurs/ fA and fB almost commute if and only if there are an analytically nilpotent extension N  E  AB, algebra homomorphisms .or lanilcurs/ sA W A ! E and sB W B ! E that lift the canonical embeddings A ! AB B, and an algebra homomorphism g W E ! D with fA D g ı sA and fB D g ı sB . Proof. The analytic bornology on .A; B/ is defined so that the algebra homomorphism fA  fB W .A; B/ Š A  B ! D is bounded if and only if fA and fB almost commute. This yields the universal property of an .A; B/. The proof that an .A; B/ is analytically nilpotent is similar to the proof of Theorem 5.11 and omitted. Using the semi-split analytically nilpotent extension an .A; B/  an .A; B/  A  B;

(5.22)

we get the last assertion as in the proof of Theorem 5.18. The corresponding extension of pro-algebras is used in [26] to construct the canonical chain homotopy equivalence HP.A ˝ B/ HP.A/ ˝ HP.B/. We do not yet have a conceptual proof of the corresponding homotopy equivalence for the analytic theory. If it exists, it is likely that (5.22) plays a role in it.

5.1.4 The category of lanilcurs The most non-trivial property of the class of analytically nilpotent algebras is that it is closed under algebra extensions. This implies that lanilcurs form a category and that analytic tensor algebras are analytically quasi-free.

5.1 Lanilcurs and the analytic tensor algebra

187

Theorem 5.23. Composites of lanilcurs are again lanilcurs, so that lanilcurs form a category. In particular, the composite map A

T A

A2 W A ! T A ! T T A ! is a lanilcur for any algebra A in Cborn or Ban. f

g

Proof. Let A  !B  ! C be two lanilcurs. We show that g ı f is a lanilcur as well. We factor f D hf i ı A . We already know that g ı hf i is a lanilcur because hf i is an algebra homomorphism. Hence we may assume without loss of generality that f D A . We can assume, in addition, that g D B D T A because g ı f D hgi ı .B ı f /. Thus we are reduced to proving that A2 is a lanilcur. This map is a section for the extension A2 WD A ı T A W T T A  A. We have a semi-split algebra extension JT A /

/ ker. 2 / A

T A

/ / JA:

Since JT A and JA are analytically nilpotent by Theorem 5.11, so is ker.A2 / by Theorem 3.27. Since the curvature of A2 factors through this analytically nilpotent kernel, it is a lanilcur by Proposition 5.13. Proposition 5.24. The following conditions are equivalent: (1) the analytic tensor algebra extension JA  T A  A splits; (2) any semi-split analytically nilpotent algebra extension N  E  A splits; (3) if f W A ! B is a homomorphism and N  E  B is a semi-split analytically nilpotent extension, then f lifts to a homomorphism fO W A ! E. Proof. It is clear that (3) ) (2) ) (1). Proposition 5.15 yields (1) ) (3). Definition 5.25. We call A analytically quasi-free if it satisfies the equivalent conditions of Proposition 5.24. Theorem 5.26. The algebra T A is analytically quasi-free. Proof. The lanilcur A2 W A ! T T A of Theorem 5.23 induces an algebra homomorphism A W T A ! T T A. The uniqueness part of the universal property of T A (Lemma 5.9) yields T A ı A D idT A because T A ı A ı A D T A ı A2 D idT A ı A . Hence the extension JT A  T T A  T A splits. The characterisation of quasi-free algebras in Theorem 4.16 shows that analytically quasi-free algebras are quasi-free because nilpotent algebras are analytically nilpotent. But there are examples of quasi-free algebras that are not analytically quasi-free by Proposition 2.49. Thus analytic quasi-freeness seems to be a rare property. We will study some interesting examples of analytically quasi-free algebras in §5.3.1. Since lanilcurs form a category, we can reformulate the definition of T : T is left adjoint to the forgetful functor from the category of algebra homomorphisms to the category of lanilcurs. This includes the following functoriality statement:

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5 Analytic cyclic homology and analytically nilpotent extensions

Proposition 5.27. The analytic tensor algebra is functorial for lanilcurs. The algebra homomorphism T f W T A ! T B induced by a lanilcur f W A ! B is characterised by the condition T f ı A D B ı f ; it also satisfies B ı T f D hf i. Proof. Since f ı B is a lanilcur, T f WD hB ı f i exists; it does the job. ! Theorem 5.28. Let D be an algebra in Cborn .or Ban/. The following are equivalent: y D, that is, if f W A ! B is a lanilcur then so is (1) lanilcurs are closed under ˝ y idD W A ˝ y D!B˝ y D; the induced map f ˝ y y ! .T A/ ˝ (2) for any algebra A, there is an algebra homomorphism ˛D W T .A ˝D/ y D with ˛D ı A˝D ˝ id D  ; this induces a natural morphism of extensions A D y y D/ / J.A ˝

/ T .A ˝ y D/

 yD / .JA/ ˝



//A˝ yD

˛D

/ .T A/ ˝ yD

//A˝ y D;

y D is analytically nilpotent whenever A is; (3) A ˝ y D is analytically nilpotent; (4) C..t//0 ˝ (5) D is locally multiplicative. Proof. Since any lanilcur f W A ! B can be factored as hf i ı A , (1) holds if and only y idD is a lanilcur, if and only if there is an algebra homomorphism ˛D as in (2) if A ˝ y idD i). Thus (1) , (2). (take ˛D WD hA ˝ Proposition 5.13 yields (3) ) (2). The converse implication (2) ) (3) follows from y D has the zero multiplication, that is, A0 ˝ y D D .A ˝ y D/0 . Example 5.3 because A0 ˝ The implication (3) ) (4) is trivial, and Theorem 3.78 yields (5) ) (3). It remains y D explicitly: it to prove (4) ) (5). SincePC..t//0 is nuclear, we can describe C..t //0 ˝ 1 n is the space of all series nD1 xn t with xn 2 D for which there are a bounded disk T 2 Sd .D/ and r > 0 with xn 2 r n T for all n 2 N1 . y D. Let S 2 Sd .D/ be a bounded disk, thenSS t is a bounded disk in C..t //0 ˝ By assumption, it is power-bounded, that is, t n ˝ S n is bounded in C..t //0 . Hence we have S n  r n T for all n 2 N1 for some T 2 Sd .D/, r > 0. This implies %.S I D/  r < 1. Thus D is locally multiplicative. More generally, we can ask the same questions as in Theorem 5.28 if we are given any commuting diagram of functors Alg.Cborn/

Forget

F

 Alg.Cborn/

/ Cborn F

Forget

 / Cborn;

or

! Alg.Ban/ F

 ! Alg.Ban/

Forget

/ ! Ban F

 Forget ! / Ban;

5.1 Lanilcurs and the analytic tensor algebra

189

y D. In this situation, the the tensor product corresponds to the case F .A/ WD A ˝ following conditions are equivalent: • the set of maps B/ is mapped under F to the set of  lanilcur.A;B/  Hom.A;   maps lanilcur F .A/; F .B/  Hom F .A/; F .B/ ; • there is an algebra homomorphism ˛A W T F .A/ ! F .T A/ with ˛A ı F .A/ D F .A / for each algebra A; • F maps analytically nilpotent algebras to analytically nilpotent algebras. The proof is the same as for (part of) Theorem 5.28. For instance, the functors C .X; / and A.Œ0; 1; / have the last property by Proposition 3.42, and hence enjoys the other properties as well. Theorem 5.28 is needed to study homotopy classes of lanilcurs. A smooth homotopy y B. of lanilcurs is, of course, a lanilcur A ! C 1 .Œ0; 1; B/ D C 1 .Œ0; 1/ ˝ Proposition 5.29. The functor T maps smooth, bounded variation, or continuous homotopies between lanilcurs f0 ; f1 W A ! B to homotopies of algebra homomorphisms with the same regularity between T f0 ; T f1 W T A ! T B. Proof. Let A 7! AŒ0; 1 be one of the cylinder functors C 1 .Œ0; 1; /, A.Œ0; 1; /, or C .Œ0; 1; / that define our notions of homotopy. A homotopy  of lanilcurs A ! BŒ0; 1 induces an algebra homomorphism T A ! T BŒ0; 1 by Proposition 5.27.   Composition with the natural map T BŒ0; 1 ! .T B/Œ0; 1 constructed above yields the required homotopy of algebra homomorphisms; it is easy to see that its endpoints are T f0 and T f1 .

5.1.5 The homotopy category of lanilcurs as a localisation ! It makes no difference here whether we consider algebras in Cborn or Ban. The category of lanilcurs itself is not a localisation of the category of algebra homomorphisms, that is, we cannot get it by inverting certain algebra homomorphisms. This changes when we pass to their homotopy categories; we will mainly work with smooth homotopies later on, but it does not really matter which notion of homotopy we use here, as long as it satisfies the analogue of Theorem 5.28 (this excludes polynomial homotopy). Of course, the homotopy category of a category is the category whose morphisms are homotopy classes of morphisms in the original category for a given homotopy relation on morphisms. Theorem 5.30. The .smooth, bounded variation, or continuous/ homotopy category of lanilcurs is the localisation of the .smooth, bounded variation, or continuous/ homotopy category of algebra homomorphisms at the maps A W T A ! A for all algebras A. The functor T descends to a fully faithful functor from the homotopy category of lanilcurs to the homotopy category of algebra homomorphisms.

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5 Analytic cyclic homology and analytically nilpotent extensions p

If N  E  A is a semi-split analytically nilpotent extension, then N Š 0 and E  A becomes a deformation retraction in the category of lanilcurs. Proof. We prove the last statement first. Since N is analytically nilpotent, so is N Œ0; 1 (see the proof of Proposition 5.29). Hence any bounded linear map into N Œ0; 1 is a lanilcur (Example 5.3), so that the linear homotopy t idN contracts N in the lanilcur category. Choose a bounded linear section s W A ! E for p, that is, p ı s D idA . Since p ı !s D !pıs D 0, the curvature of s factors through the analytically nilpotent algebra N , so that s is a lanilcur by Proposition 5.13. We connect s ı p and idA by the affine homotopy H t WD t idA C .1  t / s ı p. This is a deformation retraction, that is, p ı H t D idA for all t 2 Œ0; 1; it is a lanilcur homotopy because !H factors through the analytically nilpotent algebra N Œ0; 1. Thus p is a deformation retraction in the category of lanilcurs. As a consequence, A W T A ! A becomes invertible in the homotopy category of lanilcurs for any A; its inverse is A because A ı A D idA . By the universal property, homotopy classes of algebra homomorphisms T A ! T B correspond bijectively to homotopy classes of lanilcurs A ! T B. Since B is a homotopy equivalence in the lanilcur category, the latter correspond bijectively to homotopy classes of lanilcurs A ! B. Thus T descends to a fully faithful functor from the homotopy category of lanilcurs to the homotopy category of algebra homomorphisms. Consequently, any functor F on the homotopy category of algebra homomorphisms yields a functor F ı T on the homotopy category of lanilcurs. If F .A / is an isomorphism for all A, then F and F ı T are equivalent via F .A /, so that F itself descends to a functor on the homotopy category of lanilcurs. This shows that the latter is the localisation of the homotopy category of algebra homomorphisms at the maps A for all algebras A. If F is any functor on the homotopy category of algebras, then F ı T together with the natural transformation F .A / W F .T A/ ! F .A/ is its localisation at the class of maps A . Roughly speaking, it is the best approximation to F that descends to the homotopy category of lanilcurs. Lemma 5.31. If A is analytically quasi-free, then the set of algebra homomorphisms A ! B is a smooth deformation retract of the set of lanilcurs A ! B in a natural way. Thus any lanilcur is smoothly homotopic to an algebra homomorphism A ! B, and if two algebra homomorphisms are .smoothly/ homotopic among lanilcurs, they are .smoothly/ homotopic among algebra homomorphisms. Proof. Suppose that A is analytically quasi-free. Then A W T A  A splits by an algebra homomorphism  W A ! T A. The affine homotopy .1  t /  C t A is a lanilcur to C 1 .Œ0; 1; T A/ and thus provides a smooth homotopy of algebra homomorphisms T A ! C 1 .Œ0; 1; T A/ between A ı  and the identity map by Proposition 5.29. Thus A is a smooth deformation retract of T A via algebra homomorphisms. Since lanilcurs A ! B correspond to algebra homomorphisms T A ! B, this provides a natural, smooth deformation retraction from lanilcurs A ! B to algebra homomorphisms A ! B.

5.1 Lanilcurs and the analytic tensor algebra

191

Theorem 5.32. Let N1  E1  A1 and N2  E2  A2 be semi-split algebra extensions and let f W A1 ! A2 be an algebra homomorphism. Suppose that E1 is analytically quasi-free and N2 is analytically nilpotent. Then we can lift f to a morphism of extensions / E1 / / A1 N1 /  N2 /

 / E2

f

 / / A2 ;

the lifting is unique up to a smooth homotopy that is constant on A1 . A homotopy between two morphisms of extensions is a morphism of extensions to N2 Œ0; 1  E2 Œ0; 1  A2 Œ0; 1. Proof. Any bounded linear map E1 ! E2 that lifts f is a lanilcur because N2 is analytically nilpotent. Lemma 5.31 replaces such a lanilcur lifting by an algebra homomorphism g W E1 ! E2 . The naturality of g ensures that it still lifts f because the composite map E1 ! E2 ! A2 is an algebra homomorphism and therefore fixed by the deformation retraction. Two liftings of f are smoothly homotopic in the lanilcur category by the affine homotopy. Lemma 5.31 replaces this by a homotopy of algebra homomorphisms, whose naturality ensures that it has the same endpoints and lifts the constant homotopy const ı f W A1 ! A2 Œ0; 1. Corollary 5.33. Any semi-split algebra extension N  E  A with analytically nilpotent N and analytically quasi-free E is smoothly homotopy equivalent to the extension JA  T A  A. In particular, E and T A are smoothly homotopy equivalent. Proof. Since JA and N are analytically nilpotent and T A and E are analytically quasi-free, Theorem 5.32 provides morphisms of extensions f and g between them and smooth homotopies connecting f ı g and g ı f to the identity morphisms.

5.1.6 Localisation without homotopy Since some interesting homology theories for algebras like algebraic K-theory are not homotopy invariant, we also want to localise the category of algebras at nilpotent extensions without passing to homotopy categories. We describe the resulting localisation very briefly here. As an application, we give a very short description of HA0 and HL0 as localisations of HH0 . Our construction is closely related to a similar one for periodic cyclic homology and other purely algebraic theories by Guillermo Cortiñas ([11], [12]). We consider the localisation of the category of algebras at the projections E  A for all semi-split analytically nilpotent extensions N  E  A. Lemma 5.34. Any morphism A ! B in the localisation is of the form f ı A1 for an algebra homomorphism f W T A ! B.

192

5 Analytic cyclic homology and analytically nilpotent extensions

Proof. To begin with, morphisms in the localisation are products f1 ıp11 ı  ıfn ıpn1 with algebra homomorphisms fj and projection maps of analytically nilpotent semisplit extensions pj W Ej  Aj . Proposition 5.15 yields algebra homomorphisms gi W T Ai ! Ei with pi ı gi D Ai ; this becomes pi1 D gi ı A1 in the localisation. i 1 The relation B ı T f D f ı A for f W A ! B becomes T f ı A D B1 ı f and allows us to shift all fi to the left, leading to a representative of the form f 0 ı .An /1 with the canonical projection An W T n A ! A. Repeating the first step above, we bring this into the form f ı A1 . As a result, the morphisms A ! B in the localisation are equivalence classes of lanilcurs A ! B. We claim that the relevant equivalence relation is the equivalence relation c generated by the relation of being close (Definition 5.16). Theorem 5.35. The morphisms A ! B in the localisation at semi-split analytically nilpotent extensions are the c -equivalence classes of lanilcurs A ! B. Proof. If s0 and s1 are two sections for an analytically nilpotent extension N  E  B, then Œs0  D Œs1  in the localisation because both are one-sided inverses of the projection E ! B, which becomes invertible in the localisation. Hence the localisation identifies close homomorphisms by Theorem 5.18. Then it also identifies close lanilcurs. It remains to show that the c -equivalence classes of lanilcurs form a category in which the maps A become invertible. The last point follows because A ı A is close to idT A and A ı A D idA anyway. If two lanilcurs f0 ; f1 W A ! B are close, then so are the induced homomorphisms T f0 ; T f1 W T A ! T B. It follows immediately from the definition that composition of algebra homomorphisms respects closeness. It would be interesting to localise at analytically nilpotent extensions that are not necessarily semi-split. It should suffice for applications to consider extensions with a bounded non-linear section because such sections always exist locally. ! Definition 5.36. Let F W Alg.Cborn/ ! C (or Alg.Ban/ ! C) be some functor defined on the category of algebras. Its localisation is the universal functor LF with a natural transformation LF ! F that is invariant under semi-split analytically nilpotent extensions. Universality means that any other natural transformation FQ ! F for a functor FQ with this additional property factors uniquely through LF . Thus LF is the best approximation to F that is invariant under semi-split analytically nilpotent extensions. The following theorem seems to be the quickest definition of analytic and local cyclic homology. We will prove it in §5.3 below. ! Theorem 5.37. The functor HL0 W Alg.Ban/ ! Ab is the localisation of the commutator quotient functor A 7! HH0 .A/ D A=Œ ;  in the sense just explained. When we restrict attention to algebras in Cborn, then the localisation of HH0 is HA0 .

5.2 Analytic cyclic homology via analytic tensor algebras

193

5.2 Analytic cyclic homology via analytic tensor algebras 5.2.1 A new definition of the analytic cyclic theory The following theorem allows us to use the analytic tensor algebra to study analytic and local cyclic homology. The X-complex of an algebra is defined in Definition A.122. Theorem 5.38. There are natural chain homotopy equivalences X.T A/ HA.A/ and ! X.T A/ HL.A/ for algebras in Cborn and Ban, respectively. Joachim Cuntz and Daniel Quillen construct a similar chain homotopy equivalence  X. TA/ HP.A/ in [26, §5]. The only difference to our situation is that we must ! check carefully whether the maps that we use are bounded (or morphisms in Ban). The bimodule 1 .T A/. Following [25, §2], we use universal properties to compute 1 .T A/ as a bimodule; this yields descriptions of T A as a left and right module as well. The argument is essentially the same as for uncompleted tensor algebras in §A.4.5 and for pro-tensor algebras. Notation 5.39. We write TC A WD .T A/C . To avoid confusion between the boundary operators on .an A; ˇ/ T A and .T A/, we denote the latter by D. Theorem 5.40. The bimodule 1 .T A/ is isomorphic to the free T A-bimodule on A via   Š y A˝ y TC A  ! T A; ! ˝ x ˝  7! ! ˇ DA .x/ ˇ : TC A ˝ Proof. Let M be a T A-bimodule. Bimodule homomorphisms 1 .T A/ ! M correspond to derivations T A ! M by Proposition A.55. These correspond to sections for the square-zero extension M  T A Ë M  T A by Theorem A.53. By the universal property of T A, these correspond to lanilcurs A ! T A Ë M that lift A W A ! T A. Since analytically nilpotent algebras are closed under extensions, the projection T A Ë M  A has analytically nilpotent kernel. Hence any bounded linear map A ! T A Ë M that lifts A is a lanilcur; such bounded linear maps are of the form .A ; f / for an arbitrary bounded linear map f W A ! M . Putting these steps together, we get a natural bijection between bimodule homomorphisms 1 .T A/ ! M and bounded linear maps A ! M . Thus 1 .T A/ is isomorphic to the free bimodule on A. Inspection of the argument shows that the isomorphism has the asserted form. Corollary 5.41. As a left or right module over itself, T A is isomorphic to the free module on A via Š

y A TC A ˝ ! T A; ! ˝ x 7! ! ˇ x; Š

y TC A  A˝ ! T A; x ˝ ! 7! x ˇ !:

194

5 Analytic cyclic homology and analytically nilpotent extensions

The map

1 y odd an A Š TC A ˝ A !  .T A/;

! dx 7! ! ˇ Dx;

Š

induces an isomorphism odd ! 1 .T A/=Œ ; . an A  ! These maps are isomorphisms in Cborn or Ban. Proof. Let B be any algebra in a symmetric monoidal category. View 1 as a B-module with zero multiplication. Since 1 .B/  BC ˝ BC  BC splits as an extension of left or right B-modules, it remains a semi-split extension of right or left B-modules when we apply the functors 1 ˝B or ˝B 1. This yields 1 ˝B 1 .B/ Š ker.BC  1/ Š B;

1 .B/ ˝B 1 Š ker.BC  1/ Š B

as a right or left B-module, respectively. Moreover, the commutator quotient of a bimodule M (such as 1 .B/) is given by M=Œ ;  Š BC ˝B˝B op M . The assertions now follow because Theorem 5.40 identifies 1 .T A/ with the free bimodule on A. even This also yields an isomorphism odd an A ! an A of the form

x0 dx1 : : : dxn 7! x0 dx1 : : : dxn1 ˇ xn D x0 dx1 : : : dxn1  xn  dx0 dx1 : : : dxn1 dxn : We will not use this isomorphism. Explicit formulas for the isomorphisms. We describe the inverses of the isomorphisms in Corollary 5.41 explicitly. Since we already know that these inverses are bounded linear maps, it suffices to do this on dense subsets. We generate T A by differential forms x0 dx1 : : : dx2n or x0 and ignore the latter, trivial case. The definition of the Fedosov product yields x0 dx1 : : : dx2n D x0 dx1 : : : dx2n2 ˇ .x2n1 x2n  x2n1 ˇ x2n /: y A maps Hence the isomorphism T A Š TC A ˝ x0 dx1 : : : dx2n 7! x0 dx1 : : : dx2n2 ˝x2n1 x2n x0 dx1 : : : dx2n2 ˇx2n1 ˝x2n : y TC A maps x0 dx1 : : : dx2n 7! x0 ˝ dx1 : : : dx2n if The isomorphism T A Š A ˝ x0 2 A and dx1 : : : dx2n D .x1 x2  x1 ˇ x2 / ˇ dx3 : : : dx2n 7! x1 x2 ˝ dx3 : : : dx2n  x1 ˝ x2 dx3 : : : dx2n : y A˝ y TC A is computed in [26, §5]. Since it is The isomorphism 1 .T A/ ! TC A ˝ a left module map, it suffices to describe its value on D.x0 dx1 : : : dx2n /. The Leibniz rule yields D. dx1 dx2 / D D.x1 x2  x1 ˇ x2 / D D.x1 x2 /  x1 D.x2 /  D.x1 /x2

195

5.2 Analytic cyclic homology via analytic tensor algebras

and hence D.x0 dx1 : : : dx2n / D .Dx0 / dx1 : : : dx2n C

n X

x0 dx1 : : : dx2j 2 ˇ .D.x2j 1 x2j /

j D1

 x2j 1 D.x2j /  D.x2j 1 /x2j / ˇ dx2j C1 : : : dx2n 7 1 ˝ x0 ˝ dx1 : : : dx2n ! C

n X

x0 dx1 : : : dx2j ˝ x2j 1 x2j ˝ dx2j C1 : : : dx2n

j D1



n X

x0 dx1 : : : dx2j ˇ x2j 1 ˝ x2j ˝ dx2j C1 : : : dx2n

j D1



n X

x0 dx1 : : : dx2j ˝ x2j 1 ˝ x2j dx2j C1 : : : dx2n :

j D1

5.2.2 The complex X.T A/ Using the isomorphisms constructed above, we get X0 .T A/ D T A Š even an A;

X1 .T A/ Š 1 .T A/=Œ ;  Š odd an A:

The boundary maps in X.T A/ can be rewritten in terms of the operators b; d;  on an A. The map X1 .T A/ ! X0 .T A/ is rather easy to treat: it maps ! dx 7! ! ˇ x  x ˇ ! D !  x  x  !  d! dx C dx d! D b.! dx/  d.! dx/  d.! dx/; that is, @ D b  .1 C /d

on 2nC1 A.

(5.42)

The other boundary map D

! 1 .T A/  1 .T A/=Œ ;  Š odd @ W even an A D T A  an A y A˝ y TC A described above. A routine involves the isomorphism 1 .T A/ ! TC A ˝ computation in [26, §5] yields @D

2n X j D0

j d 

n1 X

 2j b D B 

j D0

This is the special case of the following lemma:

n1 X j D0

 2j b

on 2n A.

(5.43)

196

5 Analytic cyclic homology and analytically nilpotent extensions

Lemma 5.44. Let ! 2 2m .A/,  2 2n .A/. The isomorphism 1 .T A/=Œ ;  Š odd an A maps the class of ! d to 2n1 X

 j ı d.! ˇ / C  2n .! d/ 

j D0

n1 X

 2j ı b.! ˇ /:

j D0

Since d 2 D 0, we have d.! ˇ / D d.!  /. Hence Lemma 5.44 is equivalent to [26, Lemma 5.3]. Comparison with the cyclic bicomplex. Finally, we compare the chain complex X.T A/ and the completion of the cyclic bicomplex HA.A/ used in §2.1.2. Proof of Theorem 5.38. We use Corollary 5.41 to identify X.T A/ Š .an A; @/ with @ as in (5.42)–(5.43). The cyclic bicomplex is isomorphic to .A; B C b/. To get HA.A/ and an A, we equip A with two bornologies in Notation 2.7 and Definition 5.7, which differ by the factor bn=2cŠ in degree n. To compensate this, let c W A ! A multiply forms of degree 2n and 2n C 1 by .1/n nŠ; then c extends to an isomorphism of bornological vector spaces an A Š HA.A/. The boundary map b C B on HA.A/ translates to c 1 .B C b/c on an A; this is b  1=nC1 B on 2nC1 A and n b C B on 2n A. It remains to compare this boundary map with @. We use Theorem A.107 and, in particular, the identity . nC1  1/. n  1/ D 0 on n  A. We decompose n A as in [26, §6]; let PC and P be the spectral projections onto the generalised eigenspaces for C1 and 1, and let P? WD 1  PC  P , then A Š PC  ˚ P  ˚ P? : We must show that these projections extend to bounded operators on an A. The eigenspace for C1 has multiplicity 2 and all the other eigenspaces have multiplicity 1. Therefore, we can describe the projections PC and P explicitly as PCn ./ and P ./ on n A, where PCn and Pn are polynomials that satisfy the following conditions: PCn ./ D 0 D Pn ./ for roots of unity  of order n and n C 1 other than ˙1; PCn .1/ D 1, .PCn /0 .1/ D 0, and PCn .1/ D 0; and Pn .1/ D 0, .Pn /0 .1/ D 0, and PCn .1/ D 1. It is straightforward to construct such polynomials explicitly. Let Qn .z/ WD

1 1 zn  1 .1 C z C    C z n1 / D  : n n z1

Then Qn .1/ D 1 and Qn vanishes at other nth roots of unity. Computing derivatives at 1 as well, we see that     PCn .z/ WD Qn .z/  QnC1 .z/  1  n  1=2  .z  1/ does the job. A possible choice for Pn is Pn .z/ WD

.z n  1/.z nC1  1/ : zC1

5.3 Basic properties

197

Let n W an A ! an A be the restriction of  to n A extended by 0 on m A for m ¤ n. Since nj .x0 dx1 : : : dxn / D .1/.nC1/j dxnj C1 : : : dxn  x0 dx1 : : : dxnj for 0  j  n, the set of operators fnj j n 2 N; 0  j  ng on an A is equibounded; hence so is fnj j n 2 N; 0  j  ang for any a 2 N. If .˛n /n2N is a sequence of at most exponential growth, then multiplication by ˛n on n A is bounded. Therefore, if fn is a sequence of polynomials whose degrees P grow at most linearly and whose coefficients grow at most exponentially, then 1 nD0 fn .n / is a bounded operator on an A. Therefore, PC and P are bounded on an A. It follows easily from (5.42)–(5.43) that @ and c 1 .B C b/c agree on the range of PC . Next we consider the range of P , where we have  D 1. Since d D  nC1 ı d D .1/nC1 d and  n ı b D .1/n b on n A, we get d D 0 on forms of even degree and b D 0 on forms of odd degree. Moreover, B D 0. Hence @ and c 1 .B Cb/c become b on P odd A and 0 D b on P even A by (5.42)–(5.43). Hence they agree as well. By the way, this piece is contractible because bd C db D 1   D 2, so that d=2 is a contracting homotopy. Finally, we restrict attention to P? an A. We have bd C db D 1   and .b  .1 C /d /2 D b 2  .1 C /.bd C db/  .1 C /d 2 D .1 C /.  1/ D  2  1: The operators 1   and  2  1 are invertible on P? A; we claim that their inverses extend to bounded operators on an A. Taking this for granted for the moment, we check that P? an A is contractible for @ and c 1 .B C b/c. even The boundary map @ W P? odd an A ! P? an A is invertible with @1 D .b  .1 C /d /1 D .b  .1 C /d /. 2  1/1 I hence .P? an A; @/ is contractible. Since B projects to PC an A, it vanishes on 2n P? an A. Hence c 1 .B C b/c becomes b on P? odd an A and n b on P?  A. Let H 1 even 1 n1 1 be .1/ d on P? an A and  =n.1/ d on P? an A. Since bd Cdb D 1, this is a contracting homotopy for .P? an A; c 1 .B C b/c/. 2 It only remains to check P1that   1 and 1   are bounded on P? an A. The inverses are of the form nD1 gn .n / with suitable polynomials gn . We can take .1  PCn  Pn /=.z 2  1/ or .1  PCn  Pn /=.1  z/, respectively. It is easy to see that their degrees grow at most linearly and their coefficients grow at most exponentially. Hence we get a bounded operator on an A. This finishes the proof.

5.3 Basic properties From now on, we redefine HA.A/ and HL.A/ to be X.T A/ for algebras in Cborn ! or Ban, respectively. This does not change the resulting bivariant homology and

198

5 Analytic cyclic homology and analytically nilpotent extensions

cohomology theories by Theorem 5.38. The advantage is that our new definition is  very close to the description of HP as X. TA/. This allows us to prove results about the periodic, analytic, and local cyclic theories by literally the same arguments. We write down all statements for the analytic cyclic theory and frequently mention that the local cyclic theory behaves similarly. The analogous results for the periodic cyclic theory are stated in §4.2; many proofs in §4.2 are omitted because they are the same as in the analytic case. Our first main goal is to show X.A/ X.T A/ if A is analytically quasi-free; this allows us to compute the analytic and local cyclic theory in some simple examples and requires homotopy invariance and Goodwillie’s Theorem about invariance under nilpotent extensions. Then we prove stability with respect to algebras of nuclear operators and additivity. Excision and the behaviour for tensor products are more difficult. The local cyclic theory for locally multiplicative algebras has much better homotopy invariance and stability properties; we will prove this in §6.3.1. Homotopy invariance Theorem 5.45. If two algebra homomorphisms or lanilcurs f0 ; f1 W A ! B are homotopic via a homotopy of bounded variation, then X.T f0 / and X.T f1 / are chain homotopic. Thus HA.f0 / D HA.f1 / in HA0 .A; B/ if A; B 22 Alg.Cborn/ and ! HL.f0 / D HL.f1 / in HL0 .A; B/ if A; B 22 Alg.Ban/. Proof. Let F W A ! A.Œ0; 1; B/ be a homotopy of bounded variation between f0 and f1 . By Proposition 5.29, it induces an algebra homomorphism T A ! A.Œ0;1; T B/ that provides a homotopy between T f0 and T f1 . Since T A is quasi-free (Theorem 5.26), the Homotopy Invariance Theorem 4.27 for the X-complex yields that the chain maps X.T f0 /; X.T f1 / W X.T A/ ! X.T B/ are chain homotopic. This yields the last assertion by Theorem 5.38. Invariance under analytically nilpotent extensions p

Theorem 5.46 (Goodwillie’s Theorem). Let N  E  A be a semi-split analytically nilpotent extension. Then X.T N / is chain contractible and X.T p/ is a chain homotopy equivalence. Thus p W E ! A and 0 ! N are HA-equivalences or HLequivalences. Proof. Theorem 5.30 asserts that N is smoothly contractible and p is a smooth homotopy equivalence in the lanilcur category. Now apply Theorem 5.45. Theorem 5.47. Let N  E  A be a semi-split algebra extension with analytically nilpotent N and analytically quasi-free E. Then the canonical maps E TE ! TA induce chain homotopy equivalences X.E/ X.T E/ X.T A/. Proof. Corollary 5.33 implies that the maps E T E ! T A are smooth homotopy equivalences. Since all algebras involved are quasi-free, the Homotopy Invariance Theorem 4.27 for the X-complex yields the assertion.

5.3 Basic properties

199

Theorem 5.47 is a crucial step in the proof of the Excision Theorem. Corollary 5.48. If A is analytically quasi-free, then X.A/ X.T A/. Analytic and local cyclic homology as a localisation. Theorem 5.37 asserts, roughly ! speaking, that HA0 and HL0 are the best approximations on Alg.Cborn/ or Alg.Ban/, respectively, to the commutator quotient functor HH0 that are invariant under semi-split analytically nilpotent extensions. Now we have all the tools that we need to prove this. Similar ideas appear in [26, Proposition 4.2]. Proof of Theorem 5.37. We write down the proof for HA0 , the argument for HL0 is literally the same. By definition, the commutator quotient of an algebra A is the cokernel of the boundary map X1 .A/ ! X0 .A/. Hence we get a natural map   ker @ W X0 .T A/ ! X1 .T A/ X0 .T A/    D HH0 .T A/ ! HH0 .A/:   HA0 .A/ D @ X1 .T A/ @ X1 .T A/ Goodwillie’s Theorem 5.46 asserts that HA0 is invariant under semi-split analytically nilpotent extensions. It remains to show that any natural transformation F ! HH0 with such a functor F factors uniquely through HA0 . To begin with, A induces a natural isomorphism F .T A/ Š F .A/. Hence we get a natural transformation F .A/ Š F .T A/ ! HH0 .T A/. It remains to show that its range is contained in the subspace HA0 .A/. We write B WD T A. The split square-zero extension 1 .B/  B Ë 1 .B/  B has two canonical algebra homomorphism sections 0 WD .idB ; 0/ and 1 WD .idB ; d /. Both induce the same map F .B/ ! F .B Ë 1 B/ because F is invariant under nilpotent extensions. Hence the range of the natural transformation F .B/ ! HH0 .B/ must be contained in the kernel of .1 /  .0 / W HH0 .B/ ! HH0 B Ë 1 .B/ . We compute   HH0 B Ë 1 .B/ Š B=ŒB; B ˚ 1 .B/=ŒB; 1 .B/; and .1 /  .0 / corresponds to the map B=Œ ;  ! 1 .B/=Œ ;  induced by d ; this the is exactly the boundary map X0 .B/ ! X1 .B/. Hence   natural transformation F .B/ ! HH0 .B/ factors through the subspace H0 X.B/ . This finishes the construction of the natural transformation F ! HA0 and shows that HA0 has the right universal property. A similar argument shows that HP0 is the localisation of HH0 for pro-algebras at the pro-nilpotent extensions. Opposite algebras. Let Aop denote the opposite algebra of A, in which the order of multiplication is reversed. We wish to compare HA.A/ and HA.Aop /. These chain complexes are almost but not quite isomorphic. Given a Z=2-graded chain complex   @0 @1 @0 @1 @0 C WD    ! C0 ! C1 ! C0 ! C1 ! C0 ! C1 !    ;

200 we let

5 Analytic cyclic homology and analytically nilpotent extensions

  @0 @1 @0 @1 @0 op C WD    ! C0 ! C1 ! C0 ! C1 ! C0 ! C1 !      @0 @1 @0 @1 @0 Š    ! C0 ! C1 ! C0 ! C1 ! C0 ! C1 !    :

Multiplication by .1; C1; C1; 1/ in each degree mod 4 yields a 4-periodic isomorop phism C Š C , but there is no natural way to make this 2-periodic. Proposition 5.49. There is a natural algebra isomorphism .T A/op Š T .Aop /, which induces a chain isomorphism X.T Aop / Š X.T A/op . op

Proof. The map A W Aop ! .T A/op is a lanilcur and induces an algebra homomorphism W T .Aop / ! .T A/op . Similarly, we get an algebra homomorphism  op T A ! T .Aop / ; its opposite is inverse to because both restrict to the identity map on A. Explicitly, we have .x0 dx1 : : : dx2n / D dx2n : : : dx1  x0 : We claim that X.B op / X.B/op for any algebra B. We have 1 .B/=Π;  D 1 .B op /=Π;  because it makes no difference whether we divide out x0 x1 dx2  x0 d.x1 x2 / C x2 x0 dx1 or x1 x0 dx2  x0 d.x2 x1 / C x0 x2 dx1 for all x0 ; x1 ; x2 2 B. The map d W B ! 1 B=Π;  remains unaffected and b.x0 dx1 / D x0 x1  x1 x0 changes its sign. Thus we get X.B/op .

5.3.1 Some simple computations of analytic cyclic homology Corollary 5.48 motivates us to search for some examples of analytically quasi-free algebras and to compute their analytic cyclic homology. Examples of analytically quasi-free algebras. We have already seen that analytic tensor algebras are analytically quasi-free. So are the ordinary tensor algebras TA, for the same reason. Recall that 1 denotes C or R depending on the base field over which we work. Proposition 5.50. The base field 1 is analytically quasi-free. Proof. The following argument is based on ideas of Joachim Cuntz and Daniel Quillen ([26, §12]). Bounded algebra homomorphisms 1 ! A correspond bijectively to idempotents in A: map the idempotent e 2 A to the homomorphism  7!   e. Thus the p

analytic quasi-freeness of 1 is equivalent to the following assertion: let N  E  Q be a (semi-split) analytically nilpotent extension. Then any idempotent eP 2 Q may be lifted to an idempotent eO 2 E; our proof still works for extensions without bounded linear section.

5.3 Basic properties

201

Let e 2 E be any lifting of eP 2 Q, that is, p.e/ D e. P Then x WD e  e 2 belongs to N . We need an idempotent eO 2 E with p.e/ O D eP or, equivalently, e  eO 2 N . Our Ansatz is eO D e C .2e  1/  '.x/ for some analytic power series ' without constant term. We compute   eO 2  eO D '.x/2 C '.x/  .1  4x/  x: Since 1  4x 2 1 C N is invertible, we can rewrite the condition eO 2 D eO as '.x/2 C '.x/ 

x D 0: 1  4x

Since we want no constant term, we must choose the solution r 1 X 2n1 n 1 1 x 1 1 'D C x : C D  C .1  4x/1=2 D n 2 4 1  4x 2 2 nD1 Notice that this power series has radius of convergence 1=4 > 0. Thus eO WD e C .2e  1/

1 X 2n1 n

.e  e 2 /n

nD1

is a well-defined element of E. By construction, eO is idempotent and eO  e 2 N . The Chern–Connes character for idempotents uses explicit idempotent liftings for the extension JA ! T A ! A. We choose e D A .e/ P 2 T A, so that e  e 2 D deP de. P Hence 1 X 2n1 (5.51) .2eP  1/.de/ P 2n : eO D eP C n nD1

Proposition 5.52. An algebra A is analytically quasi-free if and only if AC is. Proof. Suppose first that A is analytically quasi-free. Let N  E  AC be a semi-split analytically nilpotent extension of AC . Since 1 is analytically quasi-free by Proposition 5.50, we may lift 1 2 AC to an idempotent eO 2 E. We get another semi-split analytically nilpotent extension eN O eO ! eE O eO  AC . Since A is assumed analytically quasi-free, we may lift the embedding A ! AC to a bounded algebra homomorphism A ! eE O e. O Since the latter algebra is unital with unit e, O we may extend this to a unital homomorphism AC ! eE O eO  E; this is the required section, so that AC is analytically quasi-free. Conversely, suppose that AC is analytically quasi-free. Let N  E  A be an analytically nilpotent extension. Then so is N  EC  AC . The latter splits by a bounded algebra homomorphism  W AC ! EC , which must map A into E, so that it splits the original extension as well.

202

5 Analytic cyclic homology and analytically nilpotent extensions

It is easy to see that analytic quasi-freeness is hereditary for free products. For the same reason, Qan .A/ is analytically quasi-free if A is, and an .A; B/ is analytically quasi-free if A and B are; this follows from the universal properties in Theorem 5.18 and Theorem 5.21. Proposition 5.53. Let F be a finite or countable set and let .Ai /i2F be a set of unital, L analytically quasi-free algebras. Then i2F Ai is analytically quasi-free. L Proof. Let A WD i2F Ai . We construct a section for an analytically nilpotent extension N  E  A. We may assume F D N or F D f1; : : : ; ng for some n 2 N. We write down the proof for F D N, the other case is similar and slightly simpler. For each i 2 N, let ei 2 Ai  A be the unit element. These idempotents are orthogonal, that is, ei ej D 0 for all i ¤ j . First we lift .ei /i2I to a sequence .eOi /i2I of orthogonal idempotents in E. We construct eOi by induction on i . By Proposition 5.50, we may lift e1 2 A to an idempotent element eO1 2 E. Suppose that we already have the orthogonal idempotents eO1 ; : : : ; eOi . Then pO WD eO1 C    C eOi is idempotent as well. 0 The O  p/ O  E is an analytically nilpotent extension of L1corner E WD .1  p/E.1 0 nDiC1 Ai . Now we use Proposition 5.50 to lift eiC1 to an idempotent eOiC1 in E , so that eOiC1 pO D 0 D pO eOiC1 . Thus eOiC1 is orthogonal to eO1 ; : : : ; eOi . Observe that eOi E eOi is an analytically nilpotent extension of Ai for each i. Since Ai is analytically quasi-free, we get L sections i W Ai ! eOi E eOi . These combine to a bounded algebra homomorphism i W A ! E because i .Ai /j .Aj / D 0 for all i ¤ j by construction. Warning 5.54. Direct sums of non-unital analytically quasi-free algebras need not remain analytically quasi-free. Consider the non-unital tensor algebra A D T1 Š t 1Œt , which is analytically quasi-free because it is a tensor algebra. If A˚A Š x 1Œx˚y 1Œy were analytically quasi-free, then so would be .A ˚ A/C Š 1Œx; y=.xy; yx/. This is the coordinate ring of an affine variety with a singularity at the point .0; 0/. Therefore, .A ˚ A/C is not even quasi-free: the nilpotent extension 1Œx; y=.xy; yx/2  1Œx; y=.xy; yx/ does not split. L Warning 5.55. Proposition 5.53 yields that n2N C is analytically quasi-free. In contrast, completed direct sums such as S.N/ or S ! .N/ with pointwise product are quasi-free but not analytically quasi-free because computations as in §2.4 show that       HP0 S.N/ Š S.N/; HA0 S.N/ Š HL0 S.N/ Š CŒN are different; this is impossible for analytically quasi-free algebras. Notation 5.56. The following algebras are related to invertible elements, isometries, and partial isometries in -algebras. The universal unital algebra generated by an invertible element is the algebra 1Œu; u1  of Laurent polynomials, which we also denote by Inv (for invertible). The universal unital algebra generated by two elements v and w subject to the relation wv D 1 is the Toeplitz algebra, which we denote by Iso (for isometry). We let Piso

5.3 Basic properties

203

(for partial isometry) be the universal unital algebra with two generators v and w and relations vwv D v and wvw D w. These algebras come equipped with canonical homomorphisms Piso  Iso  Inv. We augment them by the unique homomorphisms that send the generators v; w or u to 1 2 1 and let Inv0 , Iso0 , and Piso0 be their augmentation ideals. Theorem 5.57. The algebras Inv, Iso, and Piso and their augmentation ideals Inv0 , Iso0 , and Piso0 are analytically quasi-free. The same holds for the matrix algebras Mn for n 2 N [ f1g. Proof. We first prove the assertion for Inv. Let N  E  Inv be an analytically nilpotent extension. Since 1 is analytically quasi-free, we can lift the identity element in Inv to an idempotent e 2 E. We may replace E by the unital subalgebra eEe. Hence we may assume that E and the map E ! Inv are unital. Let V; W 2 E be liftings P1 1 of u and u . Then 1  W V 2 N , so that the geometric series nD0 .1  W V /n converges. Its limit is inverse to 1  .1  W V / D W V , so that W V is invertible. A similar argument shows that V W is invertible. Therefore, V has both a left and a right inverse and must be invertible. Now un 7! V n for n 2 Z defines the required section for our extension. Almost literally the same argument works for Iso. For Piso, we notice first that vw and wv are idempotents. Since 1 is analytically quasi-free, we can lift them to idempotents e; f 2 E. Then we lift v 2 vw Piso wv and w 2 wv Piso vw to elements V; W in eEf and f Ee. Their liftings satisfy V W 2 eEe and V W  e 2 N . Hence V W is invertible in the unital algebra eEe. The map v 7! V , w 7! W .V W /1 provides the desired lifting. The algebra Mn is generated by the orthogonal idempotents ei on the diagonal together with the matrix units ei;i C1 . To lift a representation of Mn , we first lift the orthogonal idempotents ei using Proposition 5.53. The elements v D ei;iC1 and w D ei;iC1 satisfy the relations for Piso. Now we proceed as above, but using the liftings of ei D vw and eiC1 D wv previously constructed. The formula for the Chern–Connes character for invertibles uses explicit liftings of inverses in the extension TC A  AC . Let xP 2 A be such that 1 C xP 2 AC is invertible and write its inverse in the form 1 C yP with yP 2 A; let x WD A .x/ P and y WD A .y/; P in our previous notation, we have V D 1 C x, W D 1 C y, and 1  V W D dxP dy. P We find that 1 C x is invertible in TC A with inverse .1 C x/1 D

1 X nD0

W .1  V W /n D

1 X

.1 C y/.dxP dy/ P n:

(5.58)

nD0

Proposition 5.59. If K  C is a compact subset, then the algebra O.K/ of germs of holomorphic functions near K .defined in Example 3.13) is analytically quasi-free. Proof. If N  E  O.K/ is an analytically nilpotent extension, then E is locally multiplicative by Theorem 3.20. Let x 2 E be any lifting of the identical function

204

5 Analytic cyclic homology and analytically nilpotent extensions

K ! C. Since the projection E  O.K/ is isoradial, the spectrum of x is K by Lemma 3.31. Hence the functional calculus for x (Theorem 3.15) provides the desired lifting homomorphism. Example 5.60. If K  C n for n  2, then O.K/ may have non-trivial higher Hochschild homology, which prevents it from being (analytically) quasi-free. Example 5.61. For the circle T 1  C, we have O.T 1 / D C ! .T 1 /, so that C ! .T 1 / is analytically quasi-free. I expect that C 1 .T 1 / is not, but I do not know how to prove this. Example 5.62. There are also versions of Proposition 5.59 for the Toeplitz algebra Iso: let Iso.rv ; rw / for rv ; rw > 1 be the unital complete bornological algebra whose representations correspond to pairs v; w with wv D 1 and %.v/ < rv , %.w/ < rw . Then Iso.rv ; rw / is analytically quasi-free; the argument is similar to the proof of Proposition 5.59. I do not know whether completions of M1 such as the smooth compact operators KS are analytically quasi-free. Warning 5.55 makes me suspicious about this, but since HA.KS / 1, I do not know how to prove that KS is not analytically quasi-free. Application to analytic cyclic homology. We view 1 as a chain complex supported in degree 0 with vanishing boundary map. Theorem 5.63. We have HA.1/ 1, so that HA0 .1/ Š 1 and HA1 .1/ D 0. Let e denote the unit element of 1, then HA0 .1/ Š 1 is generated by the class of eC

1 X 2n1 n

.2e  1/. de/2n 2 T 1 Š even an 1:

nD1

We have HA.Inv0 / 1Œ1, so that HA0 .Inv0 / D 0 and HA1 .Inv0 / Š 1. A generator for HA1 .Inv0 / is given by the class of 1 X

odd 0 0 x odd u1 du.du1 du/n 2  an .Inv/ Š an .Inv / Š X1 .T Inv /:

nD0

We have HA.Iso0 / 0, so that HA0 .Iso0 / D 0 D HA1 .Iso0 /. Analogous statements hold for HL. Proof. Proposition 5.50 and Theorem 5.57 assert that 1, Inv0 , and Iso0 are analytically quasi-free, so that HA.A/ X.A/ for all of them by Corollary 5.48. Since 1 .1/ is spanned by commutators, we get  X.1/ Š 1 as asserted. We get a generator by pushing the generator e of H0 X.1/ forward along a section 1 ! T 1 for 1 . The section in (5.51) yields the formula for a generator of HA0 .1/ (see also [26, §12]).

5.3 Basic properties

205

Since Inv is smooth and commutative, the commutator quotient of 1 .Inv0 / D x 1 .Inv/=Œ ;  Š CŒu; u1  du; the  .Inv/ is the space of Kähler differentials, that is,  0 isomorphism is given by f dg 7! f g du. The boundary map X1 .Inv0 / ! X0 .Inv0 / vanishes, the other one is equivalent to the map x1

CŒu; u1 0 ! CŒu; u1 ;

f 7! f 0 :

It is injective because CŒu; u1 0 is lacking constant functions, and its cokernel is spanned by the form u1 du. Hence HA0 .Inv0 / D 0 and HA1 .Inv0 / Š 1. We get a generator for HA1 .Inv0 / by pushing forward the generator u1 du of H1 X.Inv0 / along a section for T Inv0  Inv0 . Equation (5.58) yields a recipe for such a section and implies our formula for the generator of HA1 .Inv0 /; if we insist on using unreduced differential forms, we must replace du and du1 by d.u  1/ and d.u1  1/ (see also [26, §12]). We leave the computation of X.Iso0 / as an exercise. The assertions about analytic cyclic homology and cohomology follow immediately from those about HA. We also get the following special case of the Excision Theorem:   Theorem 5.64. The canonical map X.T A/ ˚ 1 ! X T .AC / is a chain homotopy equivalence. This implies HA.AC / HA.A/ ˚ 1 or HL.AC / HL.A/ ˚ 1 and corresponding isomorphisms for homology and cohomology. Proof. The algebra TC A WD .T A/C is analytically quasi-free by Proposition 5.52. Now apply Theorem 5.47 to the algebra extension JA  TC A  AC and use (4.77) to get   X T .AC / X.TC A/ X.T A/ ˚ 1: Combining Theorems 5.63 and 5.64, we get HA.Inv/ 1 ˚ 1Œ1;

HA.Iso/ 1:

5.3.2 Stability with respect to algebras of nuclear operators We prove that analytic and local cyclic homology are stable with respect to algebras of nuclear operators, using a method that goes back to [1], [31], [62]. y V ! 1, the multiplication Given a map f W W ˝ yf ˝ y idW W V ˝ yW ˝ yV ˝ y W !V ˝ yW idV ˝ y W into an associative algebra (see also §A.7.1); the notation ` reminds turns ` WD V ˝ us that the algebra of trace class operators on Hilbert space is an example of this y D construction. We suppose that there are maps W 1 ! W ,  W 1 ! V with f ı. ˝/ y W 1 ! ` is called the corner embedding associated to ; . id1 . The map WD  ˝ It is an algebra homomorphism. In terms of elements, and  correspond to 2 V , y /.  2 W with f . ; / D 1, and the corner embedding is defined by .x/ D x  . ˝

206

5 Analytic cyclic homology and analytically nilpotent extensions

y y Theorem 5.65. The corner embedding  A WD idA  ˝ W A ! A ˝ ` induces a chain y `/ . Thus it is an HA-equivalence for homotopy equivalence X.T A/ X T .A ˝ ! algebras in Cborn and an HL-equivalence for algebras in Ban. Proof. We need an inverse A for X.T A / up to chain homotopy. The main ingredient y W Š y W , which is defined by  WD f ı ˆV;W W V ˝ is the canonical trace on V ˝ y V ! 1, that is, .x ˝ y/ WD f .y ˝ x/. W ˝ It is easy to check that ` is locally multiplicative. Hence we get a canonical algebra y `/ ! .T A/ ˝ y ` by Theorem 5.28, which induces a chain map homomorphism T .A ˝     y `/ ! X .T A/ ˝ y` : X T .A ˝   y ` ! X.T A/ by the formulas Next, we map X .T A/ ˝ y 7!  .T /!; ! ˝T

y T / 7! .T / d!; !0 ˝T y 0 d.!1 ˝ y T1 / 7!  .T0 T1 /!0 d!1 d.! ˝

for !; !0 ; !1 2 T A, T; T0 ; T1 2 `. This is a well-defined chain  is a  map because  y trace. Putting things together, we get a natural chain map A W X T .A˝`/ ! X.T A/. We claim that A is inverse to X.T A / up to chain homotopy. Since  ı X.T / D id1 , we get A ı X.T A / D 1. To deal with X.T A / ı A , we use the two embeddings y ˝ y id` ; 1 WD idA ˝

y id` ˝ y W A˝ y `!A˝ y `˝ y `: 2 WD idA ˝

The naturality of  implies A˝` y ıX.T 1 / D X.T A /ıA ; the formula A ı X.T A /D1 y applied to A ˝ ` yields A˝` y ı X.T 2 / D id. We claim that the algebra homomorphisms 1 and 2 are polynomially homotopic and therefore yield chain homotopic chain maps by Theorem 4.27. This claim will finish the proof because then X.T A / ı A D A˝` y ı X.T 1 / A˝` y ı X.T 2 / D 1. y H , so that It remains to construct the homotopy 1 2 . It is of the form idA ˝ we may drop A from now on. We need some preparation to write down the homotopy. y f , and y V and W2 WD W ˝ y W with bilinear map f2 D f ˝ Let V2 WD V ˝ y W2 be the associated biprojective algebra. Exchanging two copies let `2 D V2 ˝ y `. We call a pair of maps j V W V ! V2 , of V and W , we may identify `2 Š ` ˝ W V y W j W W ! W2 with f2 ı .j ˝ j / D f an isometry from f to f2 . Any isometry y ` Š `2 , then the induces an algebra homomorphism ` ! `2 . When we identify ` ˝ maps 1 and 2 are of this form: we get them from the isometries i1 D Œi1V ; i1W  and i2 D Œi2V ; i2W  with y  W V ! V2 ; i1V WD idV ˝

y idV W V ! V2 ; i2V WD  ˝

y W W ! W2 ; i1W WD idW ˝

y idW W W ! W2 : i2W WD ˝

We construct a polynomial homotopy of isometries from i1 to i2 ; this yields the desired polynomial homotopy of algebra homomorphisms between 1 and 2 . The embeddings  W 1 ! V , W 1 ! W split via f . ; / and f .; /, so that we can decompose V Š 1 ˚ V? , W Š 1 ˚ W? . This induces decompositions y

V2 Š 1 ˚ V? ˚ V? ˚ V?˝2 ;

y

W2 Š 1 ˚ W? ˚ W? ˚ W?˝2 :

5.3 Basic properties

The isometries i1 and i2 correspond to the block matrices 0 1 0 1 0 1 B0 1C B0 V W V W C B i1 i1 B @0 0A ; i2 i2 @0 0 0 0

207

1 0 0C C 1A 0

with respect to these decompositions; here 1 stands for an identity map. y y We can forget about the blocks 1, V?˝2 and W?˝2 . Thus our task is to find a polynomial homotopy of isometries between the two isometries .V? ; W? / ! .V?2 ; W?2 / defined by the pairs of matrices    

   

1 1 0 0 ; ; ; : 0 0 1 1 We get this by concatenating the following three homotopies (which run from 0 to 1, respectively) 

  

      

 1t 1t 1 0 1 1 ; : ; ; ; ; 1 1 1 0 t t You should check that these are indeed homotopies of isometries. We have been careful to construct a polynomial homotopy because this allows the same argument to go through for the periodic cyclic homology of algebras in any Q-linear symmetric monoidal category, as claimed in §4.2. In contrast, our argument does not apply to Hochschild and cyclic homology because it uses homotopy invariance.

5.3.3 Finite additivity First we recall the well-known characterisations of additive functors with respect to finite sums (or products). Let H be a functor defined on the category of algebras and taking values in some additive category. Definition 5.66. We say that H is additive .on morphisms/ if H.f C g/ D H.f / C H.g/ for algebra homomorphisms f; g whenever f C g is again an algebra homomorphism; we say that H is additive .on objects/ if the coordinate projections induce an isomorphism H.A  B/ Š H.A/  H.B/ for all algebras A; B. Lemma 5.67. Both notions of additivity are equivalent. Proof. Suppose that H.f C g/ D H.f / C H.g/ whenever f; g; f C g are algebra homomorphisms. It follows that H.0/ D 0. Thus H maps the diagram A  AB  B of coordinate embeddings and projections to a direct sum diagram in the target additive category; hence H.A  B/ Š H.A/  H.B/.

208

5 Analytic cyclic homology and analytically nilpotent extensions

Conversely, suppose H.A  B/ Š H.A/  H.B/. If f; g; f C g are algebra homomorphisms A ! B, then we can define an algebra homomorphism .f; g/ W A  A ! B that restricts to f and g on the two factors. Then f C g D .f; g/ ı , Š

! H.A/  H.A/, the where W A ! A  A is the diagonal map. Since H.A  A/  homomorphism .f; g/ induces a map H.f; g/ W H.A/  H.A/ ! H.B/;   which is equal to H.f /; H.g/ because we know its restrictions to the two factors. Similarly, we identify H. / with .idH.A/ ; idH.A/ /t W H.A/ ! H.A/  H.A/. Composing, we find H.f C g/ D H.f / C H.g/. Proposition 5.68. The functors HA and HL are additive as functors to the homotopy ! categories of chain complexes HoKom.CbornI Z=2/ and HoKom.BanI Z=2/, respectively; so are the resulting .bivariant/ homology and cohomology theories. Proof. This is a simple special case of the Excision Theorem below. The following argument due to Masoud Khalkhali ([62]) only uses homotopy invariance and stability. Let A and B be two algebras. Then the two embeddings   a 0 j1 W A  B ! M2 .A  B/; .a; b/ 7! ; 0 b   .a; b/ 0 j2 W A  B ! M2 .A/  M2 .B/; .a; b/ 7! ; 0 0 are polynomially homotopic by rotating in B and doing nothing in M2 .A/. Hence the induced chain maps X.T  j1 / and X.T j2/ are chain homotopic by Theorem 5.45. Let B W X T M2 .A  B/ ! X T .A  B/ be the explicit trace map constructed during the proof of Theorem 5.65. Since j2 is the corner embedding, we have B ı X.T j2 / D 1. Since the range of j1 only involves diagonal matrices, we get         B ı X.T j1 / D X T .iA ı pA / C X T .iB ı pB / W X T .A  B/ ! X T .A  B/ ; and pA and pB are the coordinate where iA and iB are the coordinate  embeddings  projections. Thus X.T A/  X T .A  B/  X.T B/ is a direct sum diagram in the appropriate homotopy category of chain complexes.

5.3.4 Infinite direct sums Now we turn to infinite direct sums of algebras. Here the case of the local theory is easier and yields better results, so that we treat it first. We view HL as a functor from ! ! Alg.Ban/ to Kom.Ban/, not to some derived category of chain complexes. As such, it is compatible with direct limits, that is, we have     HL lim.Ai /i2I Š lim HL.Ai / i2I (5.69) ! !

5.3 Basic properties

209

! for any inductive system in Alg.Ban/. This follows immediately from the definition. ! Theorem 5.70. Let .Ai /i2I be a set of algebras in Ban. Then the canonical chain map M M  HL.Ai / ! HL Ai i2I

i2I

is a local chain homotopy equivalence and hence induces isomorphisms M M  Y M  Ai ; HL .Ai / Š HL Ai ; HL .Ai / Š HL i2I

L 



Q

i2I

i2I

i2I

HL .Ai ; B/ for all B.  L  We get no formula for HL B; Ai except for special B. L Proof. For a subset F  I , we let AF WD i2F Ai . An infinite direct sum is a limit of finite direct sums, that is, M M lim HL.AF /; HL.Ai / Š lim HL.Ai /: HL.AI / Š ! ! and HL

Ai ; B Š

F  I finite

i2I

F  I finite i2F

For any F , finite or not, the maps Ai ! AF for i 2 LF induce chain maps HL.Ai / ! HL.AF /, which combine to a canonical chain map i2F HL.Ai / ! HL.AF /. If F is 5.68) shows that this map is a chain finite, then the finite additivity of HL (Proposition L homotopy equivalence. Hence the map i2I HL.Ai / ! HL.AI / is an inductive

limit of chain homotopy equivalences and thus a local chain homotopy equivalence by Lemma 2.24. ! Theorem 5.71. The following assertions hold for inductive systems in Alg.Ban/: ! • Let fi W Ai ! Bi be an inductive system of algebra homomorphisms in Ban. If all fi are HL-equivalences, so is lim fi W lim Ai ! lim Bi . ! ! ! • If Ai is HL-contractible for all i 2 I , then so is lim Ai . ! • HL .lim Ai / Š lim HL .Ai /. ! ! • There is a spectral sequence that relates HL .lim Ai ; B/ to the values of the right ! derived functors of lim on the projective system i 7! HL .Ai ; B/. .It need not  converge in general because of the usual obstructions for cohomological spectral sequences./

Proof. The first two assertions follow from Lemmas 2.24 and 2.23, the third holds ! because the functor lim on Ban is exact. To get the last one, we use that X.T lim Ai / ! ! is locally chain homotopy equivalent to ho-lim X.T Ai /. The homotopy direct limit ! is a bicomplex by construction, so that we get a canonical spectral sequence. Using Hom.ho-lim Ci ; B/ Š ho-lim Hom.Ci ; B/, we can identify the E2 -term of the spectral !  sequence with R HL .Ai ; B/.

210

5 Analytic cyclic homology and analytically nilpotent extensions

Analytic cyclic homology for countable direct sums Proposition 5.72. Let .Ai /i2N be a sequence of complete bornological algebras. Then the natural chain map M M  HA.Ai / ! HA Ai i2N

i2N

is a chain homotopy equivalence and hence induces isomorphisms M

HA .Ai / Š HA

M

i2N

and HA

L

 Ai ;

i2N

Y

HA .Ai / Š HA

i2N

M

 Ai ;

i2N

 Q Ai ; B Š HA .Ai ; B/.

Proof. The proof is almost the same as for the local theory. However, some extra care is needed Lto get a global chain homotopy equivalence. Letting Bn WD A1 ˚    ˚ An , we get Ai D lim Bn and hence ! M  HA Ai Š lim HA.Bn /: ! n

i2N

Finite additivity of HA yields that the canonical chain map HA.Bn / ˚ HA.AnC1 / ! HA.BnC1 / is a chain homotopy equivalence. This map is part of a semi-split extension of chain complexes HA.Bn / ˚ HA.AnC1 /  HA.BnC1 /  Cn :

We conclude from the Puppe Exact Sequence that Cn is contractible. Therefore, the above extension splits by a chain map, so that we can write HA.BnC1 / Š HA.Bn / ˚ HA.AnC1 / ˚ Cn :

Passing to the inductive limit, we get HA.lim Ai / Š

!

HA.Ai / ˚

i2N

Since all Ci are contractible, so is HA

M

L

M i2N

i2N

M

Ci :

i2N

Ci . Thus

 M Ai HA.Ai / i2N

as desired. The assertions about analytic cyclic homology and cohomology follow.

5.3 Basic properties

211

5.3.5 Exterior products Now we consider exterior products in analytic and local cyclic homology. The existence of such products is due to Michael Puschnigg [84]. The arguments are literally the same in both cases, so that we only write them down for the analytic theory. We have explained the corresponding results for periodic cyclic homology in detail in §4.4. But in the analytic and local theory, we do not get exterior products in complete generality. Before we explain what goes wrong, we inspect those parts of the argument in §4.4 that carry over without change. Lemma 5.73. There is a natural chain homotopy equivalence y B/ HA.A/ ˝ y HA.B/ HA.A ˝ provided that both A and B are locally multiplicative and the canonical projection HA.T A  T B/  X .2/ .T A  T B/ is a chain homotopy equivalence. Proof. Proposition 4.79 yields   y HA.B/ y .T B/ X.T A/ ˝ y X.T B/ HA.A/ ˝ HP .T A/ ˝ because T A and T B are analytically quasi-free. As in the proof of Proposition 4.82, y .T B/  A ˝ y B Goodwillie’s Theorem shows that the canonical projection .T A/ ˝ is an HA-equivalence, that is,   y B/ HA .T A/ ˝ y .T B/ : HA.A ˝ Here we need that A and B are locally multiplicative and Theorem 5.28 to conclude y JB ˚ .JA/ ˝ y B is analytically nilpotent. that A ˝ y .T B/. As in the Thus it remains to compare HA and HP for the algebra .T A/ ˝ proof of Proposition 4.79, we see that the canonical map     y .T B/ ! HP .T A/ ˝ y .T B/ HA .T A/ ˝ is a chain homotopy equivalence if and only if the corresponding map     HA .T A/  .T B/ ! HP .T A/  .T B/ is one. Since the latter algebra has bidimension 2, the canonical projection     HP .T A/  .T B/ ! X .2/ .T A/  .T B/ is a chain homotopy equivalence by Theorem A.123. This yields the assertion. If an algebra D has bidimension 2, there is a connection on 2 .D/. The proof of Theorem A.123 uses such down an explicit contracting homotopy  a connection  Q to write n for the chain complex b 3 .D/  1  .D/. In some cases, contracting homonD3 topies of this form are bounded for the bornology S" .D/ defined in Notation 2.7 and

212

5 Analytic cyclic homology and analytically nilpotent extensions

therefore yield HA.D/ X .2/ .D/ as well. We have seen in Theorem 2.50 that this is the case if D is a Banach algebra. If D is an analytic tensor algebra and the connection is the standard one, then this is checked in [79, Proposition 4.3]). In our case, Michael Puschnigg writes down an explicit formula for a connection ([84, Proposition 4.3]); if A and B are locally multiplicative, he checks that the resulting contracting homotopy is bounded for the S" .D/. Already the formula for the connection is very complicated, and the boundedness computation is quite messy, so that we do not reproduce it here. It seems desirable to find a more conceptual proof. Even if the proof is messy, it yields the following theorem: Theorem 5.74 (Puschnigg). There are chain homotopy equivalences   X.T A1 / ˝ X.T A2 / X T .A1 ˝ A2 / for locally multiplicative algebras A1 and A2 , which are natural, associative, symmetric, monoidal, and compatible with boundary maps from the Excision Theorem 5.77 as explained in Theorem 4.74. These isomorphisms induce exterior product operations y A2 ; A01 ˝ y A02 /; HA .A1 ; A01 / ˝ HA .A2 ; A02 / ! HA .A1 ˝ y A2 ; A01 ˝ y A02 / HL .A1 ; A01 / ˝ HL .A2 ; A02 / ! HL .A1 ˝ for locally multiplicative algebras, which are associative, graded commutative, monoidal, and compatible with boundary maps. Although I have no counterexample, I expect that this theorem becomes false for algebras that are not locally multiplicative, for the following reason. Let A be analytiy HL.B/ 0 for any B; but cally nilpotent. Then Goodwillie’s Theorem yields HL.A/ ˝ y y B/ 0. since A ˝ B need not be analytically nilpotent, it is unclear whether HL.A ˝

5.4 Excision in analytic and local cyclic homology Now we turn to the Excision Theorems for bivariant local and analytic cyclic homology. As in the case of periodic cyclic homology, we prefer to state this in terms of chain complexes and introduce relative versions for this purpose. Recall that we redefine HA.A/ WD X.T A/. Definition 5.75. We let HArel .f / be the desuspended mapping cone   HArel .f / WD cone X.T f / W X.T A/ ! X.T B/ Œ1 for a morphism f W A ! B in Alg.Cborn/; this is part of an exact triangle HA.f /

HArel .f / ! HA.A/ ! HA.B/ ! HArel .f /Œ1

5.4 Excision in analytic and local cyclic homology

213

in HoKom.CbornI Z=2/. The relative analytic cyclic homology and cohomology are     HArel .f / WD H HArel .f / I HArel  .f / WD H HArel .f / ; rel relative bivariant theories HArel  .D; f / and HA .f; D/ are defined similarly. ! Replacing complete bornological algebras by algebras in Ban everywhere, we get the .relative/ local cyclic chain complex HLrel .f / and relative versions HLrel  .f /, rel HLrel .f /, HLrel  .D; f /, and HL .f; D/ of local cyclic homology and cohomology; ! here we use the local (bivariant) cohomology of chain complexes in Ban as in Definition 2.33.

As in §4.3, the Puppe Exact Sequence provides various cyclic six-term exact se rel rel quences involving HArel  .f /, HArel .f /, HA .D; f /, HA .f; D/, and similarly for the local theory. Our goal is to prove the following two Excision Theorems: p

i

Theorem 5.76. Let K  E  Q be a semi-split extension of complete bornological algebras. Then the canonical map HA.K/ ! HArel .p/ is a chain homotopy equivalence. This yields a natural isomorphism HA .K/ Š HArel  .p/ and an exact sequence i / HA0 .E/ p / HA0 .Q/ HA0 .K/ O @

HA1 .Q/ o

p

HA1 .E/ o



i

@

HA1 .K/

with natural connecting maps @ W HA .Q/ ! HAC1 .K/; similar statements hold for analytic cyclic cohomology and bivariant analytic cyclic homology in both variables. i

p

Theorem 5.77. Let K  E  Q be a locally semi-split extension of algebras in ! Ban .see Definition 2.35). Then the canonical map HL.K/ ! HLrel .p/ is a local chain homotopy equivalence. Hence HL .K/ Š HL .p/ and there is an exact sequence HL0 .K/ O

i

/ HL0 .E/

p

@

HL1 .Q/ o

p

HL1 .E/ o

/ HL0 .Q/ 

i

@

HL1 .K/

with natural connecting maps @ W HL .Q/ ! HLC1 .K/; similar statements hold for local cyclic cohomology and bivariant local cyclic homology in both variables. Our proof of Theorem 5.76 appeared previously in [66]; another proof is due to Michael Puschnigg ([86]). We have already explained a large portion of the proof in §4.3 in connection with the Excision Theorem for the periodic cyclic theory. There is only one step that requires a new idea: we must show that the analogue L of the

214

5 Analytic cyclic homology and analytically nilpotent extensions

 ideal L is analytically quasi-free, not just quasi-free. Before we come to this, we briefly discuss those parts of the proof that carry over easily. There is a variant of Theorem 5.77 that requires a global linear section and yields a global chain homotopy equivalence as in Theorem 5.76. The proofs for Theorem 5.76 and this variant are essentially identical and differ only in the category of algebras in which we work. We first study the case where we have a global section. Then we explain in §5.4.4 how to relax this hypothesis.

5.4.1 The common part of the proof We assume until §5.4.4 that the extension is semi-split with linear section s W Q ! E. All results in §4.3.1–§4.3.3 and Theorem 4.67 carry over to the analytic and local cyclic theory with obvious changes. We only sketch the main lines of this part of the proof. Lemma 5.78. The maps an p W an E ! an Q and an i W an K ! an E are a split ! epimorphism and a split monomorphism in Cborn or Ban, respectively. The same formulas as in the proof of Lemma 4.46 provide linear sections sL ; sR W HA.Q/ ! HA.E/ for HA.p/, which will be used much in the following. It is trivial to check that these maps are bounded or morphisms of inductive systems, respectively. Recall that the multiplication in E has no effect on the analytic bornology (or the inductive system)   that underlies an E. As in §4.3.1, we conclude that HArel .p/ ker HA.p/ . We also get thesame strengthening of the Excision Theorem: if Excision holds at all, then ker HA.p/ Š HA.K/ ˚ C with a contractible chain complex C . Corollary 5.41 yields a decomposition y E Š .TC E ˝ y K/ ˚ .TC E ˝ y Q/ T E Š TC E ˝

(5.79)

analogous to (4.52). We let L  I  T E be the closed left ideal and the closed two-sided ideal generated by K, respectively, so that y K; L Š TC E ˝

T E Š I ˚ T Q:

We can describe L more explicitly as in Lemma 4.55: it is generated by monomials e0 de1 : : : de2n with e2n 2 K. As before, we get an isomorphism y TC Q; TC E Š LC ˝ that is, TC E is isomorphic to the free left L-module on TC Q. The isomorphism is the same as in Lemma 4.56. Its boundedness is easy to check. Moreover, L is free as a left or right L-module as in (4.59) and Lemma 4.60. Again, it is easy to see that the formulas in the proof of Lemma 4.60 also define an

5.4 Excision in analytic and local cyclic homology

215

y LC Š L, where G is the closed linear span of monomials of the isomorphism G ˝ form ds.q1 / ds.q1 / : : : ds.q2n /  k

or

ds.q1 / ds.q1 / : : : ds.q2n1 / dk:

The proofs of Theorem 4.66 and 4.67 only use the previously constructed isomorphisms and standard facts about quasi-freeness and Hochschild homology that hold for algebras in any symmetric monoidal category. Hence we get: Theorem 5.80. The algebra L is quasi-free, and the canonical map   HH.L/ ! ker HH.T p/ HHrel .T p/ is a chain homotopy equivalence. So are the maps   HC.L/ ! ker HC.T p/ HCrel .T p/ and   X.L/ ! ker X.T p/ Xrel .T p/: The algebra L\JE is analytically nilpotent because it is a left ideal in JE – which is analytically nilpotent by Theorem 5.11 – and analytic nilpotence is hereditary for left ideals (Theorem 3.75). The following theorem will be proved in §5.4.3: Theorem 5.81. The algebra L is analytically quasi-free. Hence L \ JE  L  K is a semi-split, analytically quasi-free extension of K with analytically nilpotent kernel. Now we can complete the proof of the Excision Theorem as in §4.3.4 by appealing to Theorem 5.47.

5.4.2 The boundary map and its naturality We describe the boundary maps in the long exact sequences in Theorem 5.76 and explain in what sense they are natural. Let  W HA.Q/ ! HA.E/ be a bounded linear section as in Lemma 5.78. Then @ ı  C  ı @ W HA.Q/ ! HA.E/Œ1   with a chain is a chain map whose range is contained in ker HA.p/  . Composing  homotopy inverse for the embedding HA.K/ ! ker HA.p/ (whose existence we still have to prove), we get a chain map HA.Q/ ! HA.K/Œ1 which defines Œ@ 2 HA1 .Q; K/. This class only depends on the given algebra extension and is independent of   our auxiliary choices: another choice for  differs by a map HA .Q/ ! ker HA.p/ and   therefore leads to a chain homotopic map HA.Q/ !  ker HA  .p/ Œ1, and all chain homotopy inverses for the embedding HA.K/ ! ker HA.p/ are chain homotopic.

216

5 Analytic cyclic homology and analytically nilpotent extensions

The boundary maps in the Puppe Exact Sequence of Theorem A.18 are given, up to signs, by composition with Œ@ (see [27]). The usual Koszul sign rules dictate that the homological boundary maps HAn .D; Q/ ! HAn1 .D; K/;

HAn .Q/ ! HAn1 .K/

are composition with .1/n Œ@, whereas the cohomological boundary maps HAn .K; D/ ! HAnC1 .Q; D/;

HAn .K/ ! HAnC1 .Q/

are composition with Œ@ regardless of n (see also [27]). Naturality means the following. Consider a morphism of extensions K1 / 

˛

K2 /

/ E1 ˇ

 / E2

/ / Q1

 / / Q2 .

Then the diagram HA.Q1 /

@1

/ HA.K1 /Œ1

@2

 / HA.K2 /Œ1

HA. /

 HA.Q2 /

HA.˛/Œ1

commutes up to chain homotopy. This follows because the chain maps HA.Q/ ! HArel .p/Œ1

HA.K/Œ1

are evidently natural and the homotopy inverse of a chain homotopy equivalence is unique up to chain homotopy. Actually, the boundary map that is, in principle, constructed during our proof makes the above diagram commute exactly and not just up to chain homotopy. Although our proof yields, in principle, a construction of the boundary map, it is not useful for computations. Often naturality can be used to compute boundary maps efficiently, replacing the original extension by another one that is easier to handle (see, for instance, §5.4.5 or [76]). ! For a semi-split algebra extension in Ban, literally the same assertions hold for HL instead of HA. If the extension is only locally semi-split, we must be more careful. We still have a locally semi-split extension of chain complexes   ker HL.p/  HL.E/  HL.Q/:   This ensures that the canonical chain map ker HL.p/ ! HLrel .p/ is a local chain homotopy equivalence because the homotopy direct limit construction turns locally semi-split extensions of chain complexes into globally semi-split extensions.

5.4 Excision in analytic and local cyclic homology

217

The Excision Theorem 5.77 shows that the embedding HL.K/ ! HLrel .p/ is a local chain homotopy equivalence, so that it has an inverse in the local homotopy category  of chain complexes. Now we get a class Œ@ 2 HL1 .Q; K/ D H1loc HL.Q/; HL.K/ :

HL.Q/ ! HLrel .p/Œ1  HL.K/:

As above, this class is natural, and the boundary maps in the Puppe Exact Sequence are up to sign given by composition with Œ@.

5.4.3 Analytic quasi-freeness of L We are going to prove Theorem 5.81. This requires constructing an algebra homomorphism L ! T L that is a section for L W T L ! L. We will use something like a universal property of L to construct some algebra homomorphisms out of L. Notation 5.82. To distinguish the Fedosov product and the ordinary product of differential forms in T L  an an E, we denote the Fedosov product in T L by }. Recall that the letters k, e, q, and x always denote elements of K, E, Q, and L, respectively. The boundary map in L is denoted by D. Definition 5.83. For a complete bornological algebra B, we let M.B/  Hom.B; B/ be the subalgebra of right B-module endomorphisms, that is, left multipliers (we do not need the more complicated two-sided multipliers); we equip M.B/ with the equibounded bornology and the canonical bounded algebra homomorphism B ! M.B/ sending b 2 B to the map x 7! b  x. Lemma 5.84. Let fK W K ! B be a linear map and let fE W E ! M.B/ be a lanilcur, such that the diagram K

fK

i

 E

fE

/B  / M.B/

commutes. We also write e F x WD fE .e/.x/ for e 2 E, x 2 B. Then there is a unique bounded algebra homomorphism f W L ! B with f ı K D fK on K and f .E .e/ ˇ x/ D e F f .x/ for all e 2 E, x 2 L; equivalently, f is a left T E-module homomorphism for the left T E-module structure on B defined by hfE i W T E ! M.B/. Proof. The lanilcur fE induces a bounded homomorphism hfE i W T E ! M.B/, which we view as a T E; B-bimodule structure on B; we write ! F x WD hfE i.!/.x/. y K is a free left T E-module, fK W K ! B extends uniquely to Since L Š TC E ˝ a left T E-module homomorphism f W L ! B, ! ˇ k 7! ! F fK .k/. This map is an

218

5 Analytic cyclic homology and analytically nilpotent extensions

algebra homomorphism as well because     f .!1 ˇ k1 /  f .!2 ˇ k2 / D !1 F fK .k1 /  !2 F fK .k2 /    D !1 F fK .k1 /  !2 F fK .k2 / D !1 F k1 F !2 F fK .k2 / D .!1 ˇ k1 ˇ !2 / F fK .k2 / D f .!1 ˇ k1 ˇ !2 ˇ k2 /: Lemma 5.84 is useful to construct the homomorphism L ! T L. But many important homomorphisms such as the composite map L ! T L ! T K or the embedding L ! LC do not arise in this fashion. Hence we do not quite have a universal property. We let fK be the obvious linear map L ı K W K ! T L. This ensures that the homomorphism L ! T L restricts to the homomorphism 2i hA

T K ! T .T K/  T L: We always view K as a subset of L and T L in this fashion. Before we describe fE , we reformulate the problem. y TC L as a right T L-module. The universal Corollary 5.41 yields T L Š L ˝ property of free modules yields M.T L/ Š Hom.L; T L/; by adjoint associativity,   Hom E; Hom.L; T L/ Š Hom.2/ .E  L; T L/: Consequently, giving a bounded linear map fE W E ! M.B/ is equivalent to giving a bounded bilinear maps F W E  L ! T L; the only way to extend e F to a left multiplier is e F .x1 } x2 / D .e F x1 / } x2 . This yields e F .x0 Dx1 : : : Dx2n / D .e F x0 / } Dx1 : : : Dx2n D .e F x0 / Dx1 : : : Dx2n ; e F .Dx1 Dx2 : : : Dx2n / D e F .x1 ˇ x2  x1 } x2 / } Dx3 : : : Dx2n   (5.85) D e F .x1 ˇ x2 / Dx3 : : : Dx2n  .e F x1 / } x2 Dx3 : : : Dx2n : The compatibility between fK and fE dictates k F x D k } x D k ˇ x  Dk DxI We also require e F x  e ˇ x 2 JL for all e 2 E, x 2 L because this ensures that the resulting homomorphism L ! T L lifts the identity morphism on L. Warning 5.86. The tempting formula s.q/ F x D s.q/ ˇ x does not work because the curvature of the resulting map fE W E ! M.T L/ is not analytically nilpotent.

219

5.4 Excision in analytic and local cyclic homology

Construction of the map fE . The definition of F requires some preparation. Recall that L is a free right L-module on the closed linear span G of monomials of the form ds.q1 / ds.q1 / : : : ds.q2n /  k

or

ds.q1 / ds.q1 / : : : ds.q2n1 / dk

Š

y LC be the inverse of the multiplication map !G˝ (see Lemma 4.60). We let ˛ W L  y LC ! L. There is a standard recipe for connections on free modules. In our G ˝ case, this yields a connection y L 1 .L/ Š L  1 .L/  1 .L/ rW L ! L˝ by composing ˛ with the canonical map y LC  L ˝ y LC ! 1 .L/;

W G ˝

x0 ˝ x1 7! x0 Dx1 ;

with the convention D1 D 0; that is, r.g ˇ x/ WD g Dx and r.g/ D 0 for all g 2 G , x 2 L. The connection property r.x ˇ y/ D r.x/ ˇ y C x Dy

(5.87)

for all x; y 2 L follows because D W LC ! 1 .L/ is a connection. As usual, we extend r to a graded connection r W L  L ! L  L;

r.x0 Dx1 : : : Dxn / D r.x0 / Dx1 : : : Dxn I

this satisfies r.1 ˇ 2 / D r.1 / ˇ 2  .1/j1 j 1 D2 . We have r ı r D 0, that is, the standard connection on a free module is flat. Now we put e F x WD e ˇ x  Dr.e ˇ x/: (5.88) Since K  G , we have r.k ˇ x/ D k Dx, so that k F x D k ˇ x  Dk Dx D k } x; thus our maps fK and fE satisfy the compatibility conditions in Lemma 5.84. We claim that the resulting map fE W E ! M.T L/ is a lanilcur, so that Lemma 5.84 applies and yields a bounded algebra homomorphism f W L ! T L. It follows from the definition that   L e F .x } / D L .e F x/ ˇ L ./ D e ˇ x ˇ L ./ D e ˇ L .x } / for all e 2 E, x 2 L,  2 TC L. This implies L ı f D idL . Hence L is analytically quasi-free; it only remains to prove the claim that fE is a lanilcur. The connection property of r (5.87) implies   Dx1 Dx2  Dr.x1 ˇ x2 / C Dr.x1 / } x2 D r.x1 / Dx2 (5.89) for all x1 ; x2 2 L. Hence (5.85) yields e F Dx1 Dx2 D e F .x1 ˇ x2 /  .e F x1 / } x2 D e ˇ x1 ˇ x2  Dr.e1 ˇ x1 ˇ x2 /  .e ˇ x1 / } x2 C Dr.e1 ˇ x1 / } x2 D D.e ˇ x1 / Dx2  Dr.e1 ˇ x1 ˇ x2 / C Dr.e1 ˇ x1 / } x2 D r.e ˇ x1 / Dx2 :

220

5 Analytic cyclic homology and analytically nilpotent extensions

As a result, e F x0 Dx1 : : : Dx2n D e ˇ x0 Dx1 : : : Dx2n  Dr.e ˇ x0 / Dx1 : : : Dx2n ; (5.90) e F Dx1 Dx2 : : : Dx2n D r.e ˇ x1 / Dx2 : : : Dx2n : To deal with such expressions efficiently, we first modify our setup slightly. We identify L  .L/ Š even L Š TL as bornological vector spaces via x0 Dx1 : : : Dx2n 7! x0 Dx1 : : : Dx2n ;

x0 Dx1 : : : Dx2n1 7! Dx0 Dx1 : : : Dx2n :

This is compatible with the analytic bornologies on even L Š TL and L  .L/; that is, T L Š L  an .L/. We split odd L  an .L/ D L  even an .L/ ˚ L  an .L/I

this corresponds to the decomposition of T L into forms with or without leading D. The operator r extends to a bounded operator on Lan .L/ and has degree 1; therefore, odd it restricts to maps Leven an .L/  Lan .L/, which we denote by r as well. Define M! W L  an .L/ ! L  an .L/;

 7! ! ˇ 

for ! 2 L. With this notation, we can rewrite the operator fE .e/ in (5.90) as   r ı Me Me fE .e/ D : r ı Me 0 For the curvature !F .e1 ; e2 / D fE .e1 e2 /  fE .e1 / ı fE .e2 /, a routine computation using Me1 e2  Me1 ı Me2 D Mde1 de2 yields   Mde1 de2 C r ı Me1 ı r ı Me2 r ı Me1 e2  Me1 ı r ı Me2 : !F .e1 ; e2 / D r ı Mde1 de2 r ı M e1 ı r ı M e2 (5.91) Products of such matrices with ei 2 S are matrices whose entries are sums of ˙M!0 r ı M!1 ı r ı    ı M!n1 ı r ı M!n with !0 ; : : : ; !n 2 S [ S 2 [ hSi.dS dS/1 hSi; it is crucial that this set is bounded in the analytic bornology. Furthermore, the number of summands grows at most exponentially in the number n of factors; hence we can rewrite the sum as a convex combination of elements of .r ı MT /1 for some T 2 S.T E/. Lemma 5.92. If T 2 S.T E/, then .r ı MT /1 is an equibounded set of operators on L  an .L/ with respect to the analytic bornology. The lemma and the above computations show that F defines a lanilcur E ! M.T L/. This finishes the proof of the Excision Theorem 5.76 for the analytic cyclic theory.

5.4 Excision in analytic and local cyclic homology

221

Proof. According to Lemma 3.9, we must show that r ı MT -invariant subsets are    } cofinal in San L.L/ . We consider subsets of the form F0 .DF1 /1 with F0 ; F1 2 S.L/. Such a subset is invariant under r ı MT if and only if r.T ˇ F0 / } F0 DF1 : We must choose F0 carefully because it appears on both sides. Any bounded subset of E is contained in .s.SQ / [ SK /} for SQ 2 S.Q/, SK 2 S.K/. The definition of the analytic bornology allows us to choose S D s.SQ / [ SK of this form with T } .dS dS/1 hSi. Recall that r.L/  G D.L/, so that it is reasonable to choose F0  G . We put   F0 WD sR .dSQ dSQ /1 hSQ i ˇSK0 ;

  SK0 WD hs.SQ /i SK [!s .SQ ; SQ / hs.SQ /i:

  We claim that r .dS dS/1 hSi ˇ F0 } F0 DF1 for all sufficiently large F1 2 S.L/. The resulting r ı MT -invariant sets .F0 .DF1 /1 /} in San .L  L/ are cofinal because we can choose SK and SQ arbitrarily large. To prove the claim, fix ! D de1 : : : de2n he2nC1 i 2 .dS dS /1 hS i; x D sR . dq2nC2 : : : dq2nC2m1 hq2nC2m i/ ˇ k 2 F0 I our notation means that e2nC1 or q2nC2m may be missing. We compute r.! ˇ x/. Recall that r.sR ./ ˇ k ˇ !/ D sR ./ ˇ k D! for all  2 T Q, k 2 K, ! 2 L. When we bring ! ˇ x into this form, we distinguish two cases. If ej 2 K for some j , then we split de2k1 de2k D e2k1 e2k  e2k1 ˇ e2k for the first pair with e2k1 2 K or e2k 2 K. Thus ! ˇ x is equal to a sum of two terms in       sR .dSQ dSQ /1 hSQ i ˇ SK0 ˇ hS i ˇ .dS dS /1 hS i ˇ F0 and is mapped by r into the convex hull of F0 DF1 provided 2hSi ˇ .dS dS /1 hS i ˇ F0  F1 . If no ej belongs to K, then we can write ej D s.qj / for all j D 1; : : : ; 2n C 1. Since r vanishes on G , we may disregard terms in G when we compute ! ˇ x. We get !ˇx

2nC2m1 X

.1/j 22n2mCj ds.q1 / : : : ds.qj 1 / d!s .qj ; qj C1 /

j D2nC1

 ds.2qj C2 / : : : ds.2q2nC2m1 /hs.q2nC2m /i ˇ 4i mod G : We decompose the summands at !s .qj ; qj C1 / 2 K as above and get r.! ˇ x/ 2 F0 DF1 if 4hS i ˇ .4 dS dS/1 hSi  F1 . Hence r.T ˇ F0 / } F0 DF1 if F1 is sufficiently large.

222

5 Analytic cyclic homology and analytically nilpotent extensions

On forms of low degree we can compute the resulting homomorphism  W L ! T L explicitly. It maps k 7! k;

dk1 dk2 7! dk1 dk2 C Dk1 Dk2 ;

ds.q1 / dk2 7! ds.q1 / dk2 ;

ds.q1 / ds.q2 /  k3 7! ds.q1 / ds.q2 /  k3 ;

ds.q1 / dk2  k3 7! ds.q1 / dk2  k3  D.ds.q1 / dk2 / Dk3 ; dk1 ds.q2 /  k3 7! dk1 ds.q2 /  k3 C Dk1 D.s.q2 / ˇ k3 /; dk1 dk2  k3 7! dk1 dk2  k3  D.dk1 dk2 / Dk3 C Dk1 Dk2  k3 : These formulas soon get pretty complicated.

5.4.4 Excision for locally semi-split extensions ! The proof of Theorem 5.76 carries over to algebras in Ban with obvious changes and proves that the map HL.K/ ! HLrel .p/ is a chain homotopy equivalence for semi-split ! algebra extensions in Ban. This proof is, in principle, constructive. It yields universal formulas for a retraction f W HLrel .p/ ! HL.K/ and a chain homotopy h W HLrel .p/ ! HLrel .p/ between id and i ı f that depend only on the given extension K  E  Q and the section s W Q ! E. Now consider a locally semi-split extension. It is isomorphic to a direct limit of an inductive system of extensions of partial Banach algebras Kj  Ej  Qj with bounded linear sections sj W Qj ! Ej . The definition of HL.K/ and HLrel .p/ is local, that is, both are inductive limits of chain complexes that only depend on things happening within Kj  Ej  Qj for a fixed j ; it is crucial for this that our universal formulas for f and h only involve products in E of a fixed maximal length. We leave it to the reader to check this. Hence we can plug the locally defined sections sj into the universal formulas to get locally defined maps fj and hj ; now Lemma 2.24 shows that the embedding HL.K/ ! HLrel .p/ is still a local chain homotopy equivalence.

5.4.5 Some simple examples We compute the boundary maps explicitly for two particularly simple extensions: the cone extension and the Toeplitz extension. These computations are important in connection with the bivariant Chern–Connes character (see [17, Satz 6.9]). The cone extension. Let CŒ0; 1 be the algebra of smooth functions Œ0; 1 ! 1 whose nth derivatives f .n/ vanish at t D 0; 1 for all n  1. We define ideals C WD C.0; 1;

S WD C.0; 1/

5.4 Excision in analytic and local cyclic homology

223

by adding the condition f .0/ D 0 and f .0/ D f .1/ D 0, respectively. If we identify R Š .0; 1/ suitably, then S corresponds to the Schwartz space S.R/ of rapidly decreasing functions on R. The smooth cone extension is the algebra extension S  C  C, where we map C ! C by f 7! f .1/. It is shown in [17] that C is smoothly contractible. The homotopy invariance of analytic cyclic homology yields HA.C / 0. Theorem 5.63 shows that HA.C/ C. The Excision Theorem 5.76 implies HA.S / CŒ1. Similarly, the Excision Theorem 4.42 in periodic cyclic homology yields HP.S / CŒ1. Since the boundary maps in periodic and analytic cyclic homology are compatible, it follows that the canonical chain map HA.S / ! HP.S / is a chain homotopy equivalence. It is not hard to see that S is quasi-free, so that HP.S / X.S /. This yields HA.S / X.S /, although we do not know whether S is analytically quasi-free. As a result, the boundary map for the cone extension is determined completely by the map HA0 .C/ ! HA1 .S / that it induces. To compute this map explicitly, we must fix isomorphisms C Š HA0 .C/ and HA1 .S / Š C. We identify C Š HA0 .C/ using the generator Œe described in Theorem 5.63. We identify HA1 .S / Š C using the chain map Z 1 s odd 1 ' W X1 .T S / Š an S   S  ! C; s .f0 df1 / WD f0 .t /  f10 .t / dt: 0

This is a cocycle because s .df / D f .1/  f .0/ D 0 for f 2 S and s ı b D 0.   Proposition 5.93. ' @Œe D 1. Proof. The formula for s defines a linear map 1 .C / ! C. The relation s ı b D 0 remains valid on 2 .C /, but since the extension satisfies s .df / D f .1/, the largest subcomplex of X.T C / where it is a cocycle is the kernel K of the chain map X.T C / ! X.T C/ ! X.C/ Š C: The canonical map from X.T S/ to the kernel of this chain map is a chain homotopy equivalence by the Excision Theorem 5.76. We can also describe the boundary map @ W C Š HA0 .C/ ! HA1 .S / as the composition of the boundary map H0 .C/ ! H1 .K / for the extension of chain complexes K  X.T C /  C with a chain homotopy inverse X.T S/ ! K of the embedding. This follows from the construction of the boundary map in §5.4.2. In the first step, we lift 1 2 C to any f 2 C D 0an C with f .1/ D 1 and take @.f / D df 2 1an C  K1 . In the second step, we replace @.f / by a homologous cycle in X.T S/. But the cocycle ' extends to a cocycle on K , so that we can directly pair @.f / with '; it makes no difference whether we first replace @.f / by a homologous cycle. Hence we get ' ı @Œe D '. df / D s df D f .1/ D 1 as asserted. Other algebras of functions on Œ0; 1 yield variants of the smooth cone extension. The algebraic cone extension is .t  t 2 /CŒt   tCŒt   C;

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5 Analytic cyclic homology and analytically nilpotent extensions

where the projection is evaluation at 1. This is equivalent to the tensor algebra extension JC  TC  C of C. Choosing a function f 2 C with f .1/ D 1, we get a map t CŒt ! C , t n 7! f n , which is part of a morphism of extensions .t  t 2 /CŒt  /

/ t CŒt 



 /C

ev1

//C

(5.94) S /

ev1

/ / C.

  Since t CŒt is smoothly contractible, we can argue as above and find HA .t t 2 /CŒt  CŒ1; hence the vertical maps in (5.94) are HA-equivalences. Proposition 5.93 also extends to this algebraic setting. As C and tCŒt  are analytically quasi-free, we can compute their analytic cyclic homology using X.tCŒt / and X.C/. Hence the  kernel of the  evaluation map X.t CŒt / ! HA .t  t 2 /CŒt  . In this kernel, the standard X.C/ is chain homotopy equivalent to   generator of HA1 .t  t 2 /CŒt  is represented by the cycle dt ; this is a boundary in the contractible chain complex X.tCŒt /, but not in the kernel of the map to X.C/. Smooth functions on Œ0; 1 with arbitrary derivatives at 0 and 1 yield another variant of the cone extension; this leads to homotopy equivalent extension by [17].  a smoothly  We cannot replace C by C0 .0; 1 because the bivariant analytic cyclic theory is only invariant under homotopies of bounded variation. We will see in §6.3.1 that bivariant local cyclic homology is invariant under continuous homotopies as well. But   1 it is hard to write down an explicit cocycle ' 2 HL C0 .0; 1 . Here it is best to use the naturality of the boundary map; it tells us that the square HL0 .C/ HL0 .C/

   / HL1 C0 .0; 1/ O

@

@

/ HL1 .S /

commutes. The homotopy invariance result of Theorem 6.31 shows that the right vertical map is an isomorphism as well. Hence we can replace the intractable boundary map for the continuous cone extension by the one for the smooth cone extension. The Toeplitz extension. The algebraic Toeplitz extension is the extension M1  Iso  Inv, where Inv D CŒz; z 1  and Iso is the algebra with generators v; w and the relation wv D 1; we prefer to work with the non-unital version M1  Iso0  Inv0 of this extension. All algebras in this extension are analytically quasi-free by Theorem 5.63, and their analytic cyclic homology is given by HA.M1 / C;

HA.Iso0 / 0;

HA.Inv0 / CŒ1:

225

5.4 Excision in analytic and local cyclic homology

Hence the class of the boundary map in HA1 .Inv0 ; M1 / is specified by the map C Š HA1 .Inv0 / ! HA0 .M1 / Š C: We have a commuting diagram HA.M1 /

 





ker HA.p/

   ker X.p/ /

/

/ HA.Iso0 /

 / X.Iso0 /

/ / HA.Inv0 /

 / / X.Inv0 /

Š

 X.M1 /.

The two middle rows are semi-split extensions. The vertical maps are all chain homotopy equivalences either by the Excision Theorem 5.76 or because involved  the algebras  are analytically quasi-free. Inspection shows that the map ker X.p/ ! X.M1 / is an isomorphism. Hence the boundary map in analytic cyclic homology is equivalent to the boundary map for the extension of X-complexes X.M1 /  X.Iso0 /  X.Inv0 /:   0 0 1 1 x1 Now  generator z dz 2  .Inv/ D  .Inv / for H1 X.Inv / and identify  we use the H0 X.M1 / Š C using the trace. To compute the boundary map, we first lift z 1 dz x 1 .Iso/ Š X1 .Iso0 /; the boundary map in X.Iso0 / maps this to Œw; v D to w dv 2  1  vw, which is a rank-1-projection in the ideal M1 . Hence the trace maps it to 1. Summing up, the boundary map corresponds to the identity map on C if we identify HA0 .M1 / Š C using the trace and HA1 .Inv0 / Š C using z 1 dz. We can also complete the algebraic Toeplitz extension to a smooth Toeplitz extension KS  IsoS  C 1 .T /, see Example 3.48. The two extensions are related by a morphism of extensions M1 /

/ Iso



 / IsoS

KS /

/ / Inv

(5.95) / / C 1 .T /.

We claim that the vertical maps in (5.95) are HA-equivalences. The algebra KS of smooth compact operators can be obtained from the canonical bilinear map on S.N/ and the construction of §A.7.1; hence Theorem 5.65 yields HA.KS / HA.C/ C, so that the map M1 ! KS is an HA-equivalence. Smooth homotopy invariance

226

5 Analytic cyclic homology and analytically nilpotent extensions

and excision imply that the map Inv ! C 1 .T / is an HA-equivalence. Finally, the Excision Theorem 5.76 shows that the map Iso ! IsoS is an HA-equivalence. The naturality of the boundary map allows us to carry over our computation of the boundary map from the algebraic to the smooth Toeplitz extension; once we have established continuous homotopy invariance, we can complete further to the C  -Toeplitz extension. Bott periodicity. The cone and Toeplitz extension are the ingredients for a proof of Bott periodicity in bivariant K-theory (see [17, Satz 6.9]). To generate Bott periodicity, we compose the boundary maps for the cone and Toeplitz extensions. The smooth Toeplitz extensions are related because the map .0; 1/ ! T , t 7! exp.2it /, identifies S with the ideal in C 1 .T / consisting of functions that have a zero of infinite order at 1. Periodic, analytic, and local cyclic homology have a built-in periodicity of period two. Nevertheless, the proof of Bott periodicity still applies. Composing the cone and Toeplitz extension, we get a canonical element of HA0 .C; C/ Š C:   exp.2it/ @cone C HA.C/ ! HA.S /Œ1 ! HA C 1 .T / Œ1 @Iso

! HA.M1 /Œ2 D HA.M1 / HA.C/ C: Proposition 5.96. The map C ! C described above is multiplication by .2i/1 . Proof. Let us follow the standard generator of HA0 .C/. First, the boundary map of the cone extension maps it to the standard generator of HA1 .S /, which we denote  1 by dt  ; it 1 is determined by dt D 1. We choose z X C dz as our generator for H .T / Š s 1    HA1 X C 1 .T / ; this is mapped by @Iso to 1 2 C HA0 .M1 /. The numerical factor arises because the map S ! C 1 .T / maps dt 7! .2i z/1 dz. This is related to the computation z D exp.2it / H) z 1 dz D 2i dt: Hence the Bott periodicity isomorphism in analytic (or periodic or local) cyclic homology is multiplication by .2i/1 . This factor occurs frequently when we compare K-theory and cyclic homology.

5.4.6 Formulas for the chain homotopy equivalence The examples in §5.4.5 do not require an explicit general formula for the chain homotopy equivalence between HArel .p/ and HA.K/. Such formulas exist, in principle, and can be written down in low degrees. But they are too complicated for most applications. The following computations are useful, nevertheless, to estimate dimension shifts in §5.4.7.

5.4 Excision in analytic and local cyclic homology

227

We are dealing with two maps: a chain map retraction   f W ker X.T p/ ! X.T K/ and a chain homotopy     h W ker X.T p/ ! ker X.T p/ Œ1     between X.T i/ ı f and the identity map on ker X.T p/ . (Replacing ker X.T p/ by Xrel .T p/ has no significant effect.) Both f and h are constructed in two steps:   • First we contract ker X.T p/ to X.L/; this involves a retraction  f1 and a chain homotopy h1 between f1 and the identity map on ker X.T p/ . • Secondly, we contract X.L/ to X.T K/; this involves a retraction f2 and a chain homotopy h2 between f2 and the identity map on X.L/.   ker X.T X p/ h1

f1

/ X.L/ Y

f2

/ X.T K/

h2

  The chain map f WD f2 ı f1 is the desired retraction from ker X.T p/ to X.T K/, and h WD  h1 Ch2 ı f1 is the desired chain homotopy between f and the identity map on ker X.T p/ . Thus it suffices to describe the maps f1 ; f2 ; h1 ; h2 . The first step. We are going to describe a retraction   f1 W ker X.T p/ ! X.L/ and a chain homotopy     h1 W ker X.T p/ ! ker X.T p/ Œ1: These maps are implicitly constructed during the proof of Theorem 4.67. Recall that     Š ! ŒL; sL .T Q/ : ker X.T p/ Š X.L/ ˚ L DsL .T Q/  by the following list of requirements: it vanishes on the odd The map h1 is determined  part of ker X.T p/ and on L, and it satisfies   h1 Œ!; sL ./ D Œ! DsL ./ for all ! 2 L,  2 T Q; here Œ: : :  denotes the class in 1 .T E/=Œ ; . The map h1 determines f1 via f1 D id  Œ@; h1 ; this is id  @ ı h1 on even and id  h1 ı @ on odd forms. We can also characterise f1 as the unique map that acts identically on X.L/ and vanishes on L DsL .T Q/ and ŒL; sL .T Q/.

228

5 Analytic cyclic homology and analytically nilpotent extensions

We are now going to describe the values of h1 and f1 on a standard monomial e0 de1 : : : den with e0 2 KC [ s.Q/, e1 ; : : : ; en 2 K [ s.Q/; at least one ej belongs to K because we are restricting attention to ker X.T p/ . Let ej be the last entry in K, so that ej C1 ; : : : ; en 2 s.Q/. Now we distinguish several cases. Suppose first that both n and j are even. Then e0 de1 : : : den D dej C1 : : : den ˇ e0 de1 : : : dej C Œe0 de1 : : : dej ; dej C1 : : : den ˇ : The first summand dej C1 : : : den ˇ e0 de1 : : : dej belongs to L and the second one belongs to ŒL; sL .T Q/ because ej 2 K and ej C1 ; : : : ; en 2 s.Q/. Thus h1 .e0 de1 : : : den / D e0 de1 : : : dej D. dej C1 : : : Den /; f1 .e0 de1 : : : den / D dej C1 : : : den ˇ e0 de1 : : : dej D dej C1 : : : den  e0 de1 : : : dej : Now suppose that n is even and j is odd. This time, we write e0 de1 : : : den D e0 de1 : : : dej 1 ˇ .ej ej C1  ej ˇ ej C1 / ˇ dej C2 : : : den D dej C2 : : : den ˇ e0 de1 : : : dej 1 ˇ .ej ej C1 / C Œe0 de1 : : : dej 1 ˇ .ej ej C1 /; dej C2 : : : den ˇ  ej C1 dej C2 : : : den ˇ e0 de1 : : : dej 1 ˇ ej  Œe0 de1 : : : dej 1 ˇ ej ; ej C1 dej C2 : : : den ˇ : The first and third summands in the final result belong to L, the other two belong to ŒL; sL .T Q/. Thus h1 .e0 de1 : : : den / D e0 de1 : : : dej 1 ˇ .ej ej C1 / D. dej C2 : : : den /  e0 de1 : : : dej 1 ˇ ej D.ej C1 dej C2 : : : den / D e0 de1 : : : dej C1 D.dej C2 : : : den /  dej C2 : : : den ˇ e0 de1 : : : dej 1 ˇ ej Dej C1 ; f1 .e0 de1 : : : den / D dej C2 : : : den ˇ e0 de1 : : : dej 1 ˇ .ej ej C1 /  ej C1 dej C2 : : : den ˇ e0 de1 : : : dej 1 ˇ ej : This finishes the computation of f1 and h1 on even forms. Using also Lemma 5.44, we get the following: Lemma 5.97. Let ! be a monomial in 2n .E W Q/. Then h1 .!/ 2 2n1 .E W Q/ and f1 .!/ is a linear combination of terms in L \ 2n .E/ and of terms of the form x ˇ .ke/  e ˇ x ˇ k with x 2 2n2 .E/, k 2 K, e 2 E. Since h1 vanishes on odd forms, it remains to compute f1 on odd forms. We assume first that n is odd and n D j , that is, the last entry belongs to K. Recall that this last entry plays a special role because e0 de1 : : : den represents the class of e0 de1 : : : den1 Den modulo commutators. Let i < n be the last entry except en with ei 2 K, or 1 if no such entry exists. If i is even, then the Leibniz rule yields e0 de1 : : : den1 Den D e0 de1 : : : dei ˇ deiC1 : : : den1 Den D e0 de1 : : : dei D. deiC1 : : : den1 ˇ en /  en ˇ e0 de1 : : : dei D. deiC1 : : : den1 /

5.4 Excision in analytic and local cyclic homology

229

modulo commutators. The first term belongs to X.L/ because ei ; en 2 K, the second one to L DsL .T Q/ because ei 2 K and eiC1 ; : : : ; en1 2 s.Q/. By construction, f1 acts identically on X.L/ and annihilates L DsL .T Q/. Hence f1 .e0 de1 : : : den / D e0 de1 : : : dei D. deiC1 : : : den1 ˇ en /: Similarly, we get f1 .e0 de1 : : : den / D D.e0 de1 : : : den1 ˇ en / if i D 1 and f1 .e0 de1 : : : den / D e0 de1 : : : dei1 ˇ .ei eiC1 / D. deiC2 : : : den1 ˇ en /  e0 de1 : : : dei1 ˇ ei D.eiC1 deiC2 : : : den1 ˇ en / if i is odd. This finishes the computation of f1 in case n is odd and j D n. Now we suppose that j < n and that j is even. As above, we get   f1 .e0 de1 : : : den / D f1 e0 de1 : : : dej D. dej C1 : : : den1 ˇ en / : Although ej C1 ; : : : ; en 2 s.Q/, the differential form dej C1 : : : den1 ˇ en does not belong to sL .T Q/ unless s is multiplicative. Write ek D s.qk / for k D j C 1; : : : ; n. We compute dej C1 : : : den1 ˇ en D

n1 X

.1/n1k dej C1 : : : d.ek ekC1 / : : : den C .1/n1j ej C1 dej C2 : : : den

kDj C1

D

n1 X

.1/nk dej C1 : : : d!s .qk ; qkC1 / : : : den mod sL .T Q/;

kDj C1

where !s W Q  Q ! K is the curvature of s. For a summand with even k, we get   f1 e0 de1 : : : dej D. dej C1 : : : dek1 d!s .qk ; qkC1 / dekC2 : : : den /   D dekC2 : : : den ˇ e0 de1 : : : dej D dej C1 : : : dek1 d!s .qk ; qkC1 / : In the remaining cases where one or two of j and k are odd, we split dej dej C1 D ej ej C1  ej ˇ ej C1 for odd j and d!s .qk ; qkC1 / dekC2 D !s .qk ; qkC1 /ekC2  !s .qk ; qkC1 / ˇ ekC2 for odd k and then treat the resulting summands as above. For instance, if j is odd and k is even, then   f1 e0 de1 : : : dej D. dej C1 : : : dek1 d!s .qk ; qkC1 / dekC2 : : : den /   D dekC2 : : : den ˇ e0 de1 : : : dej 1 ˇ .ej ej C1 / D dej C2 : : : dek1 d!s .qk ; qkC1 /    dekC2 : : : den ˇ e0 de1 : : : dej 1 ˇ ej D ej C1 dej C2 : : : dek1 d!s .qk ; qkC1 / : We leave the formulas for odd k to the reader and only notice the following facts: Lemma 5.98. Let ! be a monomial in 2nC1 .E W Q/. Then h1 .!/ D 0 and f1 .!/ is a linear combination of terms of the following types:

230

5 Analytic cyclic homology and analytically nilpotent extensions

• l1 Dl2 with l1 ; l2 2 L and deg.l1 / C deg.l2 / D 2n, • x ˇ ke Dl  x ˇ k D.e ˇ l/ or l D.x ˇ ke/  e ˇ l D.x ˇ k/ with x 2 T E, l 2 L, k 2 K, e 2 E, and deg.l/ C deg.x/ D 2n  2, • x1 ˇ k1 e1 D.x2 ˇ k2 e2 /  x1 ˇ k1 D.e1 ˇ x2 ˇ k2 e2 /  e2 ˇ x1 ˇ k1 e1 D.x2 ˇ k2 / C e2 ˇ x1 ˇ k1 D.e1 ˇ x2 ˇ k2 / with x1 ; x2 2 T E, k1 ; k2 2 K, e1 ; e2 2 E, and deg.x1 / C deg.x2 / D 2n  4. If s is multiplicative, then we get terms of the following forms instead: • l1 Dl2 with l1 ; l2 2 L and deg.l1 / C deg.l2 / D 2n, • x ˇ ke Dl  x ˇ k D.e ˇ l/ with x 2 T E, l 2 L, k 2 K, e 2 E, and deg.l/ C deg.x/ D 2n  2. The gradings on T E and L are both defined by deg.x/ WD k for x 2 k .E/. The second step. Now we consider the deformation retraction from X.L/ to X.T K/. We use the bounded algebra homomorphism  W L ! T L constructed in the proof of the Excision Theorem 5.76 in §5.4.3. Recall that .x ˇ k/ WD x F k for all x 2 T E, k 2 K, where we view K  T L in the obvious way and define F by extending (5.90). We do not attempt to describe  more explicitly. 0 The canonical map E W T E  E restricts to a map K W L  K, which induces 0 a map T K W T L  T K; here “map” means “bounded algebra homomorphism”. 0 Equivalently, T K is the restriction of T E W T T E  T E. Let 0 ı  W L ! T K: % WD T K

Let K0 W K ! L be the restriction of E W E ! T E or, equivalently, the composition of K with the embedding T K ! L. Since  ı K0 is the canonical embedding K ! T L, we get % ı K0 D K . This implies that % restricts to the identity map on T K  L. Hence the induced chain map f2 WD X.%/ is a retraction from X.L/ to X.T K/. 0 WL!L The homotopy h2 requires a smooth homotopy from the lanilcur K0 ı K to the identity lanilcur on L. We choose the smooth homotopy H D .H t / t2Œ0;1 W L ! C 1 .Œ0; 1; L/;

H t .l/ WD t deg.l/=2  l;

0 and H1 D idL . that is, H t .l/ D t n l for l 2 L \ 2n .E/. Notice that H0 D K0 ı K The curvature of H is the map

!H D .!H;t / t2Œ0;1 W L  L ! C 1 .Œ0; 1; L/; .l1 ; l2 / 7! H.l1 ˇ l2 /  H.l1 / ˇ H.l2 / D t deg.l1 /=2Cdeg.l2 /=2 .l1  l2  t dl1 dl2  l1 ˇ l2 / D t deg.l1 /=2Cdeg.l2 /=2 .1  t / dl1 dl2 :

5.4 Excision in analytic and local cyclic homology

231

Since this factors through JE \ L, H is a lanilcur. To describe the induced algebra homomorphism hH i W T L ! C 1 .Œ0; 1; L/, we define two gradings on T L – the external grading dege and the internal grading degi – by dege .l0 Dl1 : : : Dl2n / WD 2n;

degi .l0 Dl1 : : : Dl2n / WD

2n X

deg.lj /:

j D0

With this convention, we have hH i.l0 Dl1 : : : Dl2n / D t degi .l0 Dl1 :::Dl2n /=2  .t  1/dege .l0 Dl1 :::Dl2n /=2 l0 dl1 : : : dl2n for all l0 2 LC , l1 ; : : : ; l2n 2 L. The smooth homotopy of algebra homomorphisms hH i yields a chain homotopy X.H / between the chain maps X.T L/ ! X.L/ 0 induced by T K and L ; here we use the homotopy invariance of the X-complex for quasi-free algebras (Theorem 4.27). Finally, h2 WD X.H / ı X./ is the desired chain homotopy between f2 D X.%/ and the identity map on X.L/. It remains to write down the chain homotopy X.H /. The general recipe is to take x on X0 .T L/ Š T L and H ı.id b ır/ on X1 .T L/ Š odd H ır ı D an L, with H as in the proof of Theorem 4.27, the standard right bimodule connection r (see Examx W T L ! 1 .T L/ (the bar distinguishes it ple A.93), and the universal differential D from the differential D in L). The computation in Example A.126 carries over to our context and yields   x 2nC1 X.H /.l0 Dl1 : : : Dl2nC1 / D H .l0 Dl1 : : : Dl2n / Dl Z 1 D hH t i.l0 Dl1 : : : Dl2n / ˇ HP t .l2nC1 / dt (5.99) 0 Z 1 D deg.l2nC1 / t degi .l0 Dl1 :::Dl2nC1 /=21 .t  1/n l0 dl1 : : : dl2n ˇ l2nC1 dt: 0

x proceeds as in §5.2.1. First we compute r ı D: x The computation of H ı r ı D x 0 Dl1 : : : Dl2n / D D.l x 0 /D.Dl x 1 : : : Dl2n /  r ı D.l 

n X   x 2j 1 Dl2j / Dl2j C1 : : : Dl2n : r l0 Dl1 : : : Dl2j 2 D.Dl j D1

Each summand is simplified further using the Leibniz rule:   x 2j 1 Dl2j / Dl2j C1 : : : Dl2n  r l0 Dl1 : : : Dl2j 2 D.Dl x 2j C1 : : : Dl2n / x 2j 1 ˇ l2j / D.Dl D l0 Dl1 : : : Dl2j 2 D.l x 2j 1 / D.l x 2j } Dl2j C1 : : : Dl2n /  l0 Dl1 : : : Dl2j 2 D.l x 2j C1 : : : Dl2n /: x 2j / D.Dl  l0 Dl1 : : : Dl2j 2 } l2j 1 D.l

232

5 Analytic cyclic homology and analytically nilpotent extensions

Composing this with H and abbreviating H t D hH t i, we get X.H /.l0 Dl1 : : : Dl2n / Z 1 D HP t .l0 / DH t .Dl1 : : : Dl2n / dt C

0 n Z 1 X

(5.100)

H t .l0 Dl1 : : : Dl2j 2 / ˇ HP t .l2j 1 ˇ l2j / DH t .Dl2j C1 : : : Dl2n / dt

j D1 0



n Z X

1

  H t .l0 Dl1 : : : Dl2j 2 / ˇ HP t .l2j 1 / D H t .l2j / ˇ H t .Dl2j C1 : : : Dl2n / dt

j D1 0



n Z X

1

H t .l0 Dl1 : : : Dl2j 2 / ˇ H t .l2j 1 / ˇ HP t .l2j / DH t .Dl2j C1 : : : Dl2n / dt

j D1 0

Z

D C

1

P

H t .l0 / DH t .Dl1 0 n Z 1 X

: : : Dl2n / dt

H t .l0 Dl1 : : : Dl2j 2 / ˇ HP t .Dl2j 1 Dl2j /DH t .Dl2j C1 : : : Dl2n / dt

j D1 0



n Z X

1

H t .Dl2j C1 : : : Dl2n ˇ l0 Dl1 : : : Dl2j 2 / ˇ HP t .l2j 1 / DH t .l2j / dt:

j D1 0

5.4.7 Dimension estimates in cyclic homology We can now estimate the dimension shifts in non-periodic cyclic homology that occur when we compare HC .K/ and HCrel  .p/ (see §4.3.5). We compute both HC .K/ and rel HC .p/ using the Hodge filtrations .Fn /n2N on HP.K/ and HPrel .p/ (see §A.6.3); we may also use HA or HL here, this has no effect on the relevant quotients for the Hodge filtration (except that the latter are not defined for algebras in general symmetric monoidal categories). In §5.4.6, we have described a chain map retraction   f W ker HA.p/ ! HA.K/ and a chain homotopy     h W ker HA.p/ ! ker HA.p/ Œ1   between HA.i/ ı f and the identity map on ker HA.p/ . We are going to check that both f and h map F3nC2 ! Fn for all n 2 N and that they even map F4nC2 ! F2n and F4nC3 ! F2nC1 for all n 2 N if the extension splits. This yields Theorem 4.72. It also follows that f and h aremorphisms of projective systems, so that they  define morphisms on HP.K/ and ker HP.p/ . Even more, the formulas in §5.4.6 still

5.4 Excision in analytic and local cyclic homology

233

work for pro-algebras over an arbitrary Q-linear symmetric monoidal category. From now on, we therefore switch our notation and consider an extension of pro-algebras K  E  Q over some Q-linear symmetric monoidal category, and form the      corresponding pro-tensor algebras TK, TE, TQ, and the subalgebras L  TE and   I  TE. It is clear that the constructions above extend to this setting.   The most complicated ingredient is the map  W L ! T L . It is responsible for the linear growth of the function ƒ in Theorem 4.72. To describe its behaviour with respect to the Hodge filtration, we use the internal and external degree degi and dege  of a monomial in  L defined by   dege l0 Dl1 : : : Dln WD n;

n X   deg lj I degi l0 Dl1 : : : Dln WD j D0

we have already seen this in the description of the homotopy T L ! C 1 .Œ0; 1; L/ in §5.4.6. We also define the total degree deg t and the modified total degree deg.˛/ for t ˛ > 0 by deg t D dege C degi ; deg.˛/ D ˛  dege C degi : t If ˛  1, these are related by deg t  deg.˛/  ˛  deg t : t

(5.101)

   We are mainly interested in the filtrations on  L and T L generated by these gradings.   Lemma 5.102. The section  W L ! T L , x ˇ k 7! x F k, constructed as in §5.4.3 satisfies 3 deg t .l/  deg.3/ t .l/  deg.l/  for all l 2 L . If the section s W Q ! E is multiplicative, this improves to 2 deg t .l/  deg.2/ t .l/  deg.l/: Proof. The first inequalities are just (5.101) for ˛ 2 f2; 3g. To prove deg.3/ t .l/   deg.l/, we use a more complicated grading on T L : we let (  0 if l0 2 L , .30 / .3/ deg t .l0 Dl1 : : : Dln / WD deg t .l0 Dl1 : : : Dln /  4 if l0 D 1. 0

We claim that the operator de1 de2 F D !F .e1 ; e2 / raises deg.3 / by at least 2. This follows from the description of !F .e1 ; e2 / by a 2  2-block matrix in (5.91). We observe that an application of r raises the external degree by 1 and lowers the internal degree by no more than 2 because r splits at most one pair de1 de2 . Of course, M!  for ! 2 TE does not change dege and raises degi by deg.!/. The careful definition 0 of deg.3 / ensures that each entry in (5.91) raises it by at least 2.

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5 Analytic cyclic homology and analytically nilpotent extensions

Furthermore, we have .de1 dk2 / D de1 dk2 C DpK .e1 / Dk2 ;

(5.103)

where pK W E ! K is the projection with pK ı s D 0. Since both monomials on the 0 right hand side have deg.3 / D 2, we get  0/ .30 /  deg.3 e0 de1 : : : de2n2 F. de2n1 dk2n /  2n: t .e0 de1 : : : de2n1 dk2n / D deg t  .30 / This implies deg.3/ .l/  2n for all l 2 L . In addition, we notice t .l/  deg t for later that monomials with deg.3/ D deg.l/ that occur in .l/ are of the form t l0 Dl1 : : : Dl2n because we even get deg.3/ t  2n C 4 for closed forms Dl1 : : : Dl2n that occur in .l/.  Now assume that s W Q ! E is multiplicative. Then the maps sL ; sR W TQ !    TE agree and are multiplicative. Therefore, s.Q/ ˇ G  G is annihilated by r,    so that r ı Me .g/ D D e  s ı p.e/ Dg for all g 2 G . Thus the combination r ı Me1 ı r ı Me2 raises dege by 2 and lowers degi by no more than 2.   Furthermore, the range of r is contained in G .D L /1 . To use this additional information most efficiently, we modify the grading deg.2/ t by letting 8  ˆ 0 if l0 2 L ; ˆ < 0  deg.2 / .l0 Dl1 : : : Dl2n / WD deg.2/ t .l0 Dl1 : : : Dl2n / 2 if l0 D 1 and l1 2 G ; ˆ ˆ   :4 if l D 1 and l 2 G ˇ L: 0

1

Inspection shows that each entry in (5.91) raises this degree by at least 2. For instance, the term r ı Mde1 de2 in the lower left corner does not lower degi and raises 2 dege   by 4. The correction term yields 2 because the range of r is contained in G .D L /1 ; this adds up to 2 as desired. The term Me1 rMe2 in the upper right corner does  not change dege and lowers degi by at most 2 for general elements of L , and it  does not lower degi for elements of G . Together with the change C4 or C2 in the 0 correction term, we get an increase in deg.2 / by at least 2. Since both summands on 0 the right hand side of (5.103) have deg.2 / D 2, the same argument as above yields  0 deg.2/ .l/  deg.2 / .l/  deg.l/ for all l 2 L . Furthermore, the lowest order  terms must be of the form l0 Dl1 : : : Dl2n with l0 2 L .     Recall that the chain map retraction f from ker X. Tp/ to X. TK/ is f D f2 ı f1  0  0  with f2 D X. TK / ı X./. The map X. TK / is the obvious projection X. T L / !   0 X. TK/ that vanishes on forms with degi > 0. Thus X. TK / maps even forms with  deg t  2n to forms in TK with deg  2n. Suppose now that x 2 2n .E W Q/. Then    f1 .x/ 2 2n2 .E W Q/ \ L by Lemma 5.97 and hence 3 deg t  ı f1 .x/  2n  2

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235

by Lemma 5.102. It follows that 3 deg f .x/  2n  2. Since deg f .x/ is an even integer,   f 6n2 .E W Q/  2n .K/ for all n 2 N1 . (5.104) even If s is multiplicative, then a similar computation yields   2n f 4n .K/ for all n 2 N. even .E W Q/  

(5.105)

With modest effort, we could now control the function ƒ in Theorem 4.72 by ƒ.n/  n C const for general extensions and ƒ.n/  n=2 C const for split extensions, with a reasonably sized constant like, say, 10. The complicated computations below control this constant and are mainly worthwhile because the final result is optimal, as shown by the work of Michael Puschnigg [86], [89]. Lemma 5.106. Suppose that f and h map mC1 .E W Q/ to Fn .K/ and Fn .E W Q/, respectively, for some n  m. Then there exist other maps f and h with the same properties that map Fm .E W Q/ to Fn .K/ and Fn .E W Q/, respectively.  mC1  mC1 Proof. Recall that Fm .E W Q/ differs from  .E W Q/ by b  .E W Q/ ;  mC1  we may replace this by .b C B/  .E W Q/ because B raises the degree. If  b C B were the boundary map of X. TE/, we could argue that  a chain map that maps mC1 .E W Q/ to the subcomplex Fn K maps .b C B/ mC1 .E W Q/ to h ı .b C B/ D .b C B/ ı h C f  id would yield .b C B/.Fn K/   Fn K; similarly,  h ı .b C B/ mC1 .E W Q/  Fn .E W Q/ because n  m. The boundary map in the X-complex is not quite b C B, but the projection to the -homogeneous subspace puts this right (see the proof of Theorem 5.38). Thus, if we replace f by 0 and h by the standard contracting homotopy outside the -homogeneous subspace, then the new maps map Fm ! Fn as asserted.   Disregarding b 6nC3 .E W Q/ by Lemma 5.106, (5.104) shows that f maps the even part of F6nC2 .E W Q/ to 2nC2 K  F2nC1 .K/  F2n .K/: This remains so for F6nC5 .E W Q/. A similar computation shows that f maps the even part of F4nC2 .E W Q/ F4nC3.E W Q/ to F2nC1  .K/   F2n .K/ if s is multiplicative. To tackle the subspaces b 6nC3 .E W Q/ and b 4nC3 .E W Q/ that we have disregarded here, we must control f on the odd part as well. Lemma 5.107. f maps F6nC5 .E W Q/ to F2nC1 .K/.

 6nC6 Proof. .E W  We have already tackled the even part, and this also takes care of b  odd Q/ by Lemma 5.106. Hence it remains to study f .x/ for a monomial x 2  .E W Q/ with deg.x/  6n C 7. By Lemma 5.98, f1 .x/ is a linear combination of monomials   l0 Dl1 with l0 2 L C , l1 2 L , and deg.l0 / C deg.l1 /  6n C 2; the more precise information in Lemma 5.98 is not yet needed. Then X./ıf1 .x/ is the class of .l0 / D.l1 /

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5 Analytic cyclic homology and analytically nilpotent extensions

 odd   in 1 . T L /=Œ ;  Š  L , where we use the isomorphism described in Lemma 5.44. Lemma 5.102 implies that deg t .l0 / C deg t .l1 /  2n C 2. Lemma 5.44 shows that  odd  this becomes a sum of monomials in  . L / with deg t  2n C 1, and the part with deg t D 2n C 1 is of the form b.X / with deg t X D 2n C 2. Projecting down from     L to K, we end up in F2nC1 .K/ as desired.   Without further work, we only get f F6nC2 .E W Q/  F2n1 .K/, which is not quite good enough. Before we improve this estimate, we consider the case of split extensions, where ideas. Almost exactly the same argument as  we need no additional  above yields f F4nC3 .E W Q/  F2nC1 .K/ if the section s is an algebra homomorphism.   Lemma 5.108. If s is multiplicative, then f F4nC2 .E W Q/  F2n .K/. Proof. It remains to prove f .y/ 2 F2n .K/ for y 2 4nC3 .E W Q/. First, Lemma 5.98 shows that f1 .y/ is a linear combination of terms of the form l0 Dl1 with deg l0 C   deg l1  4n C 2 and x ˇ ke Dl  x ˇ k D.e ˇ l/ with x 2 TC E, l 2 L C , k 2 K, e 2 E, and deg x C deg l  4n. The same argument as above shows that f2 maps terms of the first form to F2nC1 .K/  F2n .K/. When we rewrite .x ˇ ke/ D.l/   .x ˇ k/ D.e ˇ l/ as an element of odd L according to Lemma 5.44, then the terms j 2n that come from  ı d.! ˇ / and  .! d/ have deg t  2n C 1 and are therefore  0 / to F2n .K/. The critical terms involve mapped by X. TK   b .x ˇ ke/ } .l/  .x ˇ k/ } .e ˇ l/ D b ı .x ˇ ke ˇ l  x ˇ k ˇ e ˇ l/ D b ı .x ˇ dk de ˇ l/: Here we should be careful because the sum over  2j ıb.! ˇ/ in Lemma 5.44 depends on the external degree of . But it turns out that dege l0 Cdege l1 D 2n for all monomials l0 Dl1 that appear in X./ ı f1 .y/ and have deg t D 2n. Since we may assume that x and l above are homogeneous, the same summands  2j ı b occur for all relevant terms.  0 / ı b ı .x ˇ dk de ˇ l/ 2 Finally, deg t .x ˇ dk de ˇ l/  2n C 2 yields X. TK F2nC1 .K/.   The proof of f F6nC2 .E W Q/  F2n .K/ requires the following addendum to Lemma 5.102: Lemma 5.109. Let x 2 2n E, k 2 K, e 2 E, and let ˛ D 2 if s is multiplicative and ˛ D 3 otherwise. Then .x ˇ ke  e ˇ x ˇ k/ is a linear combination of terms with  deg.˛/  2nC2 and terms of the form b.y/ for y 2 odd TE with deg.˛/ t t .y/  2nC˛.   Here we use the obvious extension of deg.˛/ to . TE/ . L /. t     Proof. The definition of F W E  T L ! T L extends literally to E  . I ˇ T TE/ !      . I ˇ T TE/, and the resulting map TE ! M. I ˇ T TE/ has the same properties

5.4 Excision in analytic and local cyclic homology

237

  as its restriction to TE ! M. T L /. This extension is useful because we can write .x ˇ ke  e ˇ x ˇ k/ D x F ke  e F x F k D x F .k } e C dk de C Dk De/  e F x F k D .x F k/ } e C x F .dk de C Dk De/  e F x F k D Œx F k; eˇ  D.x F k/ De C x F .dk de C Dk De/ C e ˇ .x F k/  e F .x F k/: The first term Œx F k; eˇ D b.x F k De/ has the required form because deg.˛/ t .x F k/  deg.x/. The terms D.x F k/ De and x F.dk de CDk De/ have deg.˛/  deg.x/C2 bet   cause of Lemma 5.102, which remains valid on I ˇ T TE. The proof of Lemma 5.102 shows that monomials with deg.˛/ D 2 deg.x/ that appear in x F k must be of the form t l0 Dl1 : : : Dl2n . On such monomials, e ˇ  e F raises deg.˛/ by at least 2. t   Lemma 5.110. f F6nC2 .E W Q/  F2n .K/. Proof. It remains to look at f .y/ for y 2 6nC3 .E W Q/. First, f1 .y/ is a linear combination of special expressions z as in Lemma 5.98. We apply X./ to these expressions and then bring the result into standard form using Lemma 5.44. In all cases in Lemma 5.98, the terms involving d in Lemma 5.44 are harmless, and the P critical terms that involve b contain the sum jND0  2j ı b.! ˇ / for the same upper   bound N . When we apply the multiplication map 1 . T L / ! T L , ! D 7! ! ˇ , to X./.z/ with z as in Lemma 5.98, then some simplifications as in the proof of Lemma 5.108 occur, and we remain with two possible cases: either b ı .l/ with deg l  6n C 2 or b ı .x ˇ ke  e ˇ x ˇ k/ with deg x  6n. The first case produces  only terms in T L with deg t  2n C 1; the second case does so too by Lemma 5.109 because b ı b D 0.   This finishes the proof that f F3nC2 .E W Q/  Fn .K/ in general and   f F4nC2 .E W Q/  F2n .K/;

  f F4nC3 .E W Q/  F2nC1 .K/

in the split case, as asserted in Theorem 4.72. Next we consider the homotopy h D h1 C h2 ı f1 . The first piece h1 is harmless by Lemmas 5.97 and 5.98. The other piece requires some more care than the estimation of f D f2 ı f1 because the homotopy h2 D X.H / ı X./ may involve a further degree loss. Inspection of the formula for h2 shows:  Lemma 5.111. Let X 2 T L be homogeneous for dege and degi , and let 2m WD deg t .X/. If degi .X / D 0, then X.H /.X / 2 F2m1 .E W Q/; if degi .X / D 2, then X.H /.X/ 2 F2m2 .E W Q/; if degi .X /  4, then X.H /.X / 2 F2m3 .E W Q/.

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5 Analytic cyclic homology and analytically nilpotent extensions

Proof. Equation (5.100) describes X.H / on even forms. Lemma 5.44 shows that the R1 terms 0 HP t .l0 / DH t .Dl1 : : : Dl2n / dt and Z

1

H t .l0 Dl1 : : : Dl2j 2 / ˇ HP t .Dl2j 1 Dl2j /DH t .Dl2j C1 : : : Dl2n / dt

0

in (5.100) always belong to F2m1 .E W Q/. Thus it remains to study Z 1 H t .Dl2j C1 : : : Dl2n ˇ l0 Dl1 : : : Dl2j 2 / ˇ HP t .l2j 1 / DH t .l2j / dt:

(5.112)

0

 This always belongs to F2m3 .E W Q/ by Lemma 5.44. Since HP vanishes on K  TK, (5.112) vanishes if degi .X / D 0. Now suppose degi .X / D 2. Then all but one entry lj in (5.112) belong to K; if (5.112) is non-zero, then this must be l2j 1 , so that l2j 2 K. Thus (5.112) is already of the form !Dk with k 2 K and hence belongs to F2m2 .E W Q/. Now we can show that h maps the even part of F6nC5 .E W Q/ to F2nC1 .K/. The point is that  ı f1 produces forms with deg.3/ t  6n C 4, which implies deg t  2n C 4 or both deg t D 2n C 2 and degi D 0. In either case, Lemma 5.111 shows that X.H / maps such forms to F2nC1 .E W Q/ as needed.   When we apply h to the even part of F6nC2 .E W Q/ without b 6nC3 .E W Q/ , we get terms with deg.3/ t  6n C 2, which implies deg t  2n C 4 or both deg t D 2n C 2 and degi 2 f0; 2g. This still suffices to conclude that X.H / maps such forms to F2n .E W Q/. In the split case, we must show that h maps even forms of degree at least 4n C 4 to F2nC1 .E W Q/. But the argument above only yields a map to F2n .E W Q/. To gain the extra filtration step, we use that the terms in X./ ı f1 4nC4 .E W Q/   of lowest degree are in the range of b on I ˇ  TE by Lemma 5.109. Since H   extends to a polynomial homotopy T TE ! CŒt  ˝ TE, X.H / also extends to a map   X. T TE/ ! X. TE/. Now the following lemma, whose proof we omit, finishes the estimate of h on even forms:   Lemma 5.113. X.H / ı b maps terms in I ˇ  TE with deg t  n C 2 to Fn .E W Q/ for all n 2 N. Finally, we study h on odd forms. Recall that h D h1 C X.H / ı X./ ı f1 . On odd forms, X.H / is described by (5.99). Notice that X.H /.y/ D 0 if degi .y/ D 0 and that deg X.H /.y/  deg t .y/  1. As above, we see that X./ ı f1 maps odd forms of degree  6n C 7 to linear combinations of differential forms with deg t  2n C 3 and differential forms of the form b.y/ with deg.3/ t .y/  6nC4; in the latter case, we either have deg t .y/  2nC4 – so that deg t b.y/  2n C 3 – or deg t .y/ D 2n C 2 and degi .y/ D 0 – so that b.y/ is annihilated by X.H /. This implies that h maps F6nC5 .E W Q/ to F2nC1 .E W Q/.

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Almost the same argument shows that h maps F4nC3 .E W Q/ to F2nC1 .E W Q/ if s is multiplicative. Similar arguments as above show that X./ ı f1 maps odd forms of degree  6n C 3 to linear combinations of terms y0 and b.y1 / with deg t .y0 /  2n C 3 or .deg t y0 ; degi y0 / D .2n C 1; 0/ and deg t y1 D 2n C 2. Since X.H / kills forms with degi D 0 and maps forms with deg t y  2n C 3 to F2nC1 .E W Q/, Lemma 5.113  shows that the resulting forms are mapped to F2n .E W Q/. Thus h F6nC2 .E W Q/  F2n .E W Q/. If s is multiplicative, almost the same argument shows that h maps F4nC2 .E W Q/ to F2n .E W Q/. This finishes the proof of Theorem 4.72.

Chapter 6

Local homotopy invariance and isoradial subalgebras

A crucial property of the local cyclic theory is its invariance under passage to isoradial subalgebras. More precisely: isoradial homomorphisms with approximably dense range are HL-equivalences; it is crucial here to allow local chain homotopy equivalences. Our proof follows ideas of Michael Puschnigg ([88]). We have described three categories of chain complexes and functors !  !  !  Kom. C / ! HoKom. C / ! HoKom. C /loc !  in §2.3. The latter two categories are the localisations of Kom. C / at chain homotopy equivalences and local chain homotopy equivalences, respectively. We carry this over ! to algebras in Ban. ! We have already met the homotopy category of algebras HoAlg.Ban/; it has the ! same objects as Alg.Ban/ but homotopy classes of algebra homomorphisms as mor! phisms. There are several variants of HoAlg.Ban/ depending on the notion of homotopy that is used; we prefer smooth homotopies because continuous homotopies do not work for cyclic homology. Let ŒA; B denote the set of morphisms A ! B in the homotopy category of algebras. ! In order to define local homotopy equivalences of algebras, we identify Alg.Ban/ with a category of inductive systems: ! ! Alg.Ban/ Š Alg fp .Ban/; ! where Alg fp .Ban/ is the full subcategory of finitely presented objects in Alg.Ban/. An algebra homomorphism f W A ! B is called a local homotopy equivalence if it induces bijections ŒX; A ! ŒX; B for all finitely presented algebras X . It is called an approximate local homotopy equivalence if T f W T A ! T B is a local homotopy equivalence. Theorem 4.27 implies that X.T f / is a chain homotopy equivalence if T f is a smooth homotopy equivalence. Similarly, X.T f / D HL.f / is a local chain homotopy equivalence if T f is a local homotopy equivalence, that is, f is an approximate local homotopy equivalence. The main point is that isoradial homomorphisms with approximably dense range are approximate local homotopy equivalences; this yields the invariance of the local cyclic theory under passage to isoradial subalgebras. The proof uses a simple concrete description of (approximate) local homotopy equivalences, which also shows that the notion of homotopy that we use (smooth or continuous) does not matter as long as the algebras involved are locally multiplicative.

6.1 Local homotopy equivalences

241

Much of this theory still works for bornological algebras. But a technical problem ! forces us to work in the larger category Ban: the completion functor in Cborn, which enters in the construction of T A, is not local; hence local constructions that work in the uncompleted version of T A need not “extend” to the completions. We can avoid this difficulty if we require the uncompleted version of T A to be subcomplete and hence to embed in its completion; equivalently, T .diss A/ Š diss.T A/. This covers most cases of interest by Theorem 2.13. But the general theory of local homotopy equivalences works more smoothly for inductive systems.

6.1 Local homotopy equivalences Theorems 3.10 and 3.70 imply that the categories of locally multiplicative algebras ! in Ban and Cborn are equivalent to the categories of inductive systems of Banach algebras and of reduced inductive systems of Banach algebras, respectively. In §6.1.1, ! we investigate the structure of general algebras in Ban, which is considerably more complicated. We need this analysis to define local homotopy equivalences between such algebras. Since our main applications deal only with locally multiplicative algebras, you may skip this section if you do not feel at home with inductive systems. Even if we were only interested in locally multiplicative algebras, our first steps towards asymptotic morphisms after Definition 6.41 force us to consider certain algebras that are not locally multiplicative.

6.1.1 Local structure of algebras in categories of inductive systems ! We describe Alg.Ban/ as a category of inductive systems, proceeding as in the proof of Theorem 2.15, which describes the local structure of chain complexes. But now the finitely presented objects become more complicated. !  Finitely presented algebras. We work in the more general setting of algebras in C for any additive symmetric monoidal category C with cokernels and kernels. !  We first construct special finitely presented objects in C using finite presentations. !  We will see eventually that this yields all finitely presented objects of Alg. C /. The !  !  !  forgetful functor Forget Alg. C / ! C has a left adjoint functor that sends B 22 C to its tensor algebra TB (see Definition A.59), which is defined by the adjointness relation Alg.TB; Y / Š Hom.B; Y / !  and always exists because C has direct sums and these are compatible with ˝. If !  B 22 C, then Hom.B; / commutes with inductive limits in C , so that TB is finitely !  presented in Alg. C /. We get more finitely presented objects by adding relations:

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6 Local homotopy invariance and isoradial subalgebras

!  Definition 6.1. The algebra U.B; %/ 22 Alg. C / generated by B 22 C with relations % W R ! TB for R 22 C is defined by the condition !  Hom.U.B; %/; Y / Š ff 2 Alg.TB; Y / j f ı % D 0g for all Y 22 Alg. C /: !  Let Alg fp .C/  Alg. C / be the full subcategory of algebras of this form. !  It is clear from the universal property that U.B; %/ is finitely presented in Alg. C /. To construct it, equip the cokernel of the map idTB ˝ % ˝ idTB W TB ˝ R ˝ TB ! TB !  in C with the canonical algebra structure. Partial algebras. Now we simplify the presentation of U.B; %/ and relate it to a partial algebra in C, that is, an object B of C with a partially defined multiplication map m W R ! B for some subobject R ! B ˝ B; we require no associativity because this will not be relevant. LN ˝n The map % W R ! TB must factor through the direct sum BN WD nD1 B for some N 2 N because the underlying inductive system of TB is the direct limit of these subspaces. The canonical map BN ! TB induces a split surjective algebra homomorphism TBN ! TB. The multiplication in TB restricts to a map BN 1 ˝B ! BN  TBN ; let %0N W BN 1 ˝ B ! BN ˚ BN ˝ BN  TBN be the difference of this map and the canonical maps BN 1 ˝B ! BN ˝BN  TBN . It is easy to see that U.B; %/ Š U.BN ; %0N ˚ %/: The improvement is that %0N is a map into BN ˚ BN ˝ BN . Thus we may always choose finite presentations with this special property. Now consider U.B; %/ for a map % W R ! B ˚ B ˝ B. Let R0 WD ker.PB˝B ı %/, where PB˝B W B ˚ B ˝ B ! B ˝ B is the coordinate projection. Then % restricts to a map R0 ! B. Our presentation implies that any algebra homomorphism out of U.B; %/ vanishes on R0 . Hence we may replace B by B=R0 . This yields another presentation for the same algebra with the additional property that PB˝B ı % W R ! B˝B is a monomorphism, that is, R is a subobject of B˝B. The map PB ı% W R ! B is a partially defined multiplication map with domain of definition R  B ˝ B. A map f from B to an algebra D induces an algebra homomorphism on U.B; %/ if and only if it is (partially) multiplicative in the sense that the following diagram commutes: 

R FF / B ˝ B FF FF mult FFF # B

f ˝f

/D˝D mult

f

 / D.

(6.2)

6.1 Local homotopy equivalences

243

!  The passage from partial algebras in C to finitely presented algebras in C is not fully faithful: there are more algebra homomorphisms U.B1 ; %1 / ! U.B2 ; %2 / than partial algebra homomorphisms .B1 ; %1 / ! .B2 ; %2 / because any finitely presented algebra admits lots of non-equivalent presentations. The structure theorem

!  Theorem 6.3. An object of Alg. C / is finitely presented if and only if it belongs to ! !  Alg fp .C/, and the functor lim Alg fp .C/ ! Alg. C / is an equivalence of categories. ! Proof. We have already seen that objects of Alg fp .C/ are finitely presented. We claim !  that any A 22 Alg. C / is an inductive limit of objects of Alg fp .C/. This implies that ! !  the categories Alg. C / and Alg fp .C/ are equivalent. The tensor algebra TA of an algebra comes with a canonical algebra homomorphism A W TA  A, which is a semi-split surjection ([17]). Let JA be its kernel, so that we have a semi-split extension JA  TA  A. Write Forget.A/ D .Ai /i2I ;

Forget.JA/ D .Bj /j 2J

!  in C , then TA Š lim .TAi /i2I . The map JA ! TA is described by maps %ji W Bj ! ! TAi for all j 2 J and sufficiently large i. Each of them yields a finitely presented alge! bra U.Ai ; %ji /. These form an inductive system in Alg fp .C/, whose limit is isomorphic to A. !  It remains to observe that any finitely presented object of Alg. C / belongs to Alg fp .C/. By Example 2.19, this follows if Alg fp .C/ is closed under retracts. We omit the proof because we are not going to use this, anyway. ! Theorem 6.3 applies to algebras in Cborn  Ban as well. But it seems hard ! to determine which inductive systems in Alg fp .Ban/ belong to bornological algebras because the process of adding relations always creates unreduced inductive systems. A complete bornological algebra is a direct limit of a canonical reduced inductive system of partial Banach algebras: take subsets of the form .S [ S  S /~ for S 2 Sc .A/ and use the product defined on AS . But the category of reduced inductive systems of partial Banach algebras is equivalent to the category of partial algebras in Cborn, which is larger than Alg.Cborn/. Hence we do not get an equivalence of categories. Remark 6.4. For algebras in Cborn, we can only expect Hom.A; / to behave well for reduced inductive systems. A complete bornological algebra is called finitely generated if Hom.A; / commutes with reduced inductive limits. It is easy to see that A is finitely generated if and only if there is a bounded subset S 2 S.A/ such that the sets S n for n 2 N generate the bornology S.A/. It is almost trivial that Alg.Cborn/ is equivalent to the category of reduced inductive systems of finitely generated bornological algebras. But this result is not useful because finitely generated bornological algebras are not yet local enough.

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6 Local homotopy invariance and isoradial subalgebras

6.1.2 Definition of local homotopy equivalences ! Let ŒA; B for two algebras A and B in Ban denote the set of homotopy classes of algebra homomorphisms A ! B. We do not specify the notion of homotopy (smooth, bounded variation, continuous) because this has little effect on the resulting theory. The only requirement is that homotopies between maps A ! B are maps A ! BŒ0; 1 for some cylinder functor B 7! BŒ0; 1 that commutes with inductive limits. The ! ! homotopy category of algebras HoAlg.Ban/ has the same objects as Alg.Ban/ and morphism sets ŒA; B. The following definition of local homotopy equivalences for algebras mimics the corresponding definition for chain complexes (Definition 2.22). ! Let A; B 22 HoAlg.Ban/ and write A D lim .Ai /i2I and B D lim .Bj /j 2J for ! ! inductive systems of finitely presented algebras using Theorem 6.3. The canonical functor ! ! Alg.Ban/ ! HoAlg.Ban/ preserves finite presentability, that is, the canonical map lim ŒAi ; Bj  ! ŒAi ; lim Bj  ! ! is an isomorphism for all i 2 I ; it is crucial here that the cylinder functor commutes with inductive limits. Now we get a natural transformation ŒA; B ! limŒAi ; B Š lim limŒAi ; Bj :   ! i

i

j

The target lim lim ŒAi ; Bj  is the space of morphisms .Ai /i2I ! .Bj /j 2J in the i !j ! category of inductive systems over HoAlg.Ban/; more precisely, we may replace ! HoAlg.Ban/ by the full subcategory with objects Alg fp .Ban/, which we denote by ŒAlg fp .Ban/. Thus we get a functor ! ! HoAlg.Ban/ ! ŒAlg fp .Ban/ that acts identically on objects. Definition 6.5. An algebra homomorphism f W A ! B is called a local homotopy ! equivalence if its image in ŒAlg fp .Ban/ is invertible. ! ! An algebra A in Ban is called locally contractible if its image in ŒAlg fp .Ban/ is zero. Of course, A is locally contractible if and only if the zero map 0 ! A is a local homotopy equivalence. Thus the local homotopy equivalences determine the locally contractible algebras. But unlike for chain complexes, we cannot describe local homotopy equivalences only in terms of locally contractible algebras. This is why we shall not use locally contractible algebras in the sequel. The following lemma is an analogue of Lemma 2.24:

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245

! Lemma 6.6. Let A and B be algebras in Ban and let f W A ! B be an algebra homomorphism. Write f D .fi / for a morphism of diagrams fi W Ai ! Bi , i 2 I , with finitely presented Ai and Bi .this is possible by Remark 1.133), and let ˛i W Ai ! A;

˛ij W Ai ! Aj ;

ˇi W Bi ! B;

ˇij W Bi ! Bj

be the canonical maps. The following are equivalent: • f is a local homotopy equivalence; • ŒX; f  W ŒX; A ! ŒX; B is invertible for all X 22 Alg fp .Ban/; • for each i 2 I there are algebra homomorphisms gi W Bi ! A;

hA i W Ai ! AŒ0; 1;

hB i W Bi ! BŒ0; 1

with ev0 ı hA i D ˛i ;

ev1 ı hA i D gi ı fi ;

ev0 ı hB i D ˇi ;

ev1 ı hB i D f ı gi I

• for each i 2 I there are j 2 Ii and algebra homomorphisms gi W Bi ! Aj ;

hA i W Ai ! Aj Œ0; 1;

hB i W Bi ! Bj Œ0; 1

with j ev0 ı hA i D ˛i ;

ev1 ı hA i D gi ı fi ;

j ev0 ı hB i D ˇi ;

ev1 ı hB i D fj ı gi :

Being a local homotopy equivalence is hereditary for inductive systems of algebra homomorphisms. Proof. The first assertion implies the second one because ŒX; A D lim ŒX; Ai  ! for finitely presented X. When we apply this to ˇi , we get gi W Bi ! A with f ı gi ˇi . Since f ı gi ı fi ˇi ı fi D f ı ˛i W Ai ! B, we also get gi ı fi ˛i . These B two relations generate the homotopies hA i and hi . Thus the second assertion implies the third one. Since the cylinder functor commutes with inductive limits, the definition of morphisms of inductive systems yields factorisations of these maps through Aj and Bj for some j 2 Ii and hence the fourth assertion. The existence of such maps means ! that f induces an isomorphism in ŒAlg fp .Ban/ between .Ai / and .Bi /, that is, f is a local homotopy equivalence. Hence all statements are equivalent. Finally, the last characterisation is evidently hereditary for inductive limits. To specialise this to complete bornological algebras, we let S .2/ WD .S [ S  S /~ for S 2 Sc .A/.

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6 Local homotopy invariance and isoradial subalgebras

Lemma 6.7. Let A and B be .complete/ bornological algebras and f W A ! B a bounded algebra homomorphism. Then diss f W diss A ! diss B is a local homotopy equivalence if and only if for each pair of .complete/ bounded disks S  A, T  B with f .S/  T there are bounded linear maps gT W BT .2/ ! A;

hS W AS .2/ ! AŒ0; 1;

hT W BT .2/ ! BŒ0; 1

that are multiplicative on T , S , and T , respectively, such that ev0 ı hS .x/ D x; ev1 ı hS .x/ D gT ı f .x/; ev0 ı hT .y/ D y; ev1 ı hT .y/ D f ı gT .y/ for all x 2 AS .2/ , y 2 BT .2/ . y AS ! AS .2/ defines a partial Proof. The restriction of the multiplication to  W AS ˝ 0 algebra and hence a finitely presented algebra AS . By design, algebra homomorphisms A0S ! D for a bornological algebra D correspond bijectively to bounded linear maps f W AS .2/ ! D with f .x1  x2 / D f .x1 /  f .x2 / for all x1 ; x2 2 S . The algebras A0S for S 2 Sc .A/ form an inductive system in Alg fp .Ban/ with lim A0S Š diss A. This ! shows that the assertion is a special case of Lemma 6.6.

6.1.3 From algebras to chain complexes ! Theorem 6.8. Let A and B be quasi-free algebras in Ban and let f W A ! B be an algebra homomorphism. If f is a local homotopy equivalence, so is the induced chain map X.f / W X.A/ ! X.B/. This result holds, say, for smooth or bounded variation homotopies: we need the homotopy invariance of the X-complex (Theorem 4.27). Proof. Write f as an inductive limit of algebra homomorphisms fi W Ai ! Bi between finitely presented algebras. We have X.f / D lim X.fi / and X .2/ .f / D lim X .2/ .fi / ! ! B y commutes with direct limits. Construct the maps gi , hA because ˝ i , and hi as in Lemma 6.6. As in the proof of Theorem 4.27, the homotopy between gi ı f and ˛i W Ai ! A yields a chain homotopy 2 ı X.gi / ı X.f / 2 ı X.˛i /, where

2 W X .2/ .A/ ! X.A/ denotes the canonical projection. Since A is quasi-free, 2 is a chain homotopy equivalence. Hence X .2/ .gi / ı X .2/ .f / and X .2/ .˛i / are chain homotopic maps X .2/ .Ai / ! X .2/ .A/. Similarly, X .2/ .f ı gi / and X .2/ .ˇi / are chain homotopic maps X .2/ .Bi / ! X .2/ .B/. Now the third criterion in Lemma 2.24 yields that X .2/ .f / is a chain homotopy equivalence; this step works although the chain complexes X .2/ .Ai / and X .2/ .Bi / need not be finitely presented. Finally, since A and B are quasi-free, we may replace X .2/ by X.

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247

Remark 6.9. The above proof carefully avoids the issue whether the finitely presented algebras Ai can be chosen (analytically) quasi-free. I see no reason why this should be the case, even for an analytic tensor algebra. The best result that I could obtain in this direction is that the inductive system of finitely presented algebras for an analytic ! tensor algebra is isomorphic as an inductive system of algebras in Ban to a system of algebras that are analytically quasi-free but not finitely presented.

6.2 Approximate local homotopy equivalences ! Definition 6.10. Let A and B be algebras in Ban. An algebra homomorphism f W A ! B is an approximate local homotopy equivalence if T f W T A ! T B is a local homotopy equivalence. Theorem 6.11. If f W A ! B is an approximate local homotopy equivalence, then HL.f / is a local chain homotopy equivalence. Thus f is an HL-equivalence. Proof. Recall that HL.A/ is chain homotopy equivalent to X.T A/ and that T A is always quasi-free. Hence the assertion follows from Theorem 6.8. Theorem 6.20 will soon provide lots of examples of approximate local homotopy equivalences. We must describe them more concretely to prepare for this. We do this first for locally multiplicative algebras; this simpler case suffices for Theorem 6.20. ! Definition 6.12. Let A be a Banach algebra and let B be an algebra in Ban; let y be a bounded disk. Let f W A ! B be a linear map and let !f W A ˝A y !B S  A ˝A be its curvature. We say that f has small curvature .with respect to S / if the restriction of !f to S  S has spectral radius strictly less than 1. Let M.S I A; B/ be the set of linear maps A ! B with small curvature and H.S IA; B/ the set of homotopy classes of such maps, where homotopies are elements of M SI A; BŒ0; 1 . Usually, we take S of the form c  .T ˝ T /~ for some c 2 R1 , where T is the unit ball of A. Increasing c, we get better control on the curvature. Theorem 6.13. An algebra homomorphism f W A ! B between two locally multi! plicative algebras in Ban is an approximate local homotopy equivalence if and only if it induces isomorphisms H.SI X; A/ Š H.SI X; B/ for all Banach algebras X and y X. all bounded disks S  X ˝ A

Proof. Recall that JA  T A  A is an analytically nilpotent extension with a canonical linear section A W A ! T A. We claim that A induces a bijection H.SI X; T A/ Š H.SI X; A/:

(6.14)

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6 Local homotopy invariance and isoradial subalgebras

! The extension of Theorem 3.27 to algebras in Ban yields %.hI T A/ D %.A ı hI A/ for any bounded map h W S ! T A. Therefore, h W X ! T A has small curvature if and only if A ı h has. Since A allows us to lift maps, the map in (6.14) is surjective. Injectivity follows by joining two maps h0 ; h1 W X ! T A with A ı h0 D A ı h1 by the affine homotopy .1  t /h0 C t h1 . Write A D lim .Ai /i2I for an inductive system of finitely presented algebras ! .Ai /i2I . The functor H.SI X; / is a homotopy functor and satisfies H.SI X; A/ Š lim H.SI X; Ai /: !

(6.15)

Hence it maps local homotopy equivalences to isomorphisms. As a result, if f is an approximate local homotopy equivalence, then the induced map T f

H.SI X; A/ Š H.SI X; T A/ ! H.S I X; T B/ Š H.S I X; B/ is bijective as desired. Conversely, suppose that these maps are bijective for all .X; S /. We must show that T f W T A ! T B is a local homotopy equivalence. We will verify the third condition ! of Lemma 6.6. First we recall the concrete description of T A; for algebras in Ban this is explained after Lemma 5.9. We write A as an inductive system of Banach algebras .Ai /i2I and equip each Ai with some submultiplicative closed unit ball Si ; let IQ WD I  N and Si;n WD n  Si for .i; n/ 2 IQ. Let AQi;n be the completion of even Ai for the gauge norm of hhSi;n ii} . The resulting inductive system of Banach spaces, equipped with the Fedosov product, is T A. By Theorem 5.12, we can write T A as an inductive limit of Banach algebras. The above system does not achieve this, but it is equivalent to it as an inductive system of Banach spaces. This is good enough for the following argument. By Remark 1.133, we may assume that f is represented by a morphism of diagrams fi W Ai ! Bi with fi .Si /  Ti , where Ti  Bi is the unit ball of Bi . We get T B Š .BQ i;n /i;n with the Fedosov product. Since H.Ti;n I Bi ; A/ Š H.Ti;n I Bi ; B/ by assumption, the canonical map Bi ! B is homotopic to f ı gQ i;n for a map gQ i;n W .Bi ; Ti;n / ! A with small curvature; B the homotopy is a map with small curvature HQ i;n W .Bi ; Ti;n / ! BŒ0; 1. The map gQ i;n ı fi W .Ai ; Si;n / ! A has the same image in H.Ti;n I Ai ; B/ as the canonical map A Ai ! A. Hence we get another map with small curvature HQ i;n W .Ai ; Si;n / ! AŒ0; 1 that connects these two. Composing with A preserves small curvature by (6.14), so that we get maps of small curvature gQi;n

A

.Bi ; Ti;n / ! A ! T A; QB H i;n

BŒ0;1

QA H i;n

AŒ0;1

.Bi ; Ti;n / ! BŒ0; 1 ! T .BŒ0; 1/ ! .T B/Œ0; 1; .Ai ; Si;n / ! AŒ0; 1 ! T .AŒ0; 1/ ! .T A/Œ0; 1I

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249

the algebra homomorphisms T .AŒ0; 1/ ! .T A/Œ0; 1 and T .BŒ0; 1/ ! .T B/Œ0; 1 come from Theorem 5.28. The formula for hf i in Lemma 5.9 already makes sense on AQi;n if we only require that !f .Si;n ; Si;n / be power-bounded. Since sets of spectral radius strictly less than 1 are power-bounded, we can apply this recipe to define maps gi;n W BQ i;n ! T A;

B Hi;n W BQ i;n ! .T B/Œ0; 1;

A Hi;n W AQi;n ! .T A/Œ0; 1:

Now we restrict these maps to suitable Banach subalgebras of AQi;n and BQ i;n . This yields algebra homomorphisms that fulfil the requirements of Lemma 6.6. Thus T f is a local homotopy equivalence, that is, f is an approximate local homotopy equivalence. Remark 6.16. Theorem 2.13 fails if we try to work with bornological algebras. The maps AQi;n ! T A need not be injective if diss.T A/ ¤ T .diss A/, and the locally defined maps in the proof of Theorem 6.13 have no reason to vanish on their kernels. Therefore, diss.T f / need not be a local homotopy equivalence. But we have diss.T A/ Š T .diss A/ for many A by Theorem 2.13. If we impose ! this restriction, then it makes no difference whether we work in Cborn or Ban. ! Theorem 6.17. Let A and B be locally multiplicative algebras in Ban. An algebra homomorphism f W A ! B is an approximate local homotopy equivalence for smooth homotopies if and only if it is so for continuous homotopies. Proof. We claim that we get the same set of equivalence classes H.S I X; A/, regardless which notion of homotopy we use. This implies the assertion  by Theorem 6.13.  We show that for any continuous homotopy h in M SI X; C.Œ0; 1; A/ there is   1 a smooth homotopy in M S I X; C .Œ0; 1; A/ with the same endpoints. We may reparametrise our homotopy so that it is constant outside Œ1=3; 2=3 and extend it to be constant for t R 0 and t  1. Let w W R ! R0 be a smooth function supported in Œ1; 1 with w.x/ dx D 1. Consider the sequence of linear maps hn W X ! C 1 .Œ0; 1; A/ defined by Z w.s/h.a/.t C 2n s/ ds; hn .a/.t / WD R

where h.a/.t/ denotes the value of h.a/ 2 C .Œ0; 1; A/ at t . Since h is bounded with respect to the bornology of uniform continuity on C .Œ0; 1; A/, the sequence .hn /n2N converges uniformly to h as maps to C .Œ0; 1; A/. Lemma 3.76 shows that hn has small curvature as a map to C .Œ0; 1; A/ for sufficiently large n. It has small curvature as a map to C 1 .Œ0; 1; A/ as well because C 1 .Œ0; 1; A/ is an isoradial subalgebra of C .Œ0; 1; A/. Since h is constant outside Œ1=3; 2=3, we have hn .a/.0/ D h.a/.0/ and hn .a/.1/ D h.a/.1/ for n  2. Now we extend our description of approximate local homotopy equivalences to algebras that are not locally multiplicative.

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6 Local homotopy invariance and isoradial subalgebras

! Definition 6.18. Let B be an algebra in Ban (or Cborn) and let .A; ; R/ be a partial algebra in Ban, that is, A and R are Banach spaces with a bounded injective map y A, and  W R ! A is a bounded linear map. We also fix a bounded disk R ! A˝ S  R. The curvature !f W R ! B of a linear map f W A ! B is the difference of the two maps 

f

!A ! B; R

y f ˝f

mult

y A ! B ˝ y B ! B R !A˝

in (6.2). We say that f has small curvature if the restriction of !f to S has spectral radius strictly less than 1. Let M.S I A; B/ be the set of linear maps A ! B with small curvature and H.SIA; B/ the set of homotopy classes of such maps, where homotopies are elements of M SI A; BŒ0; 1 . We can also control the curvature by another constant. This makes no difference eventually because we may replace S by c S for c 2 R>0 to rescale the curvature. ! Theorem 6.19. An algebra homomorphism f W A ! B between two algebras in Ban is an approximate local homotopy equivalence if and only if it induces isomorphisms H.SI X; A/ ! H.SI X; B/ for all partial algebras .X; ; R/ with fixed unit ball S as in Definition 6.18. The proof is very similar to the proof of Theorem 6.13 above and therefore omitted.

6.3 Application to isoradial homomorphisms The following theorem already appeared in [67] for complete bornological algebras. ! The same proof works for algebras in Ban. ! Theorem 6.20. Let A and B be locally multiplicative algebras in Ban and suppose that f W A ! B is an isoradial homomorphism with approximably dense range. Then f is an approximate local homotopy equivalence. Proof. We use the characterisation of approximate local homotopy equivalences in y D be a bounded disk. We Theorem 6.13. Let D be a Banach algebra and let S  D ˝ check that f induces a bijection H.SI D; A/ ! H.S I D; B/. Let h 2 M.SI D; B/. We must find h0 2 M.S I D; A/ with h f ı h0 . Write B D .Bi /i2I for an inductive system of Banach algebras. Then h factors through ˇi W Bi ! B for some i 2 I . Let hi W S ! Bi be such that ˇi ı hi D h. Since f has approximably dense range, there is a sequence of bounded linear maps n W Bi ! A such that f ı n ! ˇi uniformly. Hence f ı n ı hi converges uniformly towards h. This implies uniform convergence of the corresponding curvatures. Lemma 3.76 shows that f ı n ı hi has small curvature for sufficiently large n. So has h0n WD n ı hi

6.3 Application to isoradial homomorphisms

251

because f is isoradial. The spectral radius in BŒ0; 1 is the supremum of the pointwise spectral radii (see §3.5.1). Therefore, the affine homotopy .1  t / h C t f ı h0n has small curvature for sufficiently large n by the same argument. Hence f ı h0n h. This finishes the proof of surjectivity. Now we prove injectivity. Let D; S be as above, let h0 ; h1 2 D ! A be maps with small curvature, and let H W D ! BŒ0; 1 be a homotopy between f ı h0 and f ı h1 with small curvature. We have to prove that Œh0  D Œh1  in H.S I D; A/. The same argument as above shows that there is a sequence of homotopies Hn0 W D ! BŒ0; 1 with small curvature such that f ı Hn0 converges uniformly to H . Hence f ı .ev0 Hn0 / converges uniformly to h0 . Since f is isoradial, Lemma 3.76 yields that the affine homotopy between ev0 Hn0 and h0 has small curvature for sufficiently large n, so that h0 ev0 Hn0 . Similarly, h1 ev1 Hn0 . Hence h0 h1 as desired. Theorem 6.21. Isoradial homomorphisms with approximably dense range are HLequivalences. Proof. Combine Theorems 6.20 and 6.11. Corollary 6.22. Let A and B be complete bornological algebras and let f W A ! B be an isoradial homomorphism with approximably dense range. Suppose that diss.T A/ Š T .diss A/;

diss.T B/ Š T .diss B/:

Then HA .f / W HA .A/ ! HA .B/ is an isomorphism, and HA.f / is a local chain homotopy equivalence. If, in addition, A and B are Silva algebras with the approximation property, then HA .f / W HA .B/ ! HA .A/ is an isomorphism. Proof. The assumptions ensure that the various analytic and local cyclic invariants agree by Theorems 2.13 and 2.44. Hence the assertions follow from Theorem 6.21.

6.3.1 Examples and applications We have exhibited many examples of isoradial homomorphisms in §3.1. Now we check whether they have approximably dense range and apply Theorem 6.20. We refrain from stating the analogous results on analytic cyclic homology and cohomology that follow under suitable additional assumptions from Corollary 6.22. ! To simplify statements, we embed Cborn  Ban without mentioning the dissection functor and equip all topological algebras with the precompact bornology. Completed inductive limits, direct sums, and stabilisations Theorem 6.23. Let .Ai /i2I be a reduced inductive system of complete bornological algebras and let A D lim .Ai /i2I . Let B be another complete bornological algebra ! with an injective bounded homomorphism  W A ! B with approximably dense range.

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6 Local homotopy invariance and isoradial subalgebras 

 B are bornological embeddings for all Suppose that the composite maps Ai ! A ! i 2 I . Then  W A ! B is an HL-equivalence. The hypotheses are verified if B is the inductive limit in the category of C  -algebras of a reduced inductive system of nuclear C  -algebras .Ai /i2I . Proof. The map  is isoradial by Proposition 3.40, and has approximably dense range by assumption. Now apply Theorem 6.21. If the Ai are nuclear, so is the C  -direct limit B; hence it has the (completely positive) approximation property, so that the map A ! B has approximably dense range by Proposition 1.172. Hence Theorem 5.71 applies to the completed inductive limit B as well. L  Theorem 6.24. Let C Ai be the direct sum of a set .Ai /i2I of C  -algebras in the LC   L category of C  -algebras. Then HL i2I Ai i2I HL.Ai /. L L  Proof. The canonical map Ai ! C Ai has approximably dense range because L L  the retractions C i2I Ai  i2F Ai for finite subsets F  I converge towards the L  identity map uniformly on precompact subsets of C Ai . Theorem 6.23 shows that L  L the map Ai ! C Ai is an HL-equivalence. Finally, apply Theorem 5.70. Theorem 6.25. Let A be a C  -algebra, let H be a Hilbert space, and let A˝C  K.H /. Choose a unit vector 2 H . The resulting corner embedding A ! A˝C  K.H / is an HL-equivalence. Thus A 7! HL.A/ is a stable functor on the category of C  -algebras. Proof. Let B be a basis of H and let MB be the algebra of finite matrices in this basis, equipped with the fine bornology. Then MB Š CŒB ˝ CŒB and the multiplication comes from the canonical bilinear map CŒBCŒB ! CŒB as in §A.7. Theorem 5.65 shows that the corner embedding A ! A ˝ MB is an HL-equivalence. Since any bounded subset of MB is already contained in and bounded in Mn for some finitedimensional subalgebra Mn  MB , the map A ˝ MB ! A ˝C  K.H / is isoradial. It has approximably dense range as well (because any precompact subset of A ˝C  K.H / is supported on a separable subspace of H ). Hence it is an HL-equivalence by Theorem 6.21. Theorem 6.26. Let V be a Banach space with Grothendieck’s approximation property .for instance, a Hilbert space/. Let `1 .V / and K.V / be the Banach algebras of nuclear and compact operators on V , respectively, equipped with the precompact bornologies. Then the corner embeddings 1 ! K.V / and 1 ! `1 .V / are HL-equivalences. y Proof. The assertion about `1 .V / follows from Theorem 5.65 because `1 .V / Š V ˝ V 0 for Banach spaces with the approximation property. The map `1 .V / ! K.V / is easily seen to be isoradial, and it has approximably dense range by Proposition 1.172.

6.3 Application to isoradial homomorphisms

253

Algebras of functions ! Theorem 6.27. Let A be a locally multiplicative algebra in Ban. The canonical maps Cck .M; A/ ! C0 .M; A/ for k 2 N [ f1g and a C k -manifold M and C ! .M; A/ ! C .M; A/ for a compact C ! -manifold M are HL-equivalences; here the function spaces carry the bornology of uniform continuity of all relevant derivatives. Proof. These maps are isoradial by Corollary 3.43 (it does not matter whether we work ! in Ban instead of Cborn). The proof of Corollary 3.43 already shows that they have approximably dense range as well. Hence the assertion follows from Theorem 6.21. ! Definition 6.28. Let A and B be algebras in Ban and f0 ; f1 W A  B algebra homomorphisms; give C .Œ0; 1; B/ the bornology of uniform continuity. A uniformly continuous homotopy between f0 and f1 is an algebra homomorphism f W A ! C .Œ0; 1; B/ with f t D ev t ı f for t D 0; 1. Corollary 6.29. Let f0 ; f1 W A  B be uniformly continuously homotopic and assume that B is locally multiplicative. Then HL.f0 / D HL.f1 / in HL0 .A; B/. That is, HL for locally multiplicative algebras is invariant under uniformly continuous homotopies. Proof. The map C 1 .Œ0; 1; B/ ! C .Œ0; 1; B/ is an HL-equivalence by Theorem 6.27. The smooth homotopy invariance of HL means that ev0 and ev1 have the same class in HL0 .C 1 .Œ0; 1; B/; B/ Š HL0 .C.Œ0; 1; B/; B/: Hence HL.f0 / D HL.f1 / if there is a uniformly continuous homotopy f0 f1 . Remark 6.30. It is crucial in Corollary 6.29 to equip C .Œ0; 1/ with the bornology of uniform continuity, which agrees with the precompact bornology by the Arzelà–Ascoli Theorem. If we choose the von Neumann bornology instead, then     HL0 .vN C.Œ0; 1/; C/ Š HL0 vN C .Œ0; 1/ Š HE0 C.Œ0; 1/ Š C.Œ0; 1/0 I here C .Œ0; 1/0 is the dual space of C .Œ0; 1/, that is, the space of Radon measures on Œ0; 1. The first isomorphism uses HL.C/ C; the second follows from Theorem 2.44 and the definition of entire cyclic cohomology; the last one is due to Masoud Khalkhali ([61, Theorem 5.2]) and uses that C .Œ0; 1/ is an amenable Banach algebra. Theorem 1.36 yields that C .Œ0; 1; B/ in Corollary 6.29 is the usual Fréchet algebra of continuous functions Œ0; 1 ! B with the precompact bornology if  B is a Fréchet  algebra with the precompact bornology. Hence the functor HL Cpt. / on the category of Fréchet algebras is homotopy invariant for continuous homotopies in the usual sense. Specialising even further, we get:   Theorem 6.31. The functor A 7! HL Cpt.A/ on the category of C  -algebras is homotopy invariant in the usual C  -algebraic sense.

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6 Local homotopy invariance and isoradial subalgebras

More classes of examples Theorem 6.32. Let A be a complete, locally multiplicative bornological algebra, and let G be a locally compact group. Let  W G  A ! A be a continuous group representation by algebra automorphisms and let A1  A and A!  A be the subalgebras of smooth and real-analytic elements for this group action. The map A1 ! A is an HL-equivalence; so is the map A! ! A if it has approximably dense range. Proof. Theorem 3.46 shows that the map A1 ! A is isoradial, and its proof already shows that it has approximably dense range. The map A! ! A has approximably dense range by assumption and is isoradial by Theorem 3.46. Now apply Theorem 6.21. As in Examples 3.47 and 3.48, we get that smooth and C  -algebraic non-commutative tori and smooth and C  -algebraic Toeplitz algebras are HL-equivalent. Using the naturality of the boundary map in the Excision Theorem 5.77, we conclude that the computations of the boundary maps for the cone and Toeplitz extensions in §5.4.5 apply to the C  -algebraic versions of the cone extension and Toeplitz extension as well. Theorem 6.33. Let A be a Fréchet algebra and let A0  A be a dense Fréchet subalgebra that is strongly spectrally invariant. The map Cpt.A0 / ! Cpt.A/ is an HL-equivalence if it has approximably dense range; in particular, this applies if A has Grothendieck’s approximation property. Proof. Theorem 3.53 asserts that Cpt.A0 / is isoradial in Cpt.A/. Proposition 1.172 ensures that it has approximably dense range. Now apply Theorem 6.21.  Example 6.34. Let G be a discrete group with rapid decay and such that Cred .G/ has Grothendieck’s approximation property as a Banach space; it is known that this holds if G is a word hyperbolic group (see [14], [38], [54], [55]). Then the map from the  Jolissaint algebra to Cred .G/ is an HL-equivalence. The corresponding maps S k .G/ ! `1 .G/ are HL-equivalences for all k 2 N [ f1; !g and all groups G because `1 .G/ always has the approximation property.

Theorem 6.35. Let B be a separable, locally multiplicative Fréchet algebra with Grothendieck’s approximation property, equipped with the precompact bornology. Then B is HL-equivalent to a locally multiplicative Silva algebra with the approximation property. Hence HL .A/ Š HA .B/ and HL .A/ Š HA .B/. Proof. Theorem 3.66 shows that there is an isoradial Silva subalgebra with the approximation property. Since B has the approximation property as well, the map A ! B has approximably dense range by Proposition 1.172 and hence is an HL-equivalence.

6.4 Local and approximate local homotopy category

255

6.4 Local and approximate local homotopy category Definition 6.36. The local homotopy category and the approximate local homotopy cat! ! egory of algebras in Ban are the localisations of HoAlg.Ban/ at the local and approx! imate local homotopy equivalences, respectively. We denote them by HoAlg.Ban/loc ! and HoAlg.Ban/apple and morphisms in them by ŒA; Bloc and ŒA; Bapple . Theorem 6.11 shows that HL descends to a functor ! ! HL W HoAlg.Ban/apple ! HoKom.Ban/loc : ! ! ! ! The categories Alg.Ban/, HoAlg.Ban/, HoAlg.Ban/loc , and HoAlg.Ban/apple have the same objects but different morphisms. The main object of interest is ŒA; Bapple , but we can easily reduce this to ŒA; Bloc : ! Proposition 6.37. Let A and B be algebras in Ban. Then ŒA; Bapple Š ŒT A; T Bloc Š ŒT A; Bloc : Proof. It is clear that approximate local homotopy equivalences become invertible in ŒT A; T Bloc . Therefore, we get a natural map ŒA; Bapple ! ŒT A; T Bloc . The projection T .A / W T T A ! T A is a smooth homotopy equivalence by Corollary 5.33; thus A W T A ! A is an approximate local homotopy equivalence. Hence we get a natural map ŒT A; T Bloc ! ŒA; Bapple , composing on both sides with the isomorphisms T A A and T B B. It is easy to see from the defining properties of localisations that the maps ŒT A; T Bloc  ŒA; Bapple are inverse to each other. We map ŒT A; T Bloc ! ŒT A; Bloc by composing with B . It follows easily from the criteria of Lemma 6.6 that T preserves local homotopy equivalences, so that we get a natural transformation ŒT A; Bloc ! ŒT T A; T Bloc Š ŒT A; T Bloc ; here we use once again the homotopy equivalence of T T A and T A. It is straightforward to check that these two transformations ŒT A; T Bloc  ŒT A; Bloc are inverse to each other. We do not yet have a full description of ŒA; Bloc . These morphism spaces are harder to describe than the corresponding ones for chain complexes, although the definitions may appear similar. The main difference is that we cannot form a homotopy direct limit of algebras because homotopies are given by maps A ! BŒ0; 1 and cannot be formulated in terms of maps H.A/ ! B. In the language of model categories, this means that we cannot find cofibrant replacements. The non-additivity of the category of algebras creates further technical problems. Although we lack a full description of ŒA; Bapple , we can still construct some interesting morphisms, which we call asymptotic morphisms. To prepare the definition, we construct some useful local homotopy equivalences.

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! Notation 6.38. Let B be an algebra in Ban. Reparametrising, we may view elements of BŒ0; 1 as functions Œa; b ! B. We write BŒa; b for BŒ0; 1 parametrised on the interval Œa; b for a; b 2 R, a < b. If Œc; d   Œa; b, then we have a restriction homomorphism BŒa; b ! BŒc; d . We let BŒa; 1/ WD lim BŒa; b:  b!1

! This projective limit exists in Ban by Proposition 1.135 (but is complicated and never locally multiplicative) and inherits an algebra structure. We have restriction homomorphisms BŒa; 1/ ! BŒc; 1/ for a  c. We let Asymp.B/ WD lim BŒa; 1/I ! a!1

! of course, the inductive limit is taken in Alg.Ban/. Example 6.39. Let B D diss Cpt.B 0 / for a Fréchet space B 0 . Since the dissection and precompact bornology functors commute with projective 1.139), the  limits (Theorem  algebra BŒa; 1/ is isomorphic to diss Cpt C.Œa; 1/; B 0 / or diss Cpt C 1 .Œa; 1/; B 0 / depending on the notion of homotopy we use. Since we use functions without any growth condition at 1, this algebra is not locally multiplicative even if B is. Furthermore, its precompact bornology is not countably generated even if B is a Banach algebra. We have defined local and approximate local homotopy equivalences for non-locally multiplicative algebras to cover the example of Asymp.B/. The space of compactly supported functions is locally dense in BŒa; 1/. Therefore, lim Asymp.B/ D 0 in Alg.Cborn/. ! Lemma 6.40. The embedding const W B ! BŒa; 1/ as constant functions is a homotopy equivalence. These maps combine to a local homotopy equivalence B ! Asymp.B/. Here we assume that the homotopy relation is smooth or continuous homotopy; homotopies of bounded variation do not work since the evaluation maps A.Œ0; 1; A/ ! A need not be homotopy equivalences of bounded variation. Proof. The evaluation map eva W BŒa; 1/ ! B satisfies eva ı const D idB . We want to connect the other composition const ı eva and the identity map on BŒa; 1/ by a homotopy defined by hs f .a C t / D f .a C st/ for s 2 Œ0; 1, t 2 R0 . We omit the technical  verification that this defines a morphism of inductive systems BŒa; 1/ ! BŒa; 1/ Œ0; 1; this is complicated because the cylinder functor does not commute with projective limits. ! Definition 6.41. Let A and B be algebras in Ban. An asymptotic morphism from A to B is an algebra homomorphism T A ! Asymp.B/.

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257

It is clear that asymptotic morphisms from A to B define elements in ŒA; Bapple : compose them with the inverse of the local homotopy equivalence B ! Asymp.B/ to get a class in ŒT A; Bloc Š ŒA; Bapple ; homotopic asymptotic morphisms define the same morphism in ŒA; Bapple . It seems likely that the resulting map from homotopy classes of asymptotic morphism to ŒA; Bapple is bijective if A is a countable inductive system. Now we describe asymptotic morphisms more explicitly, in analogy with asymptotic morphisms for separable C  -algebras (see [9]). We assume that A is locally multiplicative to simplify; we write A D .Ai /i2I for an inductive system of Banach algebras Ai , and we fix unit balls Si in Ai . For each n 2 N, let AQi;n be the Banach subalgebra of T Ai whose unit ball is the convex hull of Si [ .nhSi i dSi dSi hSi i/1 . Then T A D lim AQi;n . Hence !     Alg T A; Asymp.B/ Š lim Alg AQi;n ; Asymp.B/  n;i2NI   Š lim lim Alg AQi;n ; BŒa; 1/ :  ! n;i2NI a2N

This means that for each .i; n/ 2 I  N there is an algebra homomorphism AQi;n ! BŒa.i; n/; 1/ or, equivalently, a linear map fi;n W Ai ! BŒa.i; n/; 1/ such that n  hfi;n .Si /i!fi;n .Si ; Si /hfi;n .Si /i  BŒa.i; n/; 1/ is power-bounded. The compatibility of these linear maps implies that fi;n and fi;1 have the same restriction to Œb.i; n/; 1/ for sufficiently large b.i; n/  a.i; n/. We may assume without loss of generality that a.i; n/ D b.i; n/ and hence fi;n D f1;n jŒa.i;n/;1/ . Hence we may describe an asymptotic morphism by a family of linear maps fi W Ai ! BŒai ; 1/ for some map I ! N, i 7! ai , or, equivalently, by a ! morphism f W A ! Asymp.B/ in Ban. Arguing as in §3.5.1 and using the fact that .nhSi i dSi dSi hSi i/1 is absorbed by hSj i.dN Sj /even for some j 2 I , N 2 N, we see that the curvature condition on the maps fi;n amounts to   lim % ev t !fi .Si ; Si / D 0: t!1

This makes precise what it means for fi to become asymptotically multiplicative for t ! 1 and is considerably weaker than uniform convergence ev t ı !fi ! 0. We also remark that, by the universal property of projective limits, a map to BŒa; 1/ is determined uniquely by its restrictions to Œa; b for b 2 Ra . If B is a complete bornological algebra, then a map f to BŒa; b is determined uniquely by the maps ! ev t ı f to B for all t 2 Œa; b, but this fails for algebras in Ban. In most applications, we can lift f to a linear map A ! BŒ0; 1/. This is automatic if B is locally multiplicative and A is a Silva algebra with the approximation property because we can approximate each map fi by linear maps that are globally defined; these eventually satisfy the curvature constraints by Lemma 3.76.

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6 Local homotopy invariance and isoradial subalgebras

Comparing the above notion of asymptotic morphism with the familiar one for C  -algebras, we notice three differences: first, since we have no involution, we cannot ask for compatibility with it; secondly, we control error terms only in spectral radius, not in norm; thirdly, we require asymptotic morphisms to be linear. It is easy to define an approximate local homotopy category of -algebras and approximate local -homomorphisms: simply require involutions on all algebras and restrict attention to -homomorphisms; the analytic tensor algebra inherits a unique involution for which A W A ! T A is involutive. We claim that if f W A ! Asymp.B/ is an asymptotic -homomorphism in this new sense and B is a C  -algebra, then we get uniform convergence kev t !f .S; S /k ! 0 for t ! 1 for all S 2 S.A/. Since the norm and the spectral radius agree for positive elements, the C  -condition shows that norm and spectral radius estimates are equivalent: t!1

k!f .x; y/k D k!f .x; y/  !f .x; y/ k =2 D %.!f .x; y/  !f .x; y/ / =2 ! 0: 1

1

With this modification, the only difference to asymptotic morphisms for C  -algebras is the linearity assumption on our asymptotic morphisms. There is a version of the approximate local homotopy category that allows non-linear asymptotic morphisms and uses a non-linear substitute of the tensor algebra. We do not consider this variant here because it fails to act on local cyclic homology; we could, of course, modify local cyclic homology as well, using the new tensor algebra, but it is not clear whether the excision theorem extends to this new theory. For algebras with the local approximation property, the linear and non-linear versions agree, anyway; this corresponds to the theorem that Kasparov theory and E-theory agree for nuclear C  -algebras.

Chapter 7

The Chern–Connes character

Now we study the relationship between K-theory and local and analytic cyclic homology. We begin with the bivariant Chern–Connes character ch W KK  .A; B/ ! HL .A; B/; which is defined on Kasparov’s KK-theory for separable C  -algebras (equipped with the precompact bornology). This construction of Michael Puschnigg allows us to carry over various known properties of KK to the bivariant local cyclic theory. We prove the following Universal Coefficient Theorem: if A and B are C  -algebras in the bootstrap category, then HL .A; B/ Š Hom.K .A/ ˝ C; K .B/ ˝ C/: Along the way, we show that the chain complex HL.A/ for such algebras is essentially determined by its homology. Using the invariance under passage to isoradial subalgebras, the Universal Coefficient Theorem also extends to important non-C  -algebras. The construction of the bivariant Chern–Connes character is based on formal properties of the theories involved and not constructive. We describe the resulting maps K .A/ ! HL .A/

and

K .A/ ! HL .A/

more explicitly; here K denotes Kasparov’s K-homology. Our discussion of the Chern–Connes character K .A/ ! HL .A/ in K-theory follows Joachim Cuntz and Daniel Quillen ([26, §12]) and works for general bornological algebras; the most convenient domain for such a map is the new topological K-theory top K .A/ of Joachim Cuntz and Andreas Thom ([29]), which is carried over to complete bornological algebras in [21]. We use that C and CŒt; t 1  are analytically quasi-free to define the characters of idempotents and invertible elements. Stability of HL allows top us to extend this to a natural transformation K .A/ ! HL .A/. Cycles for K .A/ for a C  -algebra A are also called Fredholm modules. The existence of a Chern–Connes character for suitable Fredholm modules was one of the main motivations to introduce cyclic cohomology ([6]). In (periodic) cyclic cohomology, we only get a character for finitely summable Fredholm modules. We recall this construction in §7.4. Although we use the X-complex instead of the .b; B/-bicomplex, our formulas agree with those in [6] up to normalisation constants. There are some important examples of Fredholm modules that are not finitely summable, but only  -summable. This is enough to construct a Chern–Connes character in entire cyclic cohomology; this was a basic motivation for Alain Connes to define the entire theory [7]. Almost simultaneously, Arthur Jaffe, Andrzej Lesniewski

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and Konrad Osterwalder ([53]) found a more explicit formula for the Chern–Connes character of -summable Fredholm modules that is closely related to heat kernel methods in index theory. More recently, Denis Perrot ([79], [80]) has studied a bivariant generalisation of this construction. The limited amount of time available prevented me from discussing these important character formulas here. Instead, we consider another character ch W K  .A/ ! HA .A/, whose distinguishing feature is that it requires no summability hypothesis whatsoever. It is based on the Chern–Connes character for finitely summable Fredholm modules and provides a factorisation through HA .A/ of the Chern–Connes character K .A/ ! HL .A/ that we get from the universal property of Kasparov theory. I do not know whether the bivariant character KK .A; B/ ! HL .A; B/ factors through HA .A; B/, but this seems unlikely. We certainly cannot replace analytic by entire cyclic cohomology because HE .A/ D 0 for any stable nuclear C  -algebra A by [61], [63].

7.1 The bivariant Chern–Connes character We construct a multiplicative natural transformation ch W KK  .A; B/ ! HL .A; B/; which is called the bivariant Chern–Connes character and due to Michael Puschnigg. As a result, HL does not distinguish between KK-equivalent C  -algebras. This is important because many K-theory computations for C  -algebras can be constructed purely inside Kasparov theory and therefore also apply to local cyclic homology. As a sample application, we mention the Pimsner–Voiculescu exact sequence for crossed products by automorphisms. This result and some similar computations have also been noted by Vahid Shirbisheh ([98]). The construction of the bivariant Chern–Connes character is a consequence of formal properties of Kasparov theory: we treat KK as a black box and only use its universal property for the construction. More recently, Joachim Cuntz has defined bivariant K-theories for much larger classes of algebras in [17], [21], [28], [29] (these do not coincide with Kasparov’s when we restrict them to C  -algebras). In this context, Joachim Cuntz constructs a bivariant Chern–Connes character with values in bivariant periodic cyclic homology. It is easy to adapt this construction to local and analytic cyclic homology as well.

7.1.1 The universal property of Kasparov theory The bivariant K-theory of Gennadi Kasparov ([57]) unifies K-theory and various versions of K-homology studied previously by Atiyah, Brown–Douglas–Fillmore, and others. It is a bifunctor KK W .sC /op  sC ! AbZ=2 ;

.A; B/ 7! KK .A; B/;

7.1 The bivariant Chern–Connes character

261

where sC denotes the category of separable C  -algebras with -homomorphisms as morphisms and AbZ=2 denotes the category of Z=2-graded Abelian groups. This generalises K-theory because we have natural isomorphisms KK .C; B/ Š K .B/

for all B 22 sC :

We usually view Kasparov theory as a category KK, following Joachim Cuntz and Nigel Higson ([15], [16], [43]). Its objects are the separable C  -algebras; the morphism space A ! B is KK0 .A; B/; the composition of morphisms is the Kasparov product ˝B W KK i .A; B/ ˝ KKj .B; C / ! KK iCj .A; C /;

.f1 ; f2 / 7! f1 ˝B f2 D f2 ı f1 :

There is a canonical functor KK W sC ! KK which acts identically on objects and maps a -homomorphism f W A ! B to its class KK.f / 2 KK0 .A; B/. We may enrich KK to a Z=2-graded category with morphism spaces KK  .A; B/. But this gives essentially no new information because we have natural isomorphisms   KK1 .A; B/ Š KK1 .C0 .R; A/; B/ Š KK1 A; C0 .R; B/ : The category KK behaves like the stable homotopy category in algebraic topology. It is additive and carries the additional structure of a triangulated category; however, we will use this structure only implicitly in the following, so that you need not care about it. The functor KK W sC ! KK has the following properties: • it is homotopy invariant (with respect to continuous homotopies); • it is C  -stable, that is, the stabilisation homomorphisms A ! A ˝C  K.`2 N/ induce isomorphisms; • it is split-exact and half-exact for extensions with a completely positive contractive linear section; • if F W sC ! C is any C  -stable, split-exact functor to an additive category C, then F admits a unique factorisation Fx W KK ! C through KK W sC ! KK; moreover, the construction F 7! Fx is natural, that is, any natural transformation F ! F 0 extends to a natural transformation Fx ! Fx0 . Briefly, we say that KK is the universal C  -stable and split-exact functor. The functor Fx has the same values on objects, but it is functorial for more morphisms. Therefore, a natural transformation Fx ! Fx0 consists of the same data as a natural transformation F ! F 0 , but it satisfies more compatibility conditions. The above formulation of the universal property uses the rather deep result of Nigel Higson that any C  -stable and split-exact functor is homotopy invariant ([44]). Since the functors that we need are known to be homotopy invariant, anyway, our applications of the universal property are independent of this fact.

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7 The Chern–Connes character

The Kasparov product can be generalised considerably by combining it with exterior products. If D 22 sC , then we have a natural transformation D W KK  .A; B/ ! KK .A ˝ D; B ˝ D/; where ˝ denotes the minimal C  -tensor product; it maps KK.f / 7! KK.f ˝ idD / for a -homomorphism f W A ! B. We can prove the existence of exterior products using the universal property. Consider the functor sC ! sC ! KK;

A 7! KK.A ˝ D/:

It is C  -stable and split-exact because the functor ˝D preserves split-exact sequences and C  -stabilisations. Hence we can extend it to a functor  which provides  KK ! KK, the maps D on KK0 . We use KK1 .A; B/ Š KK 0 A; C0 .R; B/ to extend them to KK1 .

7.1.2 Construction of the character As usual, we equip all C  -algebras with the precompact bornology to ensure that local cyclic homology has good properties. Theorem 7.1 (Puschnigg). There is a unique natural transformation ch W KK  .A; B/ ! HL .A; B/   for separable C  -algebras A and B with ch KK.f / D HL.f / for any -homomorphism f and ch.f1 ˝B f2 / D ch.f2 / ı ch.f1 / for all f1 2 KK i .A; B/, f2 2 KKj .B; C /, i; j 2 Z=2; here ı denotes the usual product in bivariant local cyclic homology. ! We view local cyclic homology as a functor HL W sC ! HoKom.BanI Z=2/loc . This functor is C  -stable by Theorem 6.25, split-exact by the Excision Theorem 5.77, and homotopy invariant by Theorem 6.31. Hence the universal property of KK provides a unique natural transformation ch0 W KK 0 .A; B/ ! HL0 .A; B/ that intertwines the Kasparov product and the product in HL and behaves correctly on classes of -homomorphisms. It remains to construct the odd part ch1 W KK 1 .A; B/ ! HL1 .A; B/: We can also describe KK1 .A; B/ as the set of homotopy classes of C  -algebra extensions B ˝ K  E  A with a completely positive contractive section. Since such extensions are semi-split, they satisfy excision in bivariant local cyclic homology

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263

(Theorem 5.77). We get a class Œ@ 2 HL1 .A; B ˝ K/ as in §5.4.2. Using C  -stability (Theorem 6.25), we map it to HL1 .A; B/. This well-defines a map  W KK 1 .A; B/ ! HL1 .A; B/ because HL1 is invariant under continuous homotopies (Theorem 6.31). We put p ch1 D 2i   W KK 1 .A; B/ ! HL1 .A; B/: We have to show that the combination of ch0 and ch1 is a multiplicative map KK  ! HL . p The factor 2i comes from the computations in §5.4.5. In KK, the composition of the cone and Toeplitz extension yields the unit element in KK 0 .C; C/, whereas in HL, it yields .2i/1p . We compensate for this factor by decorating both Œ@cone  and Œ@Iso  with factors of 2i. It remains to check that this ensures multiplicativity in complete generality. This is a rather technical argument. The naturality of the boundary map in the Excision Theorem means that ch1 is natural for algebra homomorphisms in both variables. By the universal property of KK, we get naturality for elements in KK 0 as well. Therefore, we have ch .f1 /ıch .f2 / D ch .f1 ı f2 / if at most one of the factors is odd. It remains to tackle the case where both are odd. We abbreviate SA WD C0 .R; A/ for a C  -algebra A. We shall use the cone extension   SA  C0 .1; 1; A  A and the Toeplitz extension K ˝ A  Iso0C  ˝ A  SAI here Iso0C  denotes the augmentation ideal in the Toeplitz C  -algebra. These extensions have completely positive contractive sections and therefore yield classes ŒconeA  2 KK1 .A; SA/; ŒIsoA  2 KK1 .SA; A/:   It is well known that Iso0 and C0 .1; 1 are KK-contractible, so that ŒconeA  and ŒIsoA  are invertible; even more, these two classes are inverse to each other. Therefore, any class f 2 KK 1 .SA; SB/ can be written as f1 ı ŒIsoA  or ŒconeB  ı f2 . Since the suspension is an automorphism in KK, we may assume that the source and target C  -algebras of our morphisms are suspensions. We already know that ch1 is multiplicative with respect to ch0 on both sides. Hence we get multiplicativity in complete generality once we have ch1 .IsoA / ı ch1 .coneA / D id in HL0 .A; A/ for all C  -algebras A or, equivalently, Œ@IsoA  ı Œ@coneA  D .2i/1 : Proposition 5.96 asserts this if A D C and if we replace all C  -algebras by suitable smooth subalgebras. It remains to reduce the general case to this situation.

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7 The Chern–Connes character

Since we do not have an exterior product in HL for C  -tensor products, we leave the world of C  -algebras to check this. The Toeplitz and cone extensions contain smooth subextensions, where we replace SA by the Schwartz space S.R; A/ and K ˝ A by y A, which consists the subalgebra of smooth compact operators KS .A/ WD KS ˝ of A-valued matrices of rapid decay. The naturality of the boundary maps shows that we may just as well use these smooth subextensions. But the smooth cone and Toeplitz extensions over A are the completed bornological tensor products of A with the corresponding extensions over C. These tensor products agree with the tensor ! products in Ban by Theorem 2.13. We have exterior products for such tensor products by Theorem 5.74; since these exterior products are compatible with the boundary maps in the Excision Theorem, the general case reduces to the case A D C, which is treated in Proposition 5.96. This finishes the proof of Theorem 7.1. A bivariant Chern–Connes character for general complete bornological algebras. Since our goal is to prove a Universal Coefficient Theorem for C  -algebras, we are mainly interested in the C  -algebraic version of the bivariant Chern–Connes character. Therefore, we do not say much here about a variant of this construction for arbitrary bornological algebras. Anyway, the main issue is not to construct the character but to construct a good bivariant K-theory for such algebras. This has been accomplished by Joachim Cuntz in [19] for locally convex topological algebras. This theory is carried ! over to bornological algebras in [21]. Algebras in Ban can be treated similarly. The canonical functor from the category of complete bornological algebras to the resulting bivariant K-theory kk  .A; B/ is the universal functor that is exact for semisplit extensions, stable with respect to the algebra `1 .H / of trace-class operators on a separable Hilbert space (or other algebras of a similar nature), and homotopy invariant for smooth homotopies. Since HP , HA , and HL have these properties, we get bivariant Chern–Connes characters from kk 0 to HP0 , HA0 , and HL0 . The same trick as above yields the multiplicative extension to the Z=2-graded theories (see [17]).

7.1.3 Applications of the bivariant Chern–Connes character The bivariant Chern–Connes character allows us to extend various constructions from KK-theory to bivariant local cyclic homology; this idea is also pursued by Vahid Shirbisheh in [98]. We briefly discuss some applications of the Baum–Connes Conjecture for amenable groups; for such groups, the conjecture is established by Nigel Higson and Gennadi Kasparov ([45]). We use the formulation of the Baum–Connes Conjecture in [73], which is ideal for our purposes. Let G be an amenable group. The Baum–Connes Conjecture computes the K-theory of the crossed product algebra G Ë A for any C  -algebra A equipped with a strongly continuous action of G or, briefly, G-C  -algebra (the reduced and full crossed products agree for amenable groups). Since we do not want to introduce too much machinery, we

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265

only give a rather raw statement of the Baum–Connes Conjecture for general amenable groups, which nevertheless contains all the necessary analytical information. We will soon see how it yields the Connes–Thom Isomorphism Theorem and the Pimsner– Voiculescu Exact Sequence. We use an equivariant version KKG of KK and the fact that the crossed product descends to a functor G Ë W KKG ! KK; this follows from the universal property of KKG . It is shown in [73] that the following statement is equivalent to the result of [45]: Theorem 7.2. Let G be an amenable group, let A1 and A2 be G-C  -algebras, and let H f 2 KKG  .A1 ; A2 /. If the image of f in KK  .A1 ; A2 / is invertible for all compact subgroups H  G, then f is already invertible in KKG  .A1 ; A2 /. Hence ch .G Ëf / 2 HL .G Ë A1 ; G Ë A2 / is invertible. To see the power of this theorem, we consider the special cases where G is R or Z. Thus the only compact subgroup is the trivial group. Let A be any G-C  -algebra. We let G act on R by translation; this extends to an action on .1; 1 that fixes 1. Thus we get an equivariant cone extension   C0 .R; A/  C0 .1; 1; A  A;   where we use diagonal actions of G on C0 .R; A/ and C0 .1; 1; A .   This extension determines a class  2 KKG 1 A; C0 .R; A/ (although it does not have an equivariant completely positive section); if we had introduced Kasparov’s definition   of KKG , this class would be very easy to write down. Since C0 .1; 1; A is 7.2 yields that  contractible,  becomes invertible in KK1 A; C0 .R; A/ . Theorem  induces an invertible element in HL1 G Ë A; G Ë C0 .R; A/ . Since the action of G on R is free and proper, the crossed product G Ë C0 .R; A/ can be computed explicitly. First we consider the case G D R. We have R Ë C0 .R; A/ Š K.L2 R/ ˝ A, which is KK-equivalent to A because KK is stable. We get an invertible element in HL1 .R Ë A; A/, that is, HL.R Ë A/ is locally chain homotopy equivalent to HL.A/Œ1. This implies a Connes–Thom Isomorphism Theorem HL .R Ë A/ Š HL1 .A/;

HL .R Ë A/ Š HL1 .A/

(7.3)

for local cyclic homology and cohomology, and similar statements for bivariant local cyclic homology in both variables. Now we consider the case G D Z. Restriction to Z  R provides a surjection Z Ë C0 .R; A/  Z Ë C0 .Z; A/ Š A ˝ K.`2 Z/     with kernel Z Ë C0 Z  .0; 1/; A Š C0 .0; 1/; A ˝ K.`2 Z/. The resulting extension has a completely positive contractive linear section, so that we get an associated long exact sequence for HL . Now we use stability and homotopy invariance to simplify:       HL A ˝ K.`2 Z/ Š HL .A/; HL C0 .0; 1/; A ˝ K.`2 Z/ Š HL1 .A/:

266

7 The Chern–Connes character

  Using also the isomorphism HL .G Ë A/ Š HL1 G Ë C0 .R; A/ , we get the Pimsner–Voiculescu Exact Sequence HL0 .A/ O

1˛

HL1 .Z Ë A/ o

/ HL0 .A/

HL1 .A/ o

/ HL0 .Z Ë A/ 

1˛

(7.4)

HL1 .A/

for local cyclic homology and similar results for local cyclic cohomology and the bivariant theory in both variables. Here ˛ 2 Aut.A/ denotes the action of 1 2 Z. We omit the proof that the indicated maps in (7.4) are 1  ˛ . The Excision Theorem 5.77 also yields an exact triangle 1˛

HL.A/ ! HL.A/ ! HL.Z Ë A/ ! HL.A/Œ1

! 1˛ in HoKom.Ban/loc , that is, HL.Z Ë A/ is a cone for the map HL.A/ ! HL.A/ in ! the triangulated category HoKom.Ban/loc .

7.2 The Universal Coefficient Theorem The bivariant Chern–Connes character constructed above restricts to a natural map K .A/ ˝ C ! HL .A/ for all separable C  -algebras. The Universal Coefficient Theorem for local cyclic homology asserts that this maps is an isomorphism if A belongs to the bootstrap category, that is, A is KK-equivalent to a commutative C  -algebra. In addition, our argument shows that for such A, the chain complex HL.A/ is locally homotopy equivalent to its homology, equipped with the fine bornology and vanishing boundary map. This yields a bivariant version of the Universal Coefficient Theorem:   HL .A; B/ Š Hom HL .A/; HL .B/ : We can further identify this with Hom.K .A/˝C; K .B/˝C/ if both A and B belong to the bootstrap category. Using the invariance under isoradial homomorphisms, we can extend the Universal Coefficient Theorem also to isoradial subalgebras of C  -algebras in the bootstrap category. In all these statements, it is essential to use precompact bornologies. Throughout this section, we equip all C  -algebras with the precompact bornology. Recall that the natural map HL .A/ ! HA .A/ is an isomorphism for any Fréchet algebra with the precompact bornology and hence for all A 22 sC (Theorem 2.13). Therefore, many of our computations also work for analytic instead of local cyclic homology.

7.2 The Universal Coefficient Theorem

267

7.2.1 Chain complexes determined by their homology Our first goal is to compute the bivariant theory HL .A; B/ in terms of HL .A/ and HL .B/ in certain cases. More generally, we study the same problem for the bivariant ! local cohomology of chain complexes in Ban. We only consider Z=2-graded chain complexes, but other kinds of chain complexes can also be treated. ! Let C be a chain complex in Ban with boundary map @ and let H .C / be its ! homology, which is a graded vector space. To turn it into a chain complex in Ban, take diss Fin H .C / with the zero boundary map. Then a chain map H .C / ! C is just a homogeneous linear map H .C / ! ker @. We may lift the canonical surjection ker @  H .C / to a map H .C / ! ker @, which yields a chain map H .C / ! C ; this chain map is unique up to chain homotopy. Definition 7.5. We say that C is determined by its homology if this map H .C / ! C is a local homotopy equivalence. Proposition 7.6. The following assertions are equivalent: (1) C is determined by its homology;

  (2) the natural map Hloc .C; D/ ! Hom H .C /; H .D/ is invertible for all chain complexes D; (3) Hloc .C; D/ D 0 for all D with H .D/ D 0. The class of chain complexes that are determined by their homology is a thick subcat! egory of HoKom.Ban/loc closed under direct sums.   Proof. It follows from the definitionof homology as H .D/ WD H Hom.1; D/ that H .C; D/ Š Hom .H .C /; H .D/ if C is a direct sum of shifted copies of 1 D C. Since such objects are local, we get H .C; D/ Š Hloc .C; D/ as well. If C is determined by its homology, then it is locally homotopy equivalent to a complex of this special form. This yields (1) ) (2). The implication (2) ) (3) is trivial. It remains to prove (3) ) (1). Let D be the mapping cone of the canonical map ' W H .C / ! C . Since ' induces an isomorphism on homology, the Puppe Exact Sequence shows that H .D/ D 0. By (3), we get Hloc .C; D/ D 0. The Puppe sequence for the functor Hloc .C; / yields that ' induces an isomorphism   H0loc C; H .C / Š H0loc .C; C /: Since ' is an isomorphism on homology, it also induces an isomorphism   H0loc H .C /; H .C / Š H0loc .H .C /; C /: ! TheYoneda Lemma shows that ' is an isomorphism in HoKom.Ban/loc , that is, a local homotopy equivalence. Thus (3) ) (1).

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7 The Chern–Connes character

Condition (3) is a minimal requirement in order for Hloc .C; D/ to be computable in terms of H .C / and H .D/ for all D. As a result, such a computation is possible for all D if and only if C is determined by its homology, and we simply get an isomorphism   Hloc .C; D/ Š Hom H .C /; H .D/ in this case. This is the Universal Coefficient Theorem for Z=2-graded chain complexes ! over Ban. More abstractly, the above arguments can be expressed in terms of the forgetful ! ! functor Ban ! Vect and the functor HoKom.Ban/loc ! HoKom.Vect/ it induces. The category HoKom.Vect/ is equivalent to the category of graded objects VectZ=2 by ! taking homology. The functor diss ı Fin W Vect ! Ban is adjoint to the forgetful func! tor, and this remains so for the induced functors HoKom.Vect/  HoKom.Ban/loc . We have used this adjointness to transport the Universal Coefficient Theorem from ! HoKom.Vect/ to HoKom.Ban/loc .

7.2.2 The bootstrap category Since HL .A/ is a complex vector space, the Chern–Connes character in Theorem 7.1 extends to a C-linear map K .A/ ˝Z C ! HL .A/, which we still call Chern–Connes character. This map has a good chance of being an isomorphism. We are going to establish that this is the case for a class of separable C  -algebras, namely, the bootstrap class B (see [2]). This class can be defined in several equivalent ways: • A 22 B if and only if A is KK-equivalent to a commutative separable C  -algebra. • A 22 B if and only if A satisfies a Universal Coefficient Theorem for KK  , that is, there are natural exact sequences of Z=2-graded Abelian groups     Ext K .A/; K1 .B/  KK .A; B/  Hom K .A/; K .B/ for all B 22 sC . • A 22 B if and only if KK .A; B/ D 0 for all B 22 sC with K .B/ D 0. • Let B0 be the smallest class of separable nuclear C  -algebras that contains C and is closed under countable inductive limits, extensions, and KK-equivalences. Then B consists of those separable C  -algebras that are KK-equivalent to an object of B0 . • The bootstrap category is the smallest class of C  -algebras that is closed under suspensions, direct sums, KK-equivalence, and under the formation of mapping cones for -homomorphisms; (it is checked in [73] that this already gives all of B).

7.2 The Universal Coefficient Theorem

269

Although the definition may suggest that B is rather small, it contains many interesting C  -algebras; we know no example of a separable nuclear C  -algebra that does not belong to B at the moment. Since the bootstrap category is quite large, KK-equivalence is a rather weak notion of equivalence; this is an important feature of KK-theory.

7.2.3 Statement of the Universal Coefficient Theorem Theorem 7.7. If A 22 B, then the Chern–Connes character K .A/˝Z C ! HL .A/ is invertible, and HL.A/ is determined by its homology. Proof. We temporarily call a separable C  -algebra good if the map K .A/ ˝Z C ! HL .A/ is invertible and HL.A/ is determined by its homology. It is clear from Theorem 5.63 that C is good. It follows from Theorem 6.24 that direct sums of good C  -algebras are again good because K . / ˝Z C, HL , and HL commute with direct sums. Theorem 7.1 shows that a C  -algebra is good once it is KK-equivalent to a good  one. In particular, C0 .0; 1; A is good for all A because it is KK-equivalent to 0. The Excision Theorem 5.77 and a diagram chase show that semi-split extensions of good C  -algebras remain good. For cone extensions over A, this yields that suspensions C0 .R; / of good C  -algebras are again good. Since the mapping cone of a -homomorphism f W A ! B between good C  -algebras A and B fits in a semi-split extension C0 .R; B/  cone.f /  A, it is again good. Hence the class of good C  -algebras contains C and is closed under all the operations that we need to construct B. Therefore, all C  -algebras in B are good. Corollary 7.8. If A 22 B, then there are natural isomorphisms     HL .A; B/ Š Hom HL .A/; HL .B/ Š Hom K .A/ ˝Z C; HL .B/ ! for any algebra B in Ban. If B 22 B as well, then we also have a natural isomorphism  HL .A; B/ Š Hom K .A/ ˝Z C; K .B/ ˝Z C/: Proof. Combine Theorem 7.7 with Proposition 7.6. Since the above Universal Coefficient Theorem holds in rather great generality, we seem to get along quite well with the poor man’s version   poor HL .A; B/ WD .Hom HL .A/; HL .B/ of HL .A; B/; this avoids the use of derived categories and derived functors. If A; B 22 B, then the Universal Coefficient Theorems for KK and HL yield natural isomorphisms   KK .A; B/ ˝Z C Š Hom K .A/; K .B/ ˝Z C; HL .A; B/ Š Hom.K .A/ ˝Z C; K .B/ ˝Z C/

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7 The Chern–Connes character

  because ˝Z C is exact and kills the torsion group Ext K .A/; KC1 .B/ in the Universal Coefficient Theorem for KK  .A; B/. These isomorphisms identify the bivariant Chern–Connes character with the obvious map   Hom K .A/; K .B/ ˝Z C ! Hom.K .A/ ˝Z C; K .B/ ˝Z C/: (7.9) Warning 7.10. The map in (7.9) is always injective but need not be surjective unless K .A/ ˝Z C is finite-dimensional. For a counterexample, see Proposition 2.46. The Universal Coefficient Theorem in Corollary 7.8 applies to many C  -algebras. As an indication of this, we mention the following result, which is based on work of Jean–Louis Tu and closely related to Theorem 7.2 (see [73]): Theorem 7.11. Let G be an amenable group and let A be a G-C  -algebra. If H Ë A belongs to the bootstrap category for all compact subgroups H  G, then so does G Ë A. In particular, C  .G/ always belongs to the bootstrap category.

7.3 The character for idempotents and invertibles We are going to map idempotents and invertible elements in an algebra A to HL0 .A/ and HL1 .A/, respectively, following [26, §12]. When we combine this with the canonical pairing HL .A/  HL .A/ ! C, we get a pairing between idempotents and invertibles and local cyclic cocycles of appropriate parity. The same constructions work for periodic and analytic cyclic homology. In fact, if A is a complete bornological algebra, then we have canonical maps HL .A/ ! HA .A/ ! HP .A/, so that the character in HL .A/ contains most information. ! Idempotents and invertibles in algebras in Ban are defined as follows: Definition 7.12. If A is an algebra in a symmetric monoidal category C, then an idempotent in A is an algebra homomorphism 1 ! A. If A is unital and C has countable direct sums compatible with ˝, then an invertible L element in A is an algebra homomorphism 1Œz; z 1  ! A, where 1Œz; z 1  denotes n2Z 1 with appropriate multiplication. If A is, say, a ring or a bornological algebra, then this definition is equivalent to the standard definition; an idempotent e 2 A generates the algebra homomorphism 1 ! A,  7!   e, and an invertible element x 2 A extends uniquely to an algebra homomorphism 1Œz; z 1  ! A with z 7! x. As in Notation 5.56, we write Inv WD 1Œz; z 1  and let Inv0 be its augmentation ideal. Algebra homomorphisms Inv0 ! A correspond to invertible elements in AC that map to the unit element in 1 D AC =A. An idempotent in A yields a chain map C X.T C/ ! X.T A/

7.3 The character for idempotents and invertibles

271

by Theorem 5.63, which is equivalent to an element in HL0 .A/ called ch.e/. More explicitly, it follows from Theorem 5.63 that ch.e/ is represented by ch.e/ D e C

1 X 2n1 n

.2e  1/.de/2n 2 even an A Š T A:

nD1

This construction is canonical and does not even allow for normalisation factors because   ch.e/ for  2 C is an idempotent element in T A if and only if  D 1. Let `1 .H / be the algebra of trace-class operators on the separable Hilbert space y A have a character in HL0 .A/ as well because H WD `2 .N/. Idempotents in `1 .H / ˝ HL0 is stable with respect to `1 .H / by Theorem 5.65. To get an explicit formula, we y A/ ! HL.A/ is implemented by recall that the homotopy equivalence HL.`1 .H / ˝ even y .` .H / ˝ A/ !  A, the trace map tr W even 1 an an X   a0;i0 i1 d.a1;i1 i2 / : : : d.an;in i0 /: tr .a0;i0 j0 / d.a1;i1 j1 / : : : d.an;in jn / WD i0 ;:::;in

We find ch.e/ D tr.e/ C

1 X 2n1   tr .2e  1/.de/2n 2 even an A Š T A: n nD1

y are equivalent, then Œch.e/ D Œch.e 0 / in HL0 .A/. Lemma 7.13. If e; e 0 2 `1 .H / ˝A Proof. Equivalent idempotents are smoothly homotopy equivalent, so that this follows from the homotopy invariance of HL . Now we upgrade our construction to a group homomorphism ch W K0 .A/ ! HL0 .A/ on the algebraic K-theory K0 .A/. Elements of K0 .A/ are represented by pairs of idempotents e˙ in M1 .AC / whose difference belongs to M1 .A/. Their characters ch.e˙ / in HL0 .AC / have the same image in HL0 .C/. Therefore, the difference ch.eC /  ch.e / belongs to HL0 .A/  HL0 .AC / by Theorem 5.64. Lemma 7.13 and the additivity of HL0 imply that this construction well-defines a group homomorphism ch W K0 .A/ ! HL0 .A/: Using stability of HL, we get a natural map     y `1 .H / ! HL0 A ˝ y `1 .H / ! HL0 .A/: K0 A ˝ Recent work of Guillermo Cortiñas and Andreas Thom shows that     y `p .H / y `1 .H / Š K0 A ˝ K0 A ˝

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7 The Chern–Connes character

for all p 2 .1; 1/. Hence the domain of this character is isomorphic to the topological top K-theory K0 .A/ WD kk 0 .C; A/ as defined by Joachim Cuntz and Andreas Thom in [29]. Alternatively, since `p .H / is equivalent to `1 .H / in kk and since HL admits a bivariant Chern–Connes character kk ! HL, we can extend our construction to   y `p .H / for formal reasons. In any case, we get a natural transformation K0 A ˝ top

ch W K0 .A/ ! HL0 .A/: Lemma 7.14. This character agrees with the map top

K0 .A/ Š kk 0 .C; A/ ! HL0 .C; A/ Š HL0 .A/ that we get from the universal property of kk. For separable C  -algebras it agrees with the map K .A/ ! HL .A/ constructed in Theorem 7.1. top

Proof. Our map K0 .A/ ! HL0 .A/ is a natural transformation between two half-exact stable homotopy functors on the category of bornological algebras. The universal property of kk asserts that such a natural transformation   is natural with respect to  morphisms in kk as well. That is, ch f .x/ D f ch.x/ for all f 2 kk 0 .A; A0 /, x 2 kk 0 .C; A/. When we apply this to x D 1C 2 kk 0 .C; C/, we see that it suffices to prove that our two characters agree on 1C 2 kk 0 .C; C/. This is easy to check. The same argument works for C  -algebras. Now we turn attention to invertible elements. Theorem 5.63 uses the analytic quasi-freeness of Inv0 to get explicit chain homotopy equivalences HL.Inv0 / D X.T Inv0 / X.Inv0 / CŒ1;

where CŒ1 denotes the Z=2-graded chain complex 0  C with C in degree 1. We choose u1 d.u  1/ 2 X1 .Inv0 / as generator for the homology. Recall that there is a canonical way to lift u to an invertible element in TC .Inv/. This also yields a preferred lifting of u1 d.u  1/ to a generator for the homology of X.T Inv0 /, namely, 1 X

0 u1 d.u  1/.d.u1  1/ d.u  1//n 2 1 .T Inv0 / Š odd an .Inv /:

nD0

To remove the subtractions of 1, it is useful to work with reduced differential forms 0 odd x odd over the unitalisation here. In  an .Inv/ Š an .Inv /, the formula simplifies to 1 X

u1 du.du1 du/n :

nD0

Recall that invertible elements x 2 AC that map to 1 2 C correspond to algebra homomorphisms Inv0 ! A, which yield chain maps CŒ1 HL.Inv0 / ! HL.A/:

7.4 Finitely summable Fredholm modules

273

The Chern–Connes character ch.x/ 2 HL1 .A/ is the image of a suitable generator under this map. Since we added auxiliary factors in the definition of ch1 in Theorem 7.1, we do the same here and take the image of .2i/1=2 u1 du. Many authors use other conventions here, but ours is equivalent to the normalisation of Alain Connes in [8] when translated from his .d1 ; d2 /-bicomplex to X.T A/. Explicitly, ch.x/ is represented by 1 X

x odd .2i/ =2 x 1 dx.dx 1 dx/n 2  an .AC /: 1

nD0

The map x 7! ch.x/ is multiplicative, that is, ch.x1  x2 / is homologous to ch.x1 / C ch.x2 /. Using matrix-stability of HL, we obtain an analogous group homomorphism Gl1 .A/ ! HL1 .A/. Since its range is Abelian, it descends to a group homomorphism alg

ch W K1 .A/ ! HL1 .A/: The homotopy invariance of HL1 yields that this character does not distinguish between invertible elements that are smoothly homotopic.

7.4 Finitely summable Fredholm modules Now we construct the character for finitely summable Fredholm modules, following Alain Connes [6], [8]. Our constructions and his differ only by normalisation constants, although we work with a different, chain homotopy equivalent model for HP.

7.4.1 The setup: 1-summable Fredholm modules Let V be a complete bornological vector space; we allow Fredholm modules on V instead of a Hilbert space because this creates no additional difficulties in the 1-summable case. First we recall some basic facts about finite-rank and nuclear operators (see also §A.7). We assume V 0 ¤ 0. The algebra of finite-rank endomorphisms of V can be identified with the incomplete tensor product F .V / WD V ˝ V 0 , where an elementary tensor ˝  corresponds to the operator j ihj W V ! V;

 7!  ./ 2 V:

y V 0 with the product defined Let `1 .V / be the complete bornological algebra V ˝ in §A.7. We let B.V / WD Hom.V; V / be the complete bornological algebra of bounded linear operators on V . The algebra `1 .V / is a B.V /-bimodule via T  j ihj WD jT ihj;

and

j ihj  T WD j ihT 0 j

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7 The Chern–Connes character

0 0 0 for T 2 B.V /, where  T W V ! V denotes the transpose of T . The formula tr j ihj WD . / defines a B.V /-bimodule trace on `1 .V /, that is,

tr.T  x/ D tr.x  T /

for all x 2 `1 .V /; T 2 B.V /:

(7.15)

It suffices to check this for elementary tensors x D j ihj, where it amounts to the definition h j T i D hT 0  j i of the transpose. The map F .V /  B.V / extends to a bounded linear map \ W `1 .V / ! B.V /, which is both an algebra and a B.V /-bimodule homomorphism. Its range is the space of nuclear operators on V . From now on, we require that V has the local approximation property. This implies that \ is injective, so that we may view `1 .V /  B.V /. In particular, this covers Hilbert spaces, but also many other Banach spaces like C0 .X / or Lp .X; / for p 2 Œ1; 1/ and a measure space .X; /. Definition 7.16. Let A be a complete bornological algebra. A 1-summable odd Fredholm module over A consists of a complete bornological vector space V with the local approximation property, an algebra homomorphism ' W A ! B.V /, and an operator F 2 B.V / such that F 2 D idV ;

ŒF; '.a/ 2 `1 .V / for all a 2 AI

in addition, the map A ! `1 .V /, a 7! ŒF; '.a/ is required to be bounded. A 1-summable even Fredholm module over A consists of the above data together with a grading operator 2 B.V / with 2 D idV ;

 F C F  D 0;

 '.a/ D '.a/ 

for all a 2 A:

Notation 7.17. Fix V , F (and ) as in Definition 7.16 and let B1 D B1 .V; F / WD fT 2 B.V / j ŒF; T  2 `1 .V /g: B1C D B1C .V; F; / WD fT 2 B.V / j ŒF; T  2 `1 .V /; Œ ; T  D 0g:

(7.18)

We write B1˙ if V , F (and ) are clear from the context. Since taking commutators is a derivation, B1˙ is a subalgebra of B.V /. A subset S of B1˙ is bounded if S itself and ŒF; S are bounded in B.V / and `1 .V /, respectively. This turns B1C and B1 into complete bornological algebras. By construction, 1-summable even or odd Fredholm modules are equivalent to bounded algebra homomorphisms A ! B1˙ . In order to describe B1 more explicitly, we decompose V D VC ˚ V for V˙ WD range 12 .1 ˙ F / and write operators T 2 B.V / as 2  2-block matrices accordingly. Then T 2 B1 if and only if the off-diagonal terms belong to `1 .VC ; V / and `1 .V ; VC /:   `1 .V ; VC / B.VC / B1 D : (7.19) `1 .VC ; V / B.V /

7.4 Finitely summable Fredholm modules

275

To describe B1C more explicitly, we decompose V D VC ˚ V using the grading this time; the operator F restricts to inverse isomorphisms   V˙ Š V , which we use to identify the two. Then F becomes the scalar matrix 01 10 , and we get   ˇ  TC 0 C ˇ B1 D (7.20) 2 M2 B.VC / TC  T 2 `1 .VC / : 0 T The following definitions prepare the construction of the Chern–Connes character. We drop ' from our notation and assume A  B1˙ . Consider the (inner) derivation ı W B1˙ ! `1 .V /;

ı.T / WD

i ŒF; T I 2

the factor i=2 cancels certain normalisation constants in our computations. The universal property of .A/ (Theorem A.58) yields a bounded algebra homomorphism W A ! B1˙ ;

x0 dx1 : : : dxn ! 7 x0  ıx1    ıxn : P By assumption, the restriction of to 1 A WD j1D1 j .A/ is a bounded map to `1 .V /. The assumption F 2 D 1 implies (7.21) F ı.x/ C ı.x/F D i=2.F 2 x  F xF C F xF  xF 2 / D 0 for all x 2 B1˙ ; 2 .d!/ D F .!/  .1/n .!/F for all ! 2 n .A/: (7.22) i Remark 7.23. To understand (7.22) better, we should replace B1˙ by the differential L1 ˙ graded algebra  WD B1 .V / ˚ nD1 `1 .V /  t n with t of degree 1 and ı.x/ WD i=2ŒF t; x. This is a graded derivation with ı 2 D 0, so that we get a homomorphism of DG-algebras .A/ ! .

7.4.2 The Chern–Connes character in the even case Now we restrict attention to the even case. Define 2n W 2n .A/ ! C for n 2 N by   (7.24) 2n .!/ WD c2n i  tr F .d!/ for all ! 2 2n .A/ with c2n WD

  2n.2n  2/.2n  4/    2 4n  nŠ  nŠ : D D 4n W 2n n .2n  1/.2n  3/.2n  5/    1 .2n/Š

Thus c0 D 1 and c2nC2 D c2n  .2n C 2/=.2n C 1/ for n 2 N. The sequence .c2n / p behaves like a constant one for the analytic bornology because limn!1 2n c2n D 1. Up to the choice of the constants c2n , (7.24) is Connes’s formula for the character of a 2n C 1-summable even Fredholm module [8, p. 293]. Since our main purpose here is to prepare for the character without summability conditions, we only need p-summable Fredholm modules for p D 1. We remark,

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7 The Chern–Connes character

however, that we may relax the condition ŒF; '.a/ 2 `1 .V / in Definition 7.16 and require that restricts to a bounded map p A ! `1 .V /; this suffices for k to be well-defined for all even numbers k  p  1. To ensure that k has the desired properties, we need a generalisation of the trace property of tr, namely,     tr T .!/ .dx/ D tr .dx/T .!/ for all T 2 B.V /; ! 2 p1 .A/; x 2 A: (7.25) We can use these conditions to define p-summability for Fredholm modules on Banach   spaces. Equation (7.25) holds if p1 .A/  `1 .V / or if V is a Hilbert space and ŒF; A  `p .H /. We fix some p now. As it turns out, our computations work for any p-summable Fredholm module in this somewhat technical sense. If k  p, we can use (7.22), F D  F , and (7.15) to simplify k :     ck  2 .!/F / D tr 2F .!/ D ck tr .!/ : 2 (7.26) Since anti-commutes with F and commutes with B1C , we have

k .!/ D 

ck  tr F .F .!/  2

.!/ D .1/deg ! .!/ for all homogeneous ! 2 .A/. In addition, (7.22) implies F .d!/ C .1/deg ! .d!/F D 0: Consequently, F commutes with

.d!/ for all ! 2 .A/.

Lemma 7.27. The functional 2n is a closed graded trace for all n with 2n  p  1. That is, 2n ı d D 0 and 2n vanishes on graded commutators. Hence 2n ı  D 2n and 2n ı b D 0. Proof. We have 2n ı d D 0 by definition. The commutator subspace of A is the closed linear hull of ŒA; A D range b and ŒA; dA D range 1  . Since 2n ı .1  / D 2n ı .db C bd / D 2n ı b ı d; it remains to check 2n ı b D 0, that is, 2n .Œ!; x/ D 0 if ! 2 2n .A/, x 2 A. We use (7.25) to compute   2n .Œ!; x/ D c2n tr iF . d.!x  x!//  D ic2n tr F .d!/ .x/ C F .!/ .dx/   F .dx/ .!/  F .x/ .d!/   D ic2n tr Œ .d!/; F  .x/  Œ .dx/; F  .!/ D 0:  We get 2n ı b D 0 and 2n ı B D 0, so that 2n yields a class Œ2n  2 HC2n .A/. Next we verify that the periodicity operator S W HC2n .A/ ! HC2nC2 .A/ maps Œ2n  to Œ2nC2  by writing down linear functionals h2nC1 W 2nC1 .A/ ! C with h2nC1 ı @ D

277

7.4 Finitely summable Fredholm modules

2n  2nC2 ; here @ denotes the boundary in the X-complex (up to normalisation factors, the same formulas work for b C B). Since h2nC1 lives on odd forms, the relevant part of @ is given by (5.43). Define h2nC1 .!/ D

  c2n tr iF .!/ 2n C 1

(7.28)

for all ! 2 2nC1 .A/. Lemma 7.29. Let 2nC1  p. Then h2nC1 ı D h2nC1 and h2nC1 ı@ D 2n 2nC2 . Proof. If x 2 A and ! 2 p .A/, we use .!/ 2 `1 .V /, equation (7.25), and ŒF ; ıx D 0 to compute         tr F . dx  !/ D tr ıx  F .!/ D tr F .!/ıx D tr F .! dx/ : Thus h2nC1 ı  D h2nC1 . The boundary @ from even to odd degrees is given by (5.43). Since h2nC1 ı  D h2nC1 , the desired equation h2nC1 ı @ D 2n  2nC2 is equivalent to .2n C 1/h2nC1 ı d D 2n and .n C 1/h2nC1 ı b D 2nC2 . The first equality is the definition of 2n . For the second equality, let ! 2 2nC1 .A/, x 2 A and compute  tr iF 

     ı b.! dx/ D tr iF .Œ!; x/ D tr iF x .!/  iF .!/x       D tr iF x .!/  .!/xiF D tr .!/ŒiF; x D 2 tr .! dx/ :

Putting in our normalisation constants, we get .n C 1/h2nC1 ı b D 2nC2 . Hence Œ2n  D Œ0  in HP0 .A/ for all n 2 N. We want the Chern–Connes characters to be compatible with the canonical pairings K0 .A/  K 0 .A/ ! C and HP0 .A/  HP0 .A/ ! C. Since we prove a more general statement later, anyway, we only check a trivial special case here. Example 7.30. Let p 2 B1C be a rank-one idempotent with p D p D p. Then    1  0 ch.p/ D 0 .p/ D c0 tr.iF ŒiF=2; p/ D tr .p  FpF / D tr.p/ D 1: 2 (7.31) Since this is the desired result, our normalisation constants c2n are chosen correctly.

7.4.3 The Chern–Connes character in the odd case For n 2 N with 2n C 1  p  1, we define 2nC1 W 2nC1 .A/ ! C;

  ! 7! c1 tr F .d!/

(7.32)

p with c1 WD 2i. Up to normalisation constants, this is Connes’s formula for the character of an odd Fredholm module [8, p. 293]. We need the factor c1 to get the

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7 The Chern–Connes character

right index pairing; recall thatthe Chern–Connes p character ch1 W K1 ! HA1 maps  the generator Œu of K1 CŒu; u1  to the class of . 2iu/1 du in HA1 CŒu; u1  Š C. If 2n C 1  p, we can use (7.15) and (7.22) to simplify:     2nC1 .!/ D c1 2i tr F .F .!/ C .!/F / D ic1 tr .!/ : Lemma 7.33. The functional 2nC1 is a closed graded trace, so that 2nC1 ı d D 0;

2nC1 ı  D 2nC1 ;

2nC1 ı b D 0:

Proof. The proof is similar to the proof of Lemma 7.27. As in the even case, we write down linear functionals with h2n ı @ D 2n1  2nC1 for n 2 N with 2n  p:   h2n W 2n .A/ ! C; ! 7!  12 c1 tr F .!/ : (7.34) Lemma 7.35. Let 2n  p. Then h2n ı  D h2n and h2n ı @ D 2n1  2nC1 . Proof. If x 2 A and ! 2 p1 .A/, then         tr F . dx  !/ D  tr ıxF .!/ D  tr F .!/ıx D  tr F .! dx/ by (7.22) and (7.25). Thus h2n ı D h2n . We have h2n ı.1C/d D 2h2n d D 2n1 and   h2n ı b.! dx/ D h2n .!  x  x  !/ D  12 c1 tr F .!/x  F x .!/   D ic1 tr .!/.iF=2/  x  .!/x  .iF=2/   D ic1 tr. .!/ıx/ D ic1 tr .! dx/ : Thus h2n ı b D 2nC1 and hence h2n ı @ D 2n1  2nC1 by (5.42). It follows that Œ2nC1  D Œ1  in HP1 .A/ for all n 2 N. Now we could check that the index pairings K1 .A/  K1 .A/ ! Z and HP1 .A/  HP1 .A/ ! C are compatible with the Chern–Connes characters. Since we will consider the analogous problem in greater generality later, we only do one simple special case to check our normalisation constant. Example 7.36. Let A D CŒu; u1  and let A act on V WD `2 .Z/ by un f .x/ WD f .x C n/ for all x; n 2 Z. Hence the generator u acts as a bilateral backwards shift p on V . Recall that ch1 .u/ is the class of . 2iu/1 du in HC1 .A/. Define F 2 B.V / by F .ın / WD ın if n  0 F .ın / WD ın otherwise. This yields a 1-summable odd Fredholm module over A. The index pairing of u and the Fredholm module associated to F is 1. We compute u1 ı.u/ D u1 ŒiF =2; u D iE00 ; where E00 is the rank-1 projection onto C  ı0 . Hence   1 1 h1 j ch.u/i D 1 .2i/ =2 u1 du D ic1 .2i/ =2 tr.u1 ŒiF=2; u/ D tr.p/ D 1: (7.37) Since this is the desired result, our normalisation constant c1 is chosen correctly.

7.5 The character for general Fredholm modules

279

7.5 The character for general Fredholm modules If we forget about boundedness, the character of a p-summable Fredholm module is a boundary: 1 X hnC1C2k ı @I n D kD0

here n  p and n has the same parity as our Fredholm module. Of course, this only works in the dual complex of X.TA/, which is contractible anyway. Now we apply a suitable cut-off to each hj , replacing it by hQj , and form Q WD n 

1 X

hQ nC1C2k ı @:

nD0

P1

The cut-off ensures two things: first, nD0 hQ nC1C2k is bounded on X.T A/ if our Fredholm module is p-summable, so that Œ Q D Œn  in HAn .A/ in this case; secondly, Q is bounded on X.T A/ for Fredholm modules without any summability hypothesis, so that we get a character Œ Q 2 HAn .A/. If H is a Hilbert space, then we can use the following regularisation scheme. Let .ek / be an increasing sequence of projections on H that commute with F (and with in the even case) and satisfy rank ek D k and lim ek D 1 strongly, and let ek? WD 1  ek . We put   hQ k .x0 dx1 : : : dxk / WD hk ek? x0 d.ek? x1 / : : : d.ek? xk / : This recipe is rather flexible. In our actual construction, we replace H by a separable Banach space with Grothendieck’s Approximation Property and .ek / by a sequence of nuclear operators with suitable properties, which we will explain below. The idea to construct analytic cocycles as boundaries of unbounded cochains may also be useful in other contexts.

7.5.1 Approximation property and cut-off sequences y V0 Let V be a Banach space and let F and possibly be as above. Then `1 .V / D V ˝ is a Banach algebra with the projective tensor product norm k k1 . Let K.V / be the space of compact endomorphisms of V . This is a closed ideal of B.V / with respect to the operator norm k k1 on B.V /. We assume that there is a sequence .en / in `1 .V / with the following properties: Definition 7.38. A sequence .en / in `1 .V / is called a cut-off sequence if 1. the trace norms ken k1 in `1 .V / are of at most exponential growth; 2. the operator norms ken k1 in K.V / are uniformly bounded; 3. the set of 2 V with limn!1 en . / D is dense in V ;

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7 The Chern–Connes character

4. en commutes with F (and in the even case) for all n 2 N. We let en? WD 1  en . The proof of the following lemma contains some variations on these conditions. Lemma 7.39. A Banach space V with operators F and admits a cut-off sequence if and only if it is separable and has Grothendieck’s Approximation Property. Proof. We may replace the operators .en / in a cut-off sequence by nearby operators of finite rank. That is, if there is any cut-off sequence, then there is another one with en 2 F .V / for all n 2 N. An equibounded net of operators that converges pointwise on a dense subset automatically converges uniformly on compact subsets of V by Theorem 1.18. Conversely, a sequence of operators that converges uniformly on compact subsets must be equibounded by the principle of uniform boundedness (also called Banach–Steinhaus Theorem). Hence the second and third condition together are equivalent to uniform convergence of .en / on compact subsets to the identity map. Therefore, if a cut-off sequence exists, then V is separable and has the approximation property. Conversely, if this is the case, then there is a sequence of finite-rank operators .en / satisfying the second and third condition for a cut-off sequence. To achieve the first condition, we repeat each en sufficiently often to ensure, say, ken k1 D O.2n /. To achieve the fourth condition, first replace en by 12 .en C F en F / to make it commute with F , then replace the result by 12 .en C en / to make it commute with as well. Notation 7.40. For a subset S of a Banach space W , we let kSk D supfkxkW j x 2 Sg. Lemma 7.41. Let .en / be a cut-off sequence and let S be a set of operators on V . If S is precompact in `1 .V /, then limn!1 ken? S k1 D 0. If S is precompact in K.V /, then limn!1 ken? Sk1 D 0. y  V 0 . Then S ~ S1 ˝ S2 for suitable Proof. First let S be precompact in `1 .V / D V ˝ compact disks S1  V , S2  V 0 (see Example 1.86). We have en? ı S ~ en? .S1 / ˝ S2 . Since en? ! 0 uniformly on compact subsets, we get ken? .S1 /kV ! 0 and hence ken? ı Sk1 ! 0 for n ! 1. Now let S be precompact in K.V /. Let DV  V and DK  K.V / be the unit balls. We claim that S.DV /  V is precompact if S is precompact in K.V /. Given any " > 0, we have S  F C "DK for a suitable finite set F  K.V /. Since compact operators map DV to precompact subsets, there is a finite subset F 0  V with F .DV /  F 0 C "DV . Since "DK maps DV into "DV , we get S.DV /  F 0 C 2"DV . Since this holds for all " > 0, S.DV / is precompact. The sequence .en? / converges to 0 uniformly on the precompact subset S.DV /. Hence ken? ı S.DV /kV ! 0 for n ! 1. Thus ken? ı S k1 ! 0 for n ! 1.

7.5 The character for general Fredholm modules

281

7.5.2 Fredholm modules Notation 7.42. Let V be a separable Banach space with Grothendieck’s Approximation Property, equipped with the von Neumann bornology, and let F 2 B.V / (and 2 B.V / in the even case) satisfy F 2 D id (and 2 D id and F D  F ). Let   WD fT 2 B.V / j ŒF; T  2 K.V /g; B1 .V; F / WD B1 C C WD fT 2 B.V / j ŒF; T  2 K.V /; Œ ; T  D 0g; B1 .V; F; / WD B1

equipped with the precompact bornology; these are closed Banach subalgebras of B.V /. ˙ If we replace `1 by K in (7.19)–(7.20), then we get analogous descriptions of B1 . ˙ Lemma 7.43. The subalgebras B1˙ .V; F; / ! B1 .V; F; / are isoradial and approximably dense.

Proof. We only discuss the odd case; the even case is similar. We decompose V D  VC ˚ V and describe B1 and B1 as in (7.19). Choose cut-off sequences .en˙ / on the summands V˙ ; leaving the diagonal entries fixed and multiplying the off-diagonal  entries with en˙ defines bounded linear maps B1 ! B1 that converge towards the  identity map on B1 ; this convergence is clearly uniform on precompact subsets; hence the subalgebras are approximably dense. The Leibniz rule shows that the spectral  radius of a bounded subset of B1 or B1 is the same as its spectral radius in B.V /.   Hence the subalgebra B1 ! B1 is isoradial. Definition 7.44. Let A be a complete bornological algebra. Bounded algebra homoC  morphisms from A to B1 .V; F; / or B1 .V; F / are called even and odd Fredholm modules over A, respectively. Remark 7.45. We will slightly relax these conditions in §7.5.4 in order to construct a Chern–Connes character in entire cyclic cohomology.

7.5.3 Construction of the Chern–Connes character We are going to define the Chern–Connes character in HA˙ .A/ of a Fredholm module ˙ ' W A ! B1 by pulling back certain analytic cyclic cocycles C QC W X0 .T B1 / ! C;

 Q W X1 .T B1 /!C

along '. The main point is to construct Q˙ . We will only write down Q˙ on the subspace ˙ ˙ B1˙ of an B1 Š X.T A/, which is dense by Lemma 7.43. Even more, an B1 is ˙ the completion of B1 with respect to the bornology generated by subsets of the form ˙ /. By the universal property of completions, hS i.dS /1 for S  B1˙ with S 2 S.B1 ˙ linear functionals on an B1 are equivalent to linear functionals on B1˙ that remain

282

7 The Chern–Connes character

bounded on subsets of the form hS i.dS/1 with S as above. Hence we get the required functional Q˙ from such a densely defined one if it satisfies an appropriate growth condition. The basic idea is to use a cut-off sequence .en / to regularise the functionals hk , putting   hQ k .x0 dx1 : : : dxk / WD hk e ? x0 d.e ? x1 / : : : d.e ? xk / k

k

k

for all x0 ; : : : ; xk 2 B1˙ , k 2 N; here ek? x0 may be replaced by 1 2 .B1˙ /C if x0 D 1. Definition 7.46. We let hQ  WD hQ C WD

1 X nD1 1 X

hQ 2n1 W B1C ! C;

QC WD 0  hQ  ı @ W B1C ! C;

hQ 2n W B1 ! C;

Q WD 1  hQ C ı @ W B1 ! C:

nD1

By construction, QC ı @ D 0 ı @ D 0 and Q ı @ D 1 ı @ D 0. Lemma 7.47. The functionals hQ ˙ W B1 ! C extend uniquely to bounded linear functionals on an .B1 /, respectively. Hence so do Q , and we have ŒQC  D Œ0  in HA0 .B1C / and ŒQ  D Œ1  in HA1 .B1 /. Even more, Q˙ W B1˙ ! C extend ˙ to bounded linear functionals Q˙ W an .B1 / ! C and hence yield classes ŒQ˙  2 0 ˙ HA .B1 /. The latter do not depend on the choice of the cut-off sequence .en /. Proof. Let S be a set of operators on V . Assume first that S is bounded in B1˙ ; then ı.S / is precompact in `1 .V / and S is bounded in operator norm. Since S and fen g are equibounded, we have kSk1 kek? k1  C for all k 2 N for some constant C  1. Lemma 7.41 yields "1;k WD kek? ı.S/k1 ! 0 for k ! 1. Since F commutes with ek? , we have ı.ek? x/ D ek? ıx 2 ek? ı.S / for all x 2 S . Thus ˇ  ˇ jhQ k .x0 dx1 : : : dxk /j D ˇtr ek? x0 .ek? ıx1 /    .ek? ıxk / ˇ  kek? x0 .ek? ıx1 /    .ek? ıxk /k1  C  "k1;k for all x0 ; : : : ; xk 2 S. Since "1;k < 1 for sufficiently large k, the sequence .C  "k1;k / is bounded. Therefore, hQ ˙ is bounded on hS i.dS /1 if S is bounded in B1 . Now the universal property of the completion yields the desired unique extensions of hQ ˙ . It follows that the cocycles Q are cohomologous to 0 and 1 . Let .ek / and .fk / be two cut-off sequences. We claim that the resulting linear ˙ functionals hQ ˙ .e/ and hQ ˙ .f / only differ by bounded linear functionals on an .B1 /; hence the class of Q˙ is independent of the choice of the cut-off sequence. Using that

283

7.5 The character for general Fredholm modules

ek and fk commute with F , we compute  ?    ek x0 d.ek? x1 / : : : d.ek? xk /  fk? x0 d.fk? x1 / : : : d.fk? xk / D

k X

  ek? x0  ı.ek? x1 /    ı.ek? xj 1 /ı .ek?  fk? /xj ı.fk? xj C1 /    ı.fk? xk /

j D0

D

k X

ek? x0  .ek? ıx1 /    .ek? ıxj 1 /  .fk  ek /ı.xj /  .fk? ıxj C1 /    .fk? ıxk /:

j D0 ˙ , then ı.S/ is precompact in K.V /, so that the numbers If S is bounded in B1 ˚  "1;k WD max kek? ı.S/k1 ; kfk? ı.S /k1

converge to 0 for n ! 1 by Lemma 7.41. Hence "k1;k is O.r k / for any r > 0   and decays faster than .k C 1/ kek k1 C kfk k1 , which is at most exponential by our requirement for a cut-off sequence. If x0 ; : : : ; xk 2 S , we can therefore estimate

 ?   

e x0 d.e ? x1 / : : : d.e ? xk /  f ? x0 d.f ? x1 / : : : d.f ? xk / k k k k k k k!1

 .k C 1/  kSk1 kı.S/k1  "k1 ! 0; 1;k .kek k1 C kekC1 k1 /  using the inequalities kxyk1  kxk1 kyk1 and kxyk1  kxk1 kyk1 . Hence the ˙ difference .e/  .f / is a bounded linear map an B1 ! `1 .V /; thus another Q choice of cut-off sequence replaces h˙ by a bounded perturbation. To estimate the growth of Q˙ , we first simplify its definition using hC ı @ D 1 and h ı @ D 0 ; this follows from Lemmas 7.29 and 7.35. Define l D l.e/ W B1˙ ! B1˙ ;

x0 dx1 : : : dxk 7! ek? x0 d.ek? x1 / : : : d.ek? xk /;

so that hQ ˙ D h˙ ı l. First we compute Q using (5.42): Q D 1  hC ı @ ı l C hC ı Œ@; l D .1  1 ı l/ C hC ı Œb  .1 C /d; l: We use  ı l ı d D l ı  ı d and hk ı  D hk for all k 2 N to simplify this to Q D .1  1 ı l/ C hC ı Œb; l  2hC ı Œd; l:

(7.48)

In the even case, the resulting formulas become more complicated because the relevant part of the boundary map in (5.43) has more terms. We get QC D .0  0 ı l/ C h ı Œ@; l D .0  0 ı l/ C h ı

h2nC1 X

j d 

j D0

n1 X

 2j b; l

j D0

D .0 0 ıl/C.2nC1/h ıŒd; lnh ıŒb; lCh ıŒ1; lıbı

n1 X

j D0

aj;n  2j 1

i

284

7 The Chern–Connes character

on 2n .A/ with some coefficients aj;n 2 Z that grow at most linearly for n ! 1. We can further rewrite Œ1  ; l ı b D Œdb C bd; l ı b D d Œb; lb C bŒd; lb C Œb; ldb: The crucial observation is that each of the resulting summands involves Œd; l or Œb; l and some further operators that are built out of d; b and grow at most linearly for n ! 1. Hence the following estimates still work in the even case as well. But we only give details in the odd case. First we consider the terms   .0  0 ı l/.x/ D tr iF e0 ı.x/ ;   .1  1 ı l/.x0 dx1 / D ic1 tr e0? x0 e0 ı.x1 / C e0 x0 ı.x1 / : C  They are bounded on B1 or 1 .B1 /, respectively. Recall that multiplication by n on n .A/ defines a bounded map on an .A/. The above computations show that boundedness of Q˙ follows if ı Œd; l and ı Œb; l ˙ ! `1 .V /. More explicitly, we have to check that extend to bounded maps an B1 k ı Œd; lk1 and k ı Œb; lk1 remain bounded on hS i.dS /1 if S  B1˙ is bounded ˙ in B1 . First we consider the term involving Œd; l. The quickest argument uses the independence of the cut-off sequence .ek /. It follows that we may equally well work with d ı l.e/  l.f / ı d for two different cut-off sequences .ek / and .fk /. If we choose fk D ek1 , then we get d ı l.e/ D l.f / ı d , so that this term disappears completely. For the term involving Œb; l we use a similar trick to simplify the computation and work with b ı l.e/  l.f / ı b with fk D ekC1 ; this ensures that the same operator ek is used for b ı l.e/ and l.f / ı b on k A. We compute   ı .b ı l.e/  l.f / ı b/ x0 dx1 : : : dxk

D

k1 X

.1/j ek? x0 ı.ek? x1 /    ı.ek? xj 1 /ı.ek? xj .ek?  1/xj C1 /ı.ek? xj C2 /    ı.ek? xk /

j D0

C .1/k ek? xk .ek?  1/x0 ı.ek? x1 /    ı.ek? xk1 / D

k1 X

.1/j C1 ek? x0 .ek? ıx1 /    .ek? ıxj 1 /ek? ı.xj ek xj C1 /.ek? ıxj C2 /    .ek? ıxk /

j D0

C .1/kC1 ek? xk ek x0 ı.ek? x1 /    ı.ek? xk1 /: Using the same notation as in our previous estimates, we get

ı .b ı l.e/  l.f / ı b/.x0 dx1 : : : dxk / k!1

?  .k C 1/  kek? Sk1  "k1 ! 0 1;k kek ı.Sek S /k1 

for x0 ; : : : ; xk 2 S .

7.5 The character for general Fredholm modules

285

Remark 7.49. We have checked above that our two character constructions agree for 1-summable Fredholm modules. The same result holds for p-summable Fredholm modules: to see this, choose the cut-off sequence to begin with e0 D : : : D en D 0 for some n 2 N, so that hQ k D hk for k  n. Then our formula for QC simplifies to n=21

QC D 0 

X

kD0

h2kC1 ı @ 

1 X kDn=2

hQ 2kC1 ı @ D n 

1 X

hQ 2kC1

kDn=2

for even n. As above, this yields Œn  D ŒQC  in HA0 .A/ for n-summable Fredholm modules. Similar results hold in the odd case.

7.5.4 Application to the entire theory Let A be a locally convex topological algebra and let .V; '; F; / be a Fredholm module ˙ . If ' is compact, then it is bounded over A, that is, ' is a continuous linear map A ! B1 ˙ as a map vN.A/ ! B1 ; hence we get a character in the entire theory HE .A/ in this case. We can weaken this compactness condition: an inspection of our construction shows that we only need precompactness of ŒF; '.S /  Cpt.V / for all S 2 vN.A/. Given a bornological algebra A, we can therefore slightly relax our definition of a Fredholm module, requiring von Neumann boundedness of '.S / and precompactness of ŒF; '.S/ for all S 2 S.A/; this suffices to construct the character in HA .A/. If we start with a spectral triple or, in different notation, a K-cycle or an unbounded Fredholm module (see [6]), then this weaker precompactness condition comes for free: Lemma 7.50. Let V be a Hilbert space, let D 2 B.V / be an invertible, unbounded self-adjoint operator with compact resolvent, and let F WD D  jDj1 . Let S  B.V / be a set of operators such that ŒD; a is densely defined and extends to a bounded operator on V for all a 2 S. Suppose that both S and ŒD; S  are von Neumann bounded in B.V /. Then ŒF; S  is a precompact subset of K.V /. Proof. It is well known that ŒF; a 2 K.V / for all a 2 S . A standard proof (see [35, Lemma 10.18]) involves the integral representation p Z 1 p   d D D ŒF; a D ŒD; a  ŒD; a (7.51) p : 2 2 2 2  C D  C D  C D  C D  0 The operator norm of the integrand is controlled by 1=2 kŒD; ak1 for  & 0 and by 3=2 kŒD; ak1 for  % 1; here we use jDj  " > 0. Hence the integral is uniformly convergent in operator norm for a 2 S . Since D is invertible and has compact resolvent, f1 .D/ŒD; Sf2 .D/ is precompact in K.V / for all functions f1 ; f2 2 C0 .R X f0g/. Now (7.51) yields that ŒF; S  is precompact in K.V /. Theorem 7.52. Given any spectral triple over a locally convex topological algebra A, the constructions in Lemma 7.47 well-define its Chern–Connes character in HE .A/.

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7 The Chern–Connes character

It seems plausible that the resulting character is cohomologous to the ones constructed in [7], [53]; but we have not yet attempted to prove this. ˙ Warning 7.53. Let S  B1 be bounded if S is von Neumann bounded and ŒF; S  is ˙ into a bornological algebra because the precompact. This bornology does not turn B1 multiplication is only separately bounded.

7.5.5 Properties of the character Fredholm modules are not enough to get K-homology: we also need an appropriate equivalence relation. Then the issue arises whether it is compatible with the character constructed in Lemma 7.47. The obvious choice for the equivalence relation is homotopy, which allows us to vary all the data .V; F; ; '/ at once. But it is not clear how to handle continuous homotopies of this type because the analytic theory is only invariant under homotopies with bounded variation. Once we restrict attention to C  -algebras, the situation improves because of Kasparov’s deep result that homotopy generates the same equivalence relation as operator homotopy and addition of degenerate cycles ([57]). This is why we require C  -algebras in the following theorem, which summarises the properties of our character. Theorem 7.54. For separable C  -algebras A .with the precompact bornology/, the construction in Lemma 7.47 defines a group homomorphism ch W K  .A/ WD KK .A; C/ ! HA .A/: This character is dual to the K-theory character ch W K .A/ ! HA .A/ in the sense that hch.x/ j ch.y/i D hx j yi 2 Z for all x 2 K .A/, y 2 K .A/,  D 0; 1. The canonical map HA .A/ ! HL .A/ intertwines our new character and the character ch W K .A/ ! HL .A/ that we get from the universal property of KK .Theorem 7.1/. The proof of this theorem will occupy us for the remainder of this section. Compatibility with operator homotopy. It is often irrelevant whether we consider even or odd Fredholm modules; we usually write down assertions in the even case and leave it to the reader to notice that dropping changes nothing. We first discuss additivity. Let .Vj ; 'j ; Fj ; j / for j D 1; 2 be Fredholm modules. Their direct sum is given by V WD V1 ˚ V2 ;

' WD '1 ˚ '2 W A ! B.V1 ˚ V2 /;

F WD F1 ˚ F2 ;

WD 1 ˚ 2 :

All operators that appear in our computations for the direct sum Fredholm module are diagonal block matrices, whose entries are the corresponding operators for the summands. This implies additivity ch .x1 ˚x2 / D ch .x1 /Cch .x2 / of the character.

7.5 The character for general Fredholm modules

287

A Fredholm module is degenerate if Œ'.A/; F  D 0. Since the cut-off sequence commutes with F and as well, we get Q D 0 on the level of cocycles in such a case. Hence the character is not affected by adding degenerate cycles. An operator homotopy is a homotopy of Fredholm modules of the form .V; '; F t ; /, where V , ' and remain constant and F t varies continuously. The commutator condition means that .F t / t2Œ0;1 belongs to the unital C  -algebra C.Œ0; 1; D/ with D WD fT 2 B.V / j ŒT; '.A/  K.V /g and satisfies F 2 D 1 and F D F . It makes no difference whether we consider continuous or smooth operator homotopies: Lemma 7.55. Let F be an operator homotopy as explained above. Then there is a smooth path of invertible elements U 2 C 1 .Œ0; 1; D/ with U0 D idV and F1 D U1 F0 U11 . In the even case, we can further achieve that U commutes with . Proof. Here the arguments for the even and odd case are genuinely different. We first consider the odd case. The algebra D is a closed unital subalgebra of B.V / and thus a unital Banach algebra. We can replace F by the path of idempotents e t D 12 .1 C F t / in D. Standard techniques of K-theory [2] yield a continuous path of invertibles U 0 in D with U00 D id and U t0 e0 .U t0 /1 D e t for all t 2 Œ0; 1, so that U t0 F0 .U t0 /1 D F t as well. Since the map C 1 .Œ0; 1; D/ ! C .Œ0; 1; D/ is isoradial, we can replace it by a nearby smooth path of invertible elements U with the same endpoints; more precisely, any nearby smooth path with the same endpoints is invertible. In the even case, we use the grading operator to split V D VC ˚ V . We identify VC Š V using F0 jVC W VC ! V , which is invertible because F 2 D 1. We write F0 and F as block matrices:     0 1 0 T 1 F0 D ; ; F D T 0 1 0 where T 2 C.Œ0; 1; D/ is invertible, and take   1 0 0 U WD : 0 T   It is easy to see that U 0 F0 .U 0 /1 D F and ŒU 0 ; '.A/  C Œ0; 1; K.V / . Finally, we can replace U 0 by a smooth path with the same endpoints as above. Let U be as in Lemma 7.55. The Fredholm modules defined by .V; '; F0 ; / and .V; Ad.U1 / ı '; F1 ; / are isomorphic via U and hence have the same character in HA .A/. If we choose cut-off sequences appropriately, we can even achieve that the characters agree exactly. The algebra homomorphism   Ad.U / ı ' W A ! C 1 Œ0; 1; B1C .V; F1 ; /

288

7 The Chern–Connes character

is a smooth between ' and Ad.U1 /ı', so that HA0 .'/ D HA0 .Ad.U1 /ı'/  homotopy  C in HA0 A; B1 .V; F1 ; / by Theorem 5.45. Hence both Fredholm modules have the same character. Since operator homotopy and addition of degenerate cycles do not alter the character, we conclude that it descends to a map K .A/ ! HA .A/, which is additive because the addition in K .A/ comes from the direct sum of Fredholm modules. Our arguments work in the Banach space setting as well, but they do not apply to general homotopic Fredholm modules because we lack Kasparov’s identification of homotopy and operator homotopy. Compatibility with the index pairing. Let K .A/ be the topological K-theory of A. We briefly recall the definition of the index pairing Ind W K  .A/K .A/ ! Z,  D 0; 1. We refer to [2] for a more detailed account. Remark 7.56. Since the compatibility of the character with the index pairing is a builtin property of the bivariant Chern–Connes character KK ! HL , the corresponding result for the characters in analytic cyclic (co)homology follow from the compatibility between the characters in local and analytic cyclic (co)homology, which we will prove below. Therefore, our proof of the compatibility with the index pairing is redundant. In the even case, we represent elements of K 0 .A/ and K0 .A/ by an even Fredholm module .V; '; F; / and an element a 2 Mn .A/ such that 1 C a 2 Mn .A/C is idempotent. We inflate .V; '; F; / to an even Fredholm module .Vn ; 'n ; Fn ; n / over Mn .A/ by taking Vn WD V ˝ C n , 'n WD ' ˝ idMn , Fn WD F ˝ idC n , and n D ˝ idC n . C  The grading splits the range of the even idempotent 1 C 'n .a/ as H  ˚ H . The C  compression of F is a Fredholm operator T W H ! H . We let Ind Œ.'; F; /; Œa be its index, that is,   Ind Œ.'; F; /; Œa WD dim ker T  dim coker T: In the odd case, represent elements of K1 .A/ and K1 .A/ by an odd Fredholm module .V; '; F / and an element a 2 Mn .A/ such that 1 C a 2 Mn .A/C is invertible. Let .Vn ; 'n ; Fn / be the resulting odd Fredholm module over Mn .A/. Let H be the range WD 12 .1 C Fn / and let p be the orthogonal projection onto H . Then ofthe projection  p   p 1 C 'n .a/ p W H ! H is a Fredholm operator, whose  index is Ind Œ.';F /; Œa . The index pairing is natural in the sense that Ind ; f .y/ D Ind.f . /; y/ for any algebra homomorphism f W A ! B and 2 K .A/, y 2 K .B/. The pairing between HA and HA is natural in the same sense. When we apply this to the maps ˙ that are part of the Fredholm module, we see that it suffices to check the A ! B1 compatibility of the Chern–Connes character with the index pairing for the canonical ˙ Fredholm modules over B1 . ˙ is isoradial (Lemma 7.43) and hence induces Now we use that the map B1˙  B1 an isomorphism on K-theory. Therefore, it suffices to treat the Fredholm modules over ˙ B1˙ defined by the maps B1˙  B1 . But these Fredholm modules are 1-summable by construction. Lemma 7.47 yields that the Chern–Connes characters Q˙ agree with 0

7.5 The character for general Fredholm modules

289

or 1 . Hence it suffices to check an index formula for 1-summable Fredholm modules. Our proof of this index formula is purely formal and uses computation of  a complete  ˙ K .B1 / (in the Banach space setting, this requires K B.V / D 0). C  / D Z and K .B1 / D ZŒ1. Lemma 7.57. K .B1 C  / and E1 2 K1 .B1 / during the proof. We will describe generators E0 2 K0 .B1 C by diagonal block matrices as in (7.20); this Proof. In the even case, we describe B1 yields a split algebra extension C K.VC /  B1  B.VC /:     Since K K.VC / D Z and K B.VC / D 0, the long exact sequence in K-theory C / D Z. A diagonal operator E0 D .p; 0/ for some rank-1 projection yields K .B1 p 2 K.VC / is a generator. In the odd case, we use (7.20) and get an algebra extension  K.V /  B1  Q.VC / ˚ Q.V /;

where Q.V / WD B.V /= K.V / denotes the Calkin algebra on V . We assume that both V˙ are infinite-dimensional because otherwise F  1 2 K.V / or F C 1 2 K.V /, so that any Fredholm module involving .V; F / is operator homotopic to a degener ate one. Since the embedding K.VC /  K.V / ! B1 factors through B.VC / and  K B.VC / D 0, it induces the zero map on K-theory. The long exact sequence for   / D 0, K1 .B1 / Š Z. the above extension therefore yields K0 .B1 If .V; F / is as in Example 7.36, then we choose the backwards bilateral shift as our generator E1 . Let .V; '; F; / be an even Fredholm module, so that ' is a -homomorphism C from A to B1 .V; F; /. The index pairing with .V; '; F; / agrees with the composition '

E0

C / ! Z: K0 .A/ ! K0 .B1

Similarly, the index pairing for an odd Fredholm module .V; '; F / agrees with the composite map '

E1

 / ! Z: K1 .A/ ! K1 .B1

Here E0 and E1 denote the maps that send the generators E0 and E1 to 1, respectively. Now we prove the compatibility of the characters in K-theory and K-homology with the index pairings. We have already observed that it suffices to check this for ˙ the standard Fredholm modules over B1 . We may further restrict attention to the C  generators E0 2 K0 .B1 / and E1 2 K1 .B1 /. This amounts to the formulas hch .E0 / j ch .V; id; F; /i D 1;

hch .E1 / j ch .V; id; F /i D 1;

which we have already checked (see (7.31) and (7.37)).

290

7 The Chern–Connes character

Characters in analytic and local cyclic cohomology. Now we compose the Chern– Connes character K .A/ ! HA .A/ with the canonical map HA .A/ ! HL .A/. We claim that the resulting map W K .A/ ! HL .A/ agrees with the character constructed in Theorem 7.1. Since the latter construction is purely formal, so will be our argument. Nevertheless, such formal considerations are useful; for instance, the compatibility of the characters in K-theory and K-homology with the index pairing is an immediate consequence of this fact. It is clear that is a natural transformation with respect to -homomorphisms. Since the functors HL and K on separable C  -algebras are C  -stable and split-exact, the universal property of KK yields that is natural with respect to morphisms in KK. Any element in K0 .A/ Š KK0 .A; C/ can be written as f  .1C / where f 2 KK0 .A; C/ and 1C 2 KK0 .C; C/ is the unit element. Therefore, we get .f / D f  .1C / . The same reasoning applies to the bivariant Chern–Connes character. Hence they agree on K0 .A/ for all A once they agree on the single element 1C 2 KK0 .C; C/. But this case is trivial. Inthe odd case, we can argue similarly, replacing 1C by the canonical generator in K C0 .R/ . That is, it suffices to check that the two characters agree on this single K-homology class. This is essentially equivalent to the computation in Example 7.36. This finishes the proof of Theorem 7.54.

Appendix. Algebraic preliminaries A.1 Chain complexes over additive categories Let C be an additive category and let p 2 N. We let Kom.CI Z=p/ be the category of Z=p-graded chain complexes over C, where we interpret Z=0 D Z. We only use Z-graded and Z=2-graded chain complexes in this book. If p ¤ 0, objects of Kom.CI Z=p/ are p-periodic diagrams @1

@p1

@0

@p2

@2

@1

@0

@p1

   ! C0 ! Cp1 ! Cp2 !    ! C1 ! C0 ! Cp1 !    with @2 D 0; morphisms of such diagrams are also called p-periodic chain maps. Definition A.1. The translation functors (or suspension functors) †k W Kom.CI Z=p/ ! Kom.CI Z=p/;

C 7! †k .C / D C Œk;

are defined on objects by .Cn ; @n /Œk WD .Cnk ; .1/k @nk / and on morphisms by .fn /Œk D .fnk /; notice the sign in the boundary map. Remark A.2. If p is even, then multiplication by .1/n on Cn is a p-periodic chain map between .Cn ; @n / and .Cn ; @n /. Therefore, in the cases we are interested in, the sign in the definition of † can be removed. Remark A.3. Let CZ=p be the category of Z=p-graded objects of C. Forgetting the boundary map, we get a functor Kom.CI Z=p/ ! CZ=p ; this functor has a left adjoint F W CZ=p ! Kom.CI Z=p/; which maps .Ak /k2Z=p 22 CZ=p to the periodic chain complex with F .A/k WD Ak ˚ AkC1 and boundary map 01 00 . In case p is finite or Qif C has countable direct products, then the forgetful functor CZ=p ! C, .Ak / 7! Ak , has the left adjoint C ! CZ=p , A 7! const.A/, where const.A/k WD A for all k 2 Z=p. Combining both adjointness relations, we conclude that the forgetful functor Kom.CI Z=p/ ! C has a left adjoint.

A.1.1 Chain homotopy and bivariant homology Definition A.4. Let .Ck ; @C / and .Dk ; @D / be Z=p-graded chain complexes in C. The k k mapping complex Hom.C; D/ 22 Kom.AbI Z/ is defined by Y Hom.Ck ; DkCn /; Hom.C; D/n WD k2Z=p

  n C @n .fk /k2Z=p WD @D kCn ı fk  .1/ fk1 ı @k 



292

Appendix. Algebraic preliminaries

for all n 2 Z. It is easy to check that this is indeed a chain complex. It is p-periodic if p is even and 2p-periodic if p is odd. The n-cycles in Hom.C; D/ (that is, elements of ker @n ) are exactly the chain maps C Œn ! D. We abbreviate   Hn .C; D/ WD Hn .C; D/ WD Hn Hom.C; D/ for n 2 Z. Notation A.5. If f1 ; f2 2 Hom.C; C /, then Œf1 ; f2  denotes the graded commutator Œf1 ; f2  WD f1 f2  .1/jf1 j jf2 j f2 f1 : The boundary map on Hom.C; C / maps f 7! Œf; @ D Œ@; f  for all f because @ is homogeneous of degree 1. We also denote the boundary map in Hom.C; D/ as Œ@; f  although this involves two different boundary maps @C and @D . Definition A.6. Two chain maps f0 ; f1 W C ! D are .chain/ homotopic if they are homologous in Hom.C; D/, that is, there is h 2 Hom.C; D/1 with Œ@; h D f1  f0 . We call h a .chain/ homotopy between f0 and f1 and write f0 f1 in this case. Chain homotopy is an equivalence relation on the set of chain maps C ! D; the set of equivalence classes is H0 .C; D/. Definition A.7. Suppose that C is a symmetric monoidal category. View 1 2 C as a chain complex supported in degree 0 mod p with zero boundary map. We define the homology and cohomology of a chain complex by     Hk .C / WD Hk .1; C / D Hk Hom.1; C / ; Hk .C / WD Hk .C; 1/ D Hk Hom.C; 1/ : Example A.8. If C D Ban or C D Cborn, then 1 is R or C. Hence Hom.1; C / is just C viewed as a complex of Abelian groups, and Hom.C; 1/ is the dual complex of ! bounded linear functionals on C . If C D Ban, then Hom.1; C / is isomorphic to the inductive limit complex of C in the category of vector spaces (without any separated quotient). The composition of maps defines a linear map Hom.D; E/ ˝ Hom.C; D/ ! Hom.C; E/:

(A.9)

This is a chain map if we equip the tensor product of two complexes with the total grading and boundary map, defined by deg.a ˝ b/ D deg.a/ C deg.b/;

@.a ˝ b/ WD @.a/ ˝ b C .1/deg.a/ a ˝ @.b/:

Hence the composition of chain maps descends to maps on homology Hn .D; E/ ˝ Hm .C; D/ ! HnCm .C; E/: Thus we get a Z=p-graded category with morphism spaces H .C; D/.

A.1 Chain complexes over additive categories

293

Definition A.10. The homotopy category of Z=p-graded chain complexes in C is the category HoKom.CI Z=p/ with the same objects as Kom.CI Z=p/ and morphisms H0 .C; D/. A chain complex C is called .chain/ contractible if C Š 0 in HoKom.CI Z=p/, that is, idC 0. A chain map f W C ! D is called a .chain/ homotopy equivalence if it becomes an isomorphism in HoKom.CI Z=p/. Being a homotopy equivalence means that there are a chain map g W D ! C – the homotopy inverse – and chain homotopies hC W C ! C Œ1 and hD W D ! DŒ1 with Œ@; hD  D idD  f ı g;

Œ@; hC  D idC  g ı f:

If h is a contracting homotopy for a chain complex C , then the maps @ ı h and h ı @ are idempotent. If they have range objects, then our chain complex is isomorphic to one of the form    RR  Zp1 SS Z2 RRR Z1 RRR Z0 SSS RRR SSSŠ SSS RRŠ RRŠ RRR ˚ RRR ˚ RRR ˚ S S S ˚ ˚ S SSS ˚ RR( RR) RR) id id id SSS) )   Z3 Z2 Z1 Z0

A.1.2 Mapping cones of chain maps B Definition A.11. Let A D .An ; @A n / and B D .Bn ; @n / be Z=p-graded chain complexes over C and let f D .fn / W A ! B be a chain map. The mapping cone cone.f / of f is defined by cone.f /n WD An1 ˚ Bn with boundary map  A  @n1 0 / WD W An1 ˚ Bn ! An2 ˚ Bn1 : @cone.f n fn1 @B n

The coordinate maps f W B ! cone.f / and f W cone.f / ! AŒ1 are chain maps. The diagram f

f

f

A ! B ! cone.f / ! AŒ1 in Kom.C/ is called a mapping cone triangle. Warning A.12. Some authors prefer to use cochain complexes, where the boundary map has degree C1. This entails a different definition of the mapping cone. WD cone.idA /; it Example A.13. The cone of a chain complex is defined by cone.A/  is contractible with the canonical contracting homotopy 00 id0A . A chain map D ! cone.f /Œ1 is a pair of maps g W D ! A, h W D ! BŒ1, such that g is a chain map and Œ@; h D f ı g, that is, h is a chain homotopy from f ı g to 0.

294

Appendix. Algebraic preliminaries

Remark A.14. The name “mapping cone” has its origin in algebraic topology. Let A and B be pointed topological spaces with base points ? and let f W A ! B be a continuous map with f .?/ D ?. Its mapping cone cone.f / is obtained from the disjoint union A  Œ0; 1 t B by imposing the relations .?; t / .?; 0/ .a; 0/ for all t 2 Œ0; 1, a 2 A, and .a; 1/ f .a/ for all a 2 A; here ? 2 A is the base point. The first relations merely replace A  Œ0; 1 by the smash product A ^ Œ0; 1, where Œ0; 1 has the base point 0. If A and B are CW-complexes and f is cellular, then cone.f / inherits a canonical cell decomposition, whose cells (apart from ?) are of the form ˛  Œ0; 1 and ˇ for cells ˛  A, ˇ  B. The map f induces a chain map between the cellular chain complexes of A and B; its mapping cone is exactly the cellular chain complex of cone.f /. We can now define the triangulated category structure on HoKom.CI Z=p/. Recall that a triangle in HoKom.CI Z=p/ is a diagram of the form A ! B ! C ! AŒ1; morphisms of triangles are commuting diagrams /B

A ˛

/C

ˇ

 A0

/ AŒ1

 / B0

˛Œ1

 / A0 Œ1

 / C0

in HoKom.CI Z=p/. This also defines isomorphism of triangles. Definition A.15. Triangles of the form f

f

f

! B ! cone.f / ! AŒ1 A for a chain map f W A ! B are called mapping cone triangle. A triangle in HoKom.CI Z=p/ is exact if it is isomorphic to a mapping cone triangle. Theorem A.16. The category HoKom.CI Z=p/ with the translation functor and exact triangles as defined above is a triangulated category for any additive category C. We refer to [74], [102] for the axioms of a triangulated category. We mostly avoid using this structure explicitly. Definition A.17. Let T and T 0 be triangulated categories. A triangle functor functor T ! T 0 consists of a functor F W T ! T 0 and natural isomorphisms ˆ W F †.A/ Š †0 F .A/ for all A such that F .f /

F .g/

ˆıF .h/

F .A/ ! F .B/ ! F .C / ! †0 F .A/ f

g

h

!B is an exact triangle in T 0 if A  !C !  †A is an exact triangle in T . A covariant (contravariant) functor from a triangulated category to an Abelian category is called homological (cohomological) if it maps exact triangles to exact sequences.

295

A.1 Chain complexes over additive categories

The construction of the triangulated category HoKom.CI Z=p/ is natural, that is, an additive functor F W C1 ! C2 induces a triangle functor Fx W HoKom.C1 I Z=p/ ! HoKom.C2 I Z=p/: We first define a functor Fx W Kom.C1 I Z=p/ ! Kom.C2 I Z=p/ by applying F entrywise to chain complexes. This functor commutes with the translation functor and descends to the homotopy category. Since it preserves mapping cone triangles, it maps exact triangles again to exact triangles.

A.1.3 Exact sequences The Puppe Exact Sequence is the most basic exact sequence for homology groups of chain complexes. Theorem A.18. Let f W A ! B be a chain map in Kom.CI Z=p/. For any object D, the maps in the mapping cone triangle for f yield natural long exact sequences f  f    f    ! H1 .D; A/ ! H1 .D; B/ ! H1 D; cone.f / f  f    f  f ! H0 .D; A/ ! H0 .D; B/ ! H0 D; cone.f / !    ; f

f

f

    H1 .A; D/  H1 .B; D/  H1 .cone.f /; D/ f

f

f

f

 H0 .A; D/  H0 .B; D/  H0 .cone.f /; D/     :

Let F W C ! C0 be an additive functor with values in an Abelian category C0 , and let Fx be its extension to chain complexes as above. Then we get a natural long exact sequence           ! Hn Fx.A/ ! Hn Fx.B/ ! Hn Fx cone.f /        ! Hn1 Fx.A/ ! Hn1 Fx.B/ ! Hn1 Fx cone.f / !    : Dually, if F is a contravariant functor then we get a long exact sequence        Hn Fx.B/ Hn Fx cone.f /  Hn Fx.A/        Hn1 Fx.A/ Hn1 Fx.B/ Hn1 Fx cone.f /

 :

Proof. The functor Hom.D; / maps the mapping cone triangle for f to the mapping cone triangle for the chain map Hom.D; A/ ! Hom.D; B/ induced by f . Essentially (up to signs and suspensions) the same is true for the contravariant functor Hom. ; D/. Therefore, the Puppe Exact Sequences for Hom.D; / and Hom.D; / reduce to the Puppe Exact Sequence for the homology of a chain complex of Abelian groups. Similarly, the exactness of a chain complex in an Abelian category is equivalent to the

296

Appendix. Algebraic preliminaries

exactness  of Hom.X; / for all objects of the category. Therefore, the exact sequences for H Fx. / also reduce to the case of Abelian groups. This classical case is treated in any book on homological algebra. i

p

Theorem A.19. Let I  E  Q be a semi-split extension of chain complexes. Then there are canonical chain homotopy equivalences I Œ1 ! cone.p/ and cone.i / ! Q. Proof. We only construct the chain homotopy equivalence I Œ1 cone.p/; the equivalence cone.i/ Q is obtained by a dual argument. The coordinate embedding EŒ1 ! cone.p/ is not a chain map, but its restriction to I Œ1 is one because p ı i D 0. This provides a natural map I Œ1 ! cone.p/. A section s W Q ! E induces a decomposition E Š I ˚ Q, so that cone.p/n Š In1 ˚ Qn1 ˚ Qn . The boundary map becomes 0 I 1 @n1 Œ@; sn 0 @ 0 @Q 0A n1 0 idQn1 @Q n where Œ@; s D @ ı s  s ı @. Now we decompose 0 1 Œ@; s cone.p/n Š In1 ˚ Qn1 ˚ @ @ A .Qn / Š I Œ1 ˚ cone.Q/: id Since cone.Q/ is contractible, we get I Œ1 cone.p/ as asserted. Since homology is preserved under chain homotopy equivalence, Theorem A.19 allows to carry over the various Puppe Exact Sequences in Theorem A.18 to semi-split extensions of chain complexes. We are going to describe the resulting boundary maps explicitly. The co- and contravariant cases differ by certain signs. Let s W Q ! E be i

p

a section for the extension I  E  Q. Then the map Œ@; s WD @E ı s  s ı @Q satisfies p ı Œ@; s D Œ@; p ı s D 0, so that it factors through i . The resulting map i 1 ı Œ@; s W Q ! I is a chain map ı W Q ! I Œ1. Another choice for s differs by adding a map h W Q ! I , so that we get a chain homotopic map ı C Œ@; h W Q ! I Œ1. Hence we get a well-defined class Œı 2 H1 .Q; I /. It is easy to see that the diagram Œı

/ I Œ1 QG GG GGp GG GG

#  cone.p/

commutes. Hence the boundary maps in the exact sequences are, up to sign, given by composition with Œı. These signs are introduced to comply with the Koszul sign

A.1 Chain complexes over additive categories

297

rule. In the homological case, Œı .f / D f ı ı generates the sign .1/deg f . In the cohomological case, the correct choice is to use Œı because reversing the order of multiplication turns ı D @s  s@ into s@  @s D ı. These signs have no effect on the exactness of the chain complexes, of course. Lemma A.20. The following are equivalent for a chain map f W A ! B: • f is a chain homotopy equivalence; • H .D; f / W H .D; A/ ! H .D; B/ is invertible for all D 22 HoKom.C/; • the mapping cone cone.f / is chain contractible. Proof. The first two assertions are equivalent by the Yoneda Lemma, the last two are equivalent by the Puppe Exact Sequence (Theorem A.18).

A.1.4 Tensor product and internal Hom Let C be a symmetric monoidal category with tensor product ˝, and let p 2 N be even. If p D 0, we also assume that C has countable coproducts compatible with ˝. Then the category Kom.CI Z=p/ inherits a symmetric monoidal structure. The tensor product functor ˝ W Kom.CI Z=p/  Kom.CI Z=p/ ! Kom.CI Z=p/ is defined by .C ˝ D/k WD

L n2Z=p

Cn ˝ Dkn and

n D @jCn ˝Dm WD @C n ˝ idDm C .1/ id Cn ˝ @m :

It is straightforward to check that this is again a chain complex and that ˝ defines a monoidal structure. It is symmetric, the isomorphism C ˝ D Š D ˝ C restricts to Š

! Dm ˝ C n .1/n m ˆCn ;Dm W Cn ˝ Dm  on the direct summands. Now let C be closed symmetric monoidal and let p 2 2N be even. If p D 0, we assume that C has countable products and coproducts. Then Kom.CI Z=p/ is closed symmetric monoidal as well. The internal Hom functor Hom is defined by substituting Hom for Hom in the definition of the mapping complex (Definition A.4) and noticing that the result is p-periodic. It is easy to see that this has the right adjointness property. Warning A.21. If p is odd, say, p D 1, then the signs in C ˝ D and Hom.C; D/ do not make sense, and there is no symmetric monoidal structure on Kom.CI Z=p/.

298

Appendix. Algebraic preliminaries

A.2 Basic constructions with algebras and modules Throughout this section, we fix an additive symmetric monoidal category C; we assume that morphisms in C have cokernels and that ˝ preserves them, that is, the canonical map coker.V ˝ f W V ˝ A ! V ˝ B/ ! V ˝ coker.f W A ! B/ is an isomorphism for any morphism f W A ! B and object V in C. This compatibility is automatic if C is closed because then V ˝ is a left adjoint functor, and such functors ! automatically preserve cokernels. Hence the condition holds for Cborn and Ban and for the categories of Abelian groups or of K-vector spaces for some field K. The assumption also holds for the category of complete locally convex topological vector spaces with the complete projective topological tensor product, although it is not closed. We have already defined (unital) algebras and (unital) modules in C in §1.3.8. Now we extend some basic algebraic constructions to this abstract setting. Most of our constructions are quite familiar for classical rings and similar objects like topological or bornological algebras. The abstract approach shows that this extends to algebras in ! less familiar categories such as Ban.

A.2.1 Substitutes for elements When doing algebra in symmetric monoidal categories, the first thing we miss is elements. The identity .x  y/  z D x  .y  z/ is much easier to swallow than the associativity diagram in Definition 1.105. Mildly complicated computations with such diagrams such as, say, the proof that the differential d on differential forms is a graded derivation, get very cumbersome with such diagrams – both to write and to read. The main purpose of this section is to convince you that formal computations in terms of elements work in any additive symmetric monoidal category because there is a translation process that turns them into maps and diagrams in C. Before we discuss this, we consider a more direct approach to define elements of objects in general symmetric monoidal categories. This solves our problems in some cases, but not in general. Definition A.22. Let V be an object of a symmetric monoidal category C. Elements of V are maps 1 ! V . This definition gives the expected notion of element for familiar categories like Vect, Born1=2, or the category of topological vector spaces. In each of these cases, the functor Hom.1; / maps objects to their underlying Abelian groups. We can encode some further structure because Hom.1; V / is, in a natural way, a module over the endomorphism ring Hom.1; 1/; this ring is always commutative. If x W 1 ! A and y W 1 ! A are two elements of A, then their product is the element Š

x˝y

mA

1 ! 1 ˝ 1 ! A ˝ A ! A:

A.2 Basic constructions with algebras and modules

299

Thus the elements of algebras in C form algebras over the commutative ground ring Hom.1; 1/. A unit map in the sense of Definition 1.106 is the same as a unit element in this algebra. The above approach works well for the category Cborn, but it breaks down for ! Ban. The issue is whether the forgetful functor Hom.1; / is faithful. This is the case for Cborn and implies that a map is determined by its values on elements; therefore, it suffices to check relations between maps on elements. In contrast, there exist non-trivial ! ! inductive systems in Ban or Vect whose direct limit vanishes and which, therefore, have no elements. Hence we need another more powerful approach. Let us consider a simple example: we want to check that inner derivations are derivations. We are given an algebra A, a bimodule M , and an element x W 1 ! M . In terms of elements, the resulting inner derivation is given by dx .a/ WD Œx; a D xaax for all a 2 A; we have to check the Leibniz rule dx .a  b/ D dx .a/  b C a  dx .b/. It follows from the computation dx .a b/ D x a b a b x D x a b a x b Ca x b a b x D dx .a/b Ca dx .b/: We translate this to a computation in an additive symmetric monoidal category. Each of the summands above denotes a map, where we view the free variables a; b in A simply as tensor factors A. Thus x  a and a  x denote maps A ! M because they contain one free variable a 2 A, whereas x  a  b, a  x  b, a  b  x denote maps A ˝ A ! M because they contain two free variables in A. Here we assume that x is not free; the following discussion simplifies if we view x as a free variable as well; then a ˝ x 7! x  a and a ˝ x 7! a  x become maps A ˝ M ! M , and so on. The translation goes as follows: x  a and a  x denote the maps Š

x˝idA

mMA

! 1 ˝ A ! M ˝ A ! M; A

idA ˝x

Š

mAM

A ! A ˝ 1 ! A ˝ M ! M I

more complicated expressions such as a  x  b and dx .a/  b become Š

idA ˝x˝idA

mAMA

! A ˝ 1 ˝ A ! A ˝ M ˝ A ! M; A˝A dX ˝idA

mMA

A ˝ A ! M ˝ A ! M: The bimodule condition a  .x  b/ D .a  x/  b shows that there is only one canonical multiplication map mAMA W A ˝ M ˝ A ! M (see §A.2.4). Thus we can view all expressions in the above computations as maps. Since our computation is completely formal, the equalities continue to hold in any additive symmetric monoidal category C. All the computations in terms of elements that we shall meet can be formalised in this way and therefore apply in any additive symmetric monoidal category. I do not characterise explicitly which computations can be translated in this fashion because when I tried this I merely got confused. Let me just mention an obvious limitation: we cannot deal with non-linear expressions such as aa for a free variable a.

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All summands in a formula must contain the same types of free variables (so that they all have the same domain), and each free variable can occur at most once in each summand.

A.2.2 Opposite algebras Definition A.23. Let A be an algebra in C with multiplication map m W A ˝ A ! A and let ˆA;A W A ˝ A ! A ˝ A be the symmetry that exchanges the two tensor factors. Then m ı ˆA;A is another associative product on A. The resulting algebra is called the opposite algebra of A and denoted by Aop . In terms of elements, the product in Aop is the map a ˝ b 7! a op b WD b  a. Recall that A is commutative if and only if m ı ˆA;A D m, that is, the identity map is an algebra isomorphism A Š Aop . There are also many non-commutative algebras – for example, group rings – for which A Š Aop by a non-identity isomorphism. Proposition A.24. The category Mod.Aop / of left modules over Aop is isomorphic to the category of right A-modules in C. If A is unital, so is Aop , and the category ModC .Aop / of left unital Aop -modules in C is isomorphic to the category of right unital A-modules in C.

A.2.3 Adjoining units Recall that Alg.C/ and AlgC .C/ denote the categories of algebras and unital algebras in C, respectively. Definition A.25. Let A be an algebra in C. The unital algebra generated by A is a unital algebra AC in C together with an algebra homomorphism i W A ! AC such that composition with i induces isomorphisms AlgC .AC ; B/ Š Alg.A; B/ for all B 22 AlgC .C/: This means that A 7! AC is the left adjoint functor of the embedding of categories AlgC .C/ ! Alg.C/. To construct AC explicitly, let AC WD A ˚ 1 as an object of C. The coordinate embeddings i W A ! AC and u W 1 ! AC provide the map i in Definition A.25 and the unit map of AC . To describe the multiplication, we use the isomorphism .A ˚ 1/ ˝ .A ˚ 1/ Š .A ˝ A/ ˚ .1 ˝ A/ ˚ .A ˝ 1/ ˚ .1 ˝ 1/; which follows from the additivity of ˝; the multiplication map AC ˝ AC ! AC restricts to the multiplication of A on A ˝ A, and to the canonical isomorphisms 1 ˝ A Š A, A ˝ 1 Š A, and 1 ˝ 1 Š 1 on the other direct summands. In terms of elements, this amounts to .a1 ; 1 /.a2 ; 2 / WD .a1 a2 C1 a2 C2 a1 ; 1 2 / for all a1 ; a2 2 A; 1 ; 2 2 1:

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It is straightforward to verify that AC is associative, that the coordinate embeddings A ! AC 1 are algebra homomorphisms, and that AC has the universal property of Definition A.25. The coordinate projection A ˚ 1 ! 1 provides a unital homomorphism AC ! 1 called the augmentation map of AC . The unit map 1 ! A is a section for this homomorphism. Hence we get a split algebra extension A  AC  1. Here “algebra extension” means an extension in the category Alg.C/ in the sense of Definition 1.72. Definition A.26. An augmented algebra in C is a unital algebra A in C together with a unital algebra homomorphism A ! 1 called augmentation map; its kernel is called the augmentation ideal. Morphisms of augmented algebras are unital algebra homomorphisms compatible with the augmentation maps. Proposition A.27. The map A 7! AC provides an equivalence between the categories of non-unital algebras and of augmented algebras in C. Proof. Any algebra homomorphism A ! B extends uniquely to a morphism of augmented algebras AC ! BC , and any morphism of augmented algebras AC ! BC restricts to an algebra homomorphism A ! B. Thus A 7! AC becomes a fully faithful functor. If B is an augmented algebra and A is its augmentation ideal, then the embedding A ! B extends to an isomorphism of augmented algebras AC Š B. The functor A 7! AC corresponds to the one-point compactification for locally compact topological spaces, which explains our notation. But the one-point compactification is not functorial: only proper maps X ! Y extend to maps XC ! YC . This problem disappears in categories of algebras because already X 7! C0 .X / is only functorial for proper continuous maps. Hence we prefer to view C0 as a functor on the category of pointed compact spaces. An A-module structure extends uniquely to a unital AC -module structure and, conversely, a unital AC -module structure is already determined by the A-module structure it restricts to. This provides a useful isomorphism of categories Mod.A/ Š ModC .AC /:

(A.28)

A.2.4 Bimodules Let A; B 22 Alg.C/. An A; B-bimodule in C carries both a left A-module and a right B-module structure, and these are compatible in the sense that the diagram A˝M ˝B

mAM ˝idB

idA ˝mMB

 A˝M

/M ˝B mMB

mAM

 /M

(A.29)

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commutes. If A D B, we briefly speak of A-bimodules instead of A; A-bimodules. We denote the categories of bimodules and unital bimodules by Mod.A; B/ and ModC .A; B/, respectively; we want to identify them with categories of (unital) left modules. First we consider unital A; B-bimodules; of course, “unital” means being unital over both A and B. We need the tensor product algebra A˝B, which has the multiplication idA ˝ˆB;A ˝idB

mA ˝mB

A ˝ B ˝ A ˝ B ! A ˝ A ˝ B ˝ B ! A ˝ B Š

or .a1 ˝ b1 /  .a2 ˝ b2 / WD a1  a2 ˝ b1  b2 in terms of elements. This turns A ˝ B into a (unital) algebra if A and B are. Let A and B be unital algebras with unit maps uA W 1 ! A and uB W 1 ! B, and let M be a unital A; B-bimodule. First, we translate the right B-module structure into a left B op -module structure. Then we combine the two module structures into a single multiplication map mAMB W A ˝ B op ˝ M ! M using the diagonal map in (A.29). This turns M into a unital A ˝ B op -module. Conversely, any unital A ˝ B op -module arises in this fashion: the left A- and B op -module structures are the composite maps B ˝idM A ˝ M ZZidZAZZ˝u ZZZZZZZ op 1A˝B ˝M d d d d d d d d d d d B op ˝ M uA ˝idB ˝idM

mAMB

/ M:

Hence the categories of unital A; B-bimodules and of unital A ˝ B op -modules are isomorphic, that is, ModC .A; B/ Š ModC .A ˝ B op /: For non-unital A, B, and M , a variant of (A.28) yields isomorphisms of categories Mod.A; B/ Š ModC .AC ; BC / Š ModC .AC ˝ .BC /op / Š ModC .AC ˝ .B op /C /; that is, an A; B-bimodule structure extends to a unital AC ; BC -bimodule structure. Tensor products of augmented unital algebras are again augmented in a canonical way. Let A  B WD ker.AC ˝ BC ! 1 ˝ 1 Š 1/: be the augmentation ideal of AC ˝ BC . This is related to A ˝ B by a semi-split algebra extension A ˝ B  A  B  A ˚ B; which splits on the summands A and B separately. Since Mod.A  B/ Š ModC .A ˝ B/, our computation above shows Mod.A; B/ Š Mod.A  B/:

(A.30)

Summing up, all categories of modules, whether left, right or two-sided, whether unital or not, are isomorphic to ModC .A/ for a suitable unital algebra A. Therefore, it suffices to explain constructions for left unital modules.

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A.2.5 Direct and inverse limits of modules Let I be a small category and let F W I ! ModC .A/ be a diagram of unital A-modules. Suppose that this diagram has a direct limit lim F or an inverse limit lim F in the !  underlying category C. We want to equip it with a canonical unital A-module structure and show that this yields a direct or inverse limit in the category ModC .A/. First we consider the inverse limit. Let i W lim F ! F .i / for i 22 I be the  canonical map. The maps idA ˝i

mA;F .i /

A ˝ lim F ! A ˝ F .i / ! F .i /  combine to a map A ˝ lim F ! lim F by the universal property of lim F . This    defines a unital A-module structure on lim F and turns lim F into an inverse limit in   the category ModC .A/; that is, a map h to lim F is a module homomorphism if and  only if the composite maps i ı h to F .i / are module homomorphisms for all i 22 I . Now we consider the direct limit. We assume that ˝ is compatible with direct limits in the sense that the canonical maps i W F .i / ! lim F induce an invertible map !   lim A ˝ F .i / ! A ˝ lim F .i /: ! ! The maps A ˝ F .i / ! lim F that we get by composing the module structures A ˝ !   F .i/ ! F .i/ with i induce a map A ˝ lim F Š lim A ˝ F .i / ! lim F . This ! ! ! defines a unital A-module structure on lim F with the correct universal property for a ! direct limit. The compatibility of ˝ with direct limits is automatic if C is closed because then ˝ is a left adjoint, and such functors always preserve direct limits. Hence we get: ! Proposition A.31. If C is closed and bicomplete – such as Cborn or Ban – then for any unital algebra A in C the category ModC .A/ is bicomplete and the forgetful functor ModC .A/ ! C commutes with inverse and direct limits. The assertion about the forgetful functor merely says that direct and inverse limits in ModC .A/ are the corresponding limits in C equipped with appropriate module structures. Warning A.32. Proposition A.31 fails in the category of complete locally convex topological vector spaces with the complete projective topological tensor product. Consider the Fréchet algebraLA D C 1 .Œ0; 1/, viewed as a module over itself. Equip the infinite direct sum V WD 1 iD1 A with the direct sum topology. Then the canonical module structure on V is only separately continuous, that is, V is not an A-module in C. To see that the multiplication map A ˝ V ! V , f ˝ .gn / 7! .f  gn /, is not jointly continuous, let kf kk be the supremum of the first k derivatives of f . Then .gn / 7! supkgn kn is a continuous norm on V . In order to control supkf  gn kn , we need simultaneous control on all derivatives of f (or vanishing of gn for n  0). But continuous seminorms on A only involve finitely many derivatives of f . Therefore, the multiplication A ˝ V ! V is not jointly continuous.

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Nevertheless, categories of complete topological bimodules have direct limits for all diagrams; but the forgetful functor does not commute with direct limits.

A.2.6 Balanced tensor products and internal Hom From now on, we work with unital algebras and modules unless we explicitly allow non-unital ones. Let A be a unital algebra in C. Definition A.33. Let V 22 ModC .Aop /, W 22 ModC .A/, that is, V and W are a right and a left A-module, respectively. Their A-balanced tensor product is   mVA ˝idW idV ˝mAW V ˝A W WD coker V ˝ A ˝ W ! V ˝ W 22 C: In terms of elements, we have .mVA ˝ idW  idV ˝ mAW /.v ˝ a ˝ w/ D .v  a/ ˝ w  v ˝ .a  w/: Example A.34. Let C D Cborn. Then ˝ means the completed tensor product, so that y A is more appropriate. The cokernel of a map in Cborn is the quotient by the notation ˝ the closure of its range; this ensures that V ˝A W is again a complete bornological vector space. Hence the forgetful functor Cborn ! Vect does not preserve balanced tensor products. But its adjoint Vect ! Cborn, the fine bornology functor, is symmetric monoidal; therefore, Fin.V ˝A W / Š Fin.V / ˝Fin.A/ Fin.W / for vector spaces V and W and an algebra A. Proposition A.35. The balanced tensor product in C has the following properties: • V ˝A W inherits a canonical B; C -bimodule structure if V is a B; A-bimodule and W is an A; C -bimodule; • ˝A is an additive bifunctor ModC .Aop /  ModC .A/ ! C or ModC .B; A/  ModC .A; C / ! ModC .B; C /; • there are natural associativity isomorphisms .V ˝A W / ˝B X Š V ˝A .W ˝B X / whenever such triple tensor products make sense; more precisely, if V is a C; A-bimodule, W an A; B-bimodule, and X a B; D-bimodule, then we have a natural C; D-bimodule isomorphism .V ˝A W / ˝B X Š V ˝A .W ˝B X /; • there are natural A; B-bimodule isomorphisms A ˝A V Š V; for all A; B-bimodules V .

V ˝B B Š V

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305

Proof. We omit proofs for the first three assertions, which require compatibility of ˝ with cokernels, and only check the last one. The associativity of the multiplication map m W A ˝ V ! V means that it descends to a map m Q W A ˝A V ! V . The map uA ˝idV

s0 W V ! A ˝ V  A ˝A V is a section for m, Q that is, m Q ı s0 D idV . The map .uA ˝ idV / ı m differs from the identity on A ˝ V by .m ˝ idV  idA ˝ m/ ı .uA ˝ idA˝V /; hence it descends to the identity map on A ˝A V D coker.m ˝ idV  idA ˝ m/. Let V and W be two left (or two right) A-modules. We have   V  HomA .V; W / Š ker mW   .m / W Hom.V; W / ! Hom.A ˝ V; W / ; V  where mW  and .m / denote post- and pre-composition with the module structures on W and V . This motivates the following definition:

Definition A.36. Suppose that C is closed. Let V and W be two left (or two right) A-modules. We let   V  HomA .V; W / Š ker mW   .m / W Hom.V; W / ! Hom.A ˝ V; W / : PropositionA.37. Let V and W be an A; B- and an A; C -bimodule. Then HomA .V;W / becomes a right B ˝ C -bimodule in a canonical way, and this defines an additive bifunctor HomA W ModC .A; B/op  ModC .A; C / ! ModC .C; B/: The canonical forgetful functor C ! Ab, V 7! Hom.1; V /, maps HomA .V; W / to HomA .V; W / because it is compatible with kernels, being a right adjoint functor. The adjointness between ˝ and Hom generalises to natural isomorphisms     HomB;C W; HomA .V; X / Š HomA;C .V ˝B W; X / Š HomA;B V; HomC .W; X / (A.38) if V , W , and X are A; B-, B; C -, and A; C -bimodules, respectively; here Hom.: : : / carries the appropriate induced bimodule structure. If some of V , W , or X carry further compatible module structures, then the above isomorphism is compatible with them; for instance, if V is an A ˝ D; B-bimodule, then the above isomorphism is one of right D-modules. Proof. We only sketch some parts of the proof. The bimodule structure on HomA .V; W / is constructed using generalisations of the natural maps in (1.113): .mW /

HomA .V; W / ˝ C ! HomA .V; W ˝ C / ! HomA .V; W /; m V

HomA .V; W / ˝ B ! HomA .V ˝ B; W / ˝ B ! HomA .V; W /:

The adjointness between Hom and ˝ identifies all three spaces in (A.38) with the kernel of a map Hom.V ˝ W; X / ! Hom.V ˝ W ˝ A ˝ B ˝ C; X / because Hom and hence HomB;C preserves kernels in the second variable. This yields (A.38).

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Example A.39. Let C D Cborn. Then HomA .V; W / is the subspace of A-module homomorphisms in Hom.V; W /, equipped with the subspace bornology, that is, with the equibounded bornology.

A.2.7 Free and cofree modules Definition A.40. Let V 22 C. The free unital module on V is a unital A-module F .V / such that HomA .F .V /; W / Š Hom.V; W / for all unital A-modules W . The cofree unital module on V is a unital A-module F  .V / such that HomA .W; F  .V // Š Hom.W; V / for all unital A-modules W . There are analogous definitions for right modules and bimodules. Lemma A.41. The free unital left module on V is A ˝ V with the canonical module structure. If C is closed, then the cofree unital left module on V is Hom.A; V / with the canonical module structure. If C is not closed, then cofree modules need not exist. This happens, for instance, for categories of locally convex topological modules. Proof. We prove adjointness of V 7! A ˝ V and the forgetful functor by constructing the unit and counit of adjunction (see [64]). If V is an A-module, then the module structure provides a map mAV W A ˝ V ! V , which is a module homomorphism if we equip A ˝ V with the canonical module structure mA ˝ idV W A ˝ A ˝ V ! A ˝ V . If V 22 C, then the unit map u W 1 ! A provides a canonical map u ˝ idV W V ! A ˝ V . The obvious relations mAV ı .u ˝ idV / D idV for V 22 ModC .A/ and .mA ˝ idV / ı .idA ˝ u ˝ idV / D idA˝V for V 22 C mean that these two natural transformations generate an adjointness of functors. The assertion for Hom.A; V / follows from the adjointness relation   HomA X; Hom.A; V / Š Hom.A ˝A X; V / Š Hom.X; V / for X 22 Mod.A/, V 22 C, or by an argument similar to the one above. For non-unital modules, we must use AC instead of A here: the module A ˝ V for non-unital A usually has no special properties. The free right unital A-module is V ˝ A and the free unital A; B-bimodule is A ˝ V ˝ B.

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307

A.2.8 Functoriality Now let f W A ! B be a unital algebra homomorphism between two unital algebras in C. Composing the module structure with f ˝ id, we get a functor f  W ModC .B/ ! ModC .A/: It has a left adjoint functor, namely, fŠ W ModC .A/ ! ModC .B/;

M 7! B ˝A M I

here we view B as a B; A-bimodule using f on one side. The proof is similar to the proof of Lemma A.41, which is in fact the special case of the unit homomorphism 1 ! A. The special case of the identity morphism A ! A is contained in Proposition A.35. If C is closed, then we also have a right adjoint functor f W ModC .A/ ! ModC .B/;

M 7! HomA .B; M /I

here we view B as an A; B-bimodule. The adjointness follows from (A.38) and   HomB V; HomA .B; M / Š HomA .B ˝B V; M / Š HomA .V; M /:

A.2.9 Traces and commutator quotients Let M be an A-bimodule and let  W A ˝ M ! M and % W M ˝ A ! M be its left and right module structure, respectively. Definition A.42. The commutator quotient M=Œ ;  of M is the cokernel of the map   % ı ˆA;M W A ˝ M ! M;

a ˝ x 7! a  x  x  a;

where ˆA;M W A ˝ M ! M ˝ A is the flip isomorphism. A map  W M ! V for V 22 C is called a trace if  ı  D  ı % ı ˆA;M , that is  .a  x/ D  .x  a/ for a 2 A, x 2 M. The canonical map M ! M=Œ ;  is the universal trace on M , that is, it is a trace and any trace factors uniquely through it. Definition A.43. Suppose that C is closed. Then  and % ı ˆA;M transpose to maps  W M ! Hom.A; M / and % W M ! Hom.A; M /. We define the centre of M by   Z.M / WD ker   % W M ! Hom.A; M / : Example A.44. If C D Cborn or similar, then we get Z.M / D fx 2 M j ax D xa for all a 2 Ag:

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The commutator quotient, the centre, and traces on the algebra A itself are defined by viewing it as a bimodule over itself in the standard way. These constructions are closely related to ˝A and HomA . If V and W are a right and a left A-module, respectively, then V ˝ W becomes an A-bimodule, and we have V ˝A W Š .V ˝ W /=Œ ; : If both V and W are left A-modules, then Hom.V; W / becomes an A-bimodule, and   HomA .V; W / Š Z Hom.V; W / : Conversely, we can express commutator quotients as balanced tensor products, viewing A-bimodules as left modules over A ˝ Aop : Lemma A.45. Let M be an A-bimodule viewed as a left A ˝ Aop -module and view A as a right or left A ˝ Aop -module. Then Z.M / Š HomA˝Aop .A; M /:

M=Œ ;  Š A ˝A˝Aop M; u˝idM



Proof. The maps M ! A ˝ M !  M descend to maps M=Œ ;  ! A ˝A˝Aop M ! M=Œ ; . It is not hard to check that these maps are inverse to each other. This yields the first isomorphism. By the Yoneda Lemma and (A.38), the second assertion is equivalent to the existence of a natural isomorphism HomA˝Aop .A ˝ V; M / Š  Hom V; Z.M / for all V 22 C. If f W A ˝ V ! M is a bimodule map, its composite with u ˝ idV W V ! A ˝ V factors through a map f 0 W V ! Z.M /. Conversely, if we have a map f 0 W V ! Z.M /, then  ı .idA ˝ f 0 / W A ˝ V ! M is a bimodule homomorphism. These two constructions are inverse to one another.

A.2.10 Adaptation to the non-unital case We can also apply the above theory to Mod.A/ Š ModC .AC / for a non-unital algebra A. But we must use AC instead of A. This makes no difference in the definition of ˝A and HomA because the action of the unit object on unital modules is prescribed, anyway. But the free and cofree non-unital left modules are AC ˝ V and Hom.AC ; V /, respectively. The modules A ˝ V and Hom.A; V / do not satisfy any universal property and may be rather badly behaved. Furthermore, we must use AC to construct the covariant functoriality fŠ and f , and Lemma A.45 becomes false if we use A instead of AC instead.

A.3 Non-commutative differential forms In this section, we study the bimodules n .A/ of non-commutative differential n-forms x n .A/ over a unital algebra, and the over a non-unital algebra A, the reduced version  differential graded algebras they generate. We begin by relating 1 .A/ to derivations and split square-zero extensions.

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A.3.1 Non-commutative 1-forms Definition A.46. Let A 22 AlgC .C/. We view A and A ˝ A as unital A-bimodules in the standard way, so that the multiplication map mA W A ˝ A ! A is an A-bimodule homomorphism. Thus ker.mA /  A ˝ A is an A-bimodule as well. We denote it by x 1 .A/ and call it the bimodule of reduced non-commutative differential 1-forms on A.  x 1 .AC / is called the bimodule of non-commutative If A 22 Alg.C/, then 1 .A/ WD  differential 1-forms on A. In our applications we mainly use 1 .A/ even if A is unital. We limit our discussion x 1 .A/ for unital A because it contains 1 .A/ as a special case. here to  The maps u ˝ idA W A Š 1 ˝ A ! A ˝ A;

idA ˝ u W A Š A ˝ 1 ! A ˝ A

are sections for mA . They are a right and a left module homomorphism, respectively. Let AN WD coker.u W 1 ! A/, then we have x 1 .A/ Š coker.idA ˝ u/ Š A ˝ AN  x 1 .A/ Š AN ˝ A as a right A-module. Thus 1 .A/ as a left A-module and, similarly,  is free as a left A-module and free as a right A-module – but usually not as a bimodule. N We denote elements of 1 .A/ by x0 dx1 or .dx1 /x0 with x0 2 A, x1 2 A. 1 1 x .A/ ˝ A !  x .A/ translates to The right A-module structure  N ˝ A ! A ˝ AN idA ˝ mA  mA ˝ idA W .A ˝ A/ or .x1 ˝ xS2 /  x3 WD x1 ˝ xS2 x3  x1 xS2 ˝ x3 in terms of elements. Notice that this is well-defined. The difference idA ˝ u  u ˝ idA W A ! A ˝ A of our two sections induces a map x 1 .A/: dW A! Identifying 1 .A/ Š A ˝ AN as above, d becomes u˝p

N A Š 1 ˝ A ! A ˝ A;

x 7! 1 ˝ x;

where u W 1 ! A is the unit map and p W A ! AN is the canonical projection. If we identify 1 .A/ Š AN ˝ A instead, then d becomes p ˝ u. Our next goal is to describe bimodule homomorphisms out of 1 .A/ in terms of sections for square-zero extensions. The crossed product algebra of a bimodule Definition A.47. An algebra extension I  E  A is called a square-zero extension if the multiplication map on I vanishes.

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For any algebra extension, the multiplication on E restricts to an E-bimodule structure on I . For a square-zero extension, this descends to an A-bimodule structure on I because ˝ is compatible with quotients. Each bimodule can occur in such an extension: Definition A.48. Let A 22 Alg.C/ and let M 22 Mod.A; A/. Then we let A Ë M be the algebra whose underlying object in C is A ˚ M and whose multiplication .A ˚ M / ˝ .A ˚ M / Š .A ˝ A/ ˚ .A ˝ M / ˚ .M ˝ A/ ˚ .M ˝ M / ! A ˚ M is given by 0 on M ˝ M and the multiplication maps A ˝ A ! A, A ˝ M ! M , M ˝ A ! M on the other summands. We call A Ë M the crossed product algebra of the A-bimodule M . Equip M with the zero multiplication. The coordinate embeddings A; M ! AËM and the coordinate projection A Ë M ! A are algebra homomorphisms. Hence we get a split square-zero extension M  A Ë M  A:

(A.49)

Conversely, any split square-zero extension I  E  A is isomorphic to one of the form (A.49), where M D I with the canonical A-bimodule structure. The section homomorphism A ! E gives rise to a decomposition E Š A ˚ I in C. It is easy to check that this isomorphism is an algebra isomorphism E Š A Ë M . Derivations. For a complete understanding of split square-zero extensions, we must also describe the morphisms between two extensions of the form (A.49). More precisely, we are interested in morphisms of the special form M1 f

 M2

/ A Ë M1

/A

(A.50)

g



/ A Ë M2

/ A.

If M1 D M2 , we may further ask that f D idM . Since the A-bimodule structure on an ideal is canonical, the map f must be a bimodule homomorphism. In the decomposition A Ë M‹ Š A ˚ M‹ , we must have   idA 0 with d W A ! M2 gD d f in order to make (A.50) commute. It is straightforward to check that such a matrix yields an algebra homomorphism if and only if d is a derivation in the following sense: Definition A.51. Let A 22 Alg.C/ and let M be an A-bimodule. Write  W A ˝ M ! M and % W M ˝ A ! M for the left and right module structures and m W A ˝ A ! A for the multiplication in A. A morphism f 2 Hom.A; M / is called a derivation if f ı m D  ı .idA ˝ f / C % ı .f ˝ idA /:

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311

In terms of elements, this becomes the familiar Leibniz rule f .x  y/ D x  f .y/ C f .x/  y: Let Deriv.A; M / be the space of derivations A ! M . Example A.52. Any map x W 1 ! M generates an inner derivation dx WD % ı .u ˝ idA /   ı .idA ˝ u/ W A ! M;

a 7! x  a  a  x D Œx; a:

TheoremA.53. Any split square-zero extension I  E  A in C is isomorphic to the crossed product extension I  A Ë I  A. The isomorphism is determined uniquely by the choice of a section s W A ! E for the extension; another section sQ W A ! E is a homomorphism section if and only if sQ  s W A ! I is a derivation. The space of morphisms .over A/ between two split square-zero extensions I1  A Ë I1  A and I2  A Ë I2  A is HomA .I1 ; I2 / Ë Deriv.A; I2 /. Proof. We leave it to the reader to check that   idA 0 W A Ë M1 ! A Ë M2 d f is an algebra homomorphism if and only if f is a bimodule homomorphism and d is a derivation. Now assume that A is unital and M is a unital A-bimodule. Then A Ë M is a unital algebra as well, and the coordinate embedding s0 W A ! A Ë M is a unital algebra homomorphism. If s W A ! A Ë M is another section, then our classification yields s D ˛ıs0 for an algebra isomorphism ˛ W AËM ! AËM ; therefore, s is unital as well; equivalently, any derivation satisfies d ı u D 0, where u W 1 ! A is the unit map. This also follows from the computation d.1/ D d.1  1/ D 1  d.1/ C d.1/  1 D d.1/ C d.1/. Warning A.54. This computation only applies if the bimodule M is unital. If we allow non-unital bimodules over unital algebras then we may have d ı u ¤ 0. x 1 .A/ and We mainly use derivations and square-zero extensions as a tool to study  1 .A/. The following proposition explains the link: x 1 .A/ is a derivaProposition A.55. Let A be a unital algebra. The map d W A !  tion, and it is the universal derivation into a unital bimodule in the following sense: any derivation f W A ! M into a unital bimodule M admits a unique factorisation x 1 .A/ ! M . More succinctly, f D fQ ı d with a bimodule homomorphism fQ W  1 . .A/; d / represents the derivation functor, that is, composition with d induces a natural isomorphism x 1 .A/; M / Š Deriv.A; M /: HomA;A . If A 22 Alg.C/, then d W A ! 1 .A/ is the universal derivation in a similar sense.

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Proof. It is straightforward to verify that d is a derivation as a map to A ˝ A (it is the inner derivation associated to u ˝ u W 1 Š 1 ˝ 1 ! A ˝ A); hence it is a derivation to x 1 .A/ as well; in terms of elements, the computation looks as follows:  d.a  b/ D Œa  b; 1 ˝ 1 D a  Œb; 1 ˝ 1 C Œa; 1 ˝ 1  b D a  d.b/ C d.a/  b: Conversely, let f W A ! M be a derivation into a unital A-bimodule M . Then f ı u D 0, so that f descends to a map AN ! M . This induces a left module hox 1 .A/ Š A ˝ AN ! M . The derivation property momorphism on the free module  of f shows that this is even a bimodule homomorphism.

A.3.2 Higher-dimensional forms Let A be a unital algebra in C. If M; N are two A-bimodules, then M ˝A N is an A-bimodule as well. Thus ˝A turns ModC .A; A/ into a monoidal category; but this category is not symmetric. Definition A.56. The bimodule of reduced non-commutative differential n-forms over x 0 .A/ WD A if n D 0 and a unital algebra A in C is defined by  x 1 .A/˝A n x n .A/ WD  

for n 2 N1 .

x n .AC / Š 1 .A/˝A n If A is a non-unital algebra, we let 0 .A/ WD A and n .A/ WD  for n 2 N1 . These are the bimodules of non-commutative differential n-forms on A. x 1 .A/ Š A ˝ AN as a left A-module. By induction, this implies Recall that  x n .A/ Š A ˝ AN˝n 

as a left A-module.

x n .A/ is free both as a left x n .A/ Š AN˝n ˝ A as a right A-module, so that  Similarly,  and a right module. Similarly, n .A/ Š AC ˝ A˝n ;

n .A/ Š A˝n ˝ AC :

We also define functors from the category ModC .A/ or Mod.A/ to itself by x n .A/ ˝A M; x An .M / WD  

An .M / WD n .A/ ˝A M;

x n D . x 1 /n and n D .1 /n , respectively. Since ˝A is associative, we have  A A A A 1 n that is, n-fold iteration of A gives A . Disregarding the module structure, we have x n .M / Š AN˝n ˝ M and n .M / Š A˝n ˝ M in C.  A A

A.3.3 The algebra of non-commutative differential forms x n .A/ as a graded object of ModC .A; A/. The We view the sequence of bimodules  canonical maps x n .A/ Š  x n .A/   x m .A/ ˝A  x mCn .A/ x m .A/ ˝  

A.3 Non-commutative differential forms

313

 n  x .A/ into an associative graded algedefine an associative multiplication, turning  bra. Now we assume that C has countable direct sums and that these are compatible with ˝. Then the above maps combine to an associative multiplication on x WD .A/

1 M

x n .A/: 

nD0

x 0 .A/ !  x 1 .A/ extends uniquely to a map Lemma A.57. The map d W A D  x x d W .A/ ! .A/ with the following properties: • d is a graded derivation, that is, it satisfies the graded Leibniz rule x n .A/ ˝  x k .A/; d ı m D m ı .d ˝ id/ C .1/n m ı .id ˝ d / on  this is the translation of d.x  y/ D .dx/  y C .1/n x  .dy/. • d ı d D 0. ˝n x x ! .A/ be the n-fold multiplication map. The map Proof. Let mn W .A/

x n .A/ mnC1 ı .id ˝ d ˝n / W A ˝ AN˝n !  x n .A/ once the is an isomorphism. Hence the Leibniz rule dictates the values of d on  values on A and d.A/ are known. Thus the only possible extension of d satisfying the requirements is given by u˝p˝id

˝n

AN x nC1 .A/ x n A Š 1 ˝ A ˝ AN˝n  ! A ˝ AN ˝ AN˝n Š  

or d.x0 dx1 : : : dxn / WD dx0 dx1 : : : dxn . It is clear that d 2 D 0. Since A and dA x x x generate .A/, it suffices to check the graded Leibniz rule on A˝.A/ and dA˝.A/. 1 x This reduces to the Leibniz rule for d W A !  .A/ (Proposition A.55). x Theorem A.58. Unital algebra homomorphisms .A/ ! B for a unital algebra B correspond naturally to pairs .f; ı/, where f W A ! B is a unital algebra homomorphism and ı W A ! B is a derivation relative to f , that is, with respect to the A-bimodule structure on B defined by f ; that is, d ı mA D mB ı .d ˝ f / C mB ı .f ˝ d /. Similarly, algebra homomorphisms .A/ ! B for an algebra B correspond to pairs .f; ı/ where f is an algebra homomorphism and ı is a derivation relative to f . Proof. This is a special case of a more general result about tensor algebras of bimodules x (see [25, §2]). First let g W .A/ ! B be an algebra homomorphism. Then it restricts to g g d x 0 .A/  ! B and to a map ı W A  ! B. ! 1 .A/  an algebra homomorphism f W A D  It is easy to see that ı is a derivation relative to f .

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Conversely, suppose such a pair .f; ı/ is given; equip B with the A-bimodule structure induced by f . Proposition A.55 shows that ı induces an A-bimodule homomorphism ıQ W 1 .A/ ! B. The multiplication map in B descends to an A-bimodule homomorphism mB W B ˝A B ! B because f is an algebra homomorphism. Hence we get maps Q ˝A n

mn

ı B x n .A/ Š  x 1 .A/˝A n   ! B ˝A n ! B

x ! B. for all n 2 N1 . These combine to an algebra homomorphism g W .A/ The algebra .A/ carries the richer structure of a differential graded algebra. Theorem A.58 implies that any algebra homomorphism from A to the degree-0-part of a differential graded algebra extends uniquely to a homomorphism of differential graded algebras on .A/. Thus .A/ is the universal differential graded algebra under A.

A.4 Fedosov type products We assume now that C has countable direct sums that are compatible with ˝. This allows us to construct free algebras on objects in C and free products of algebras. These can be defined easily by universal properties: Definition A.59. The free algebra or tensor algebra TV 22 Alg.C/ and the unital tensor algebra TC V 22 AlgC .C/ on V 22 C are defined by the existence of natural isomorphisms Alg.TV; B/ Š Hom.V; B/ for all B 22 Alg.C/ or AlgC .TC V; B/ Š Hom.V; B/ for all B 22 AlgC .C/, respectively. Definition A.60. The free product A  B of two algebras A and B is their coproduct in the category of algebras, that is, we have Alg.A  B; D/ Š Alg.A; D/  Alg.B; D/ for all D 22 Alg.C/. The unital free product A C B is defined similarly. It is rather easy to describe tensor algebras and free products using tensor powers of V or alternating tensor products of A and B, respectively (see [17]). We describe these algebras in a different way, using deformations of algebras of non-commutative differential forms. These constructions follow Joachim Cuntz and Daniel Quillen [25, §1]. The point is that we need variants of these constructions that have slightly different universal properties. The pro-tensor algebra and the analytic tensor algebra are supposed to be universal for “almost multiplicative” linear maps. There is a variant of AA that is universal for pairs of “close” algebra homomorphisms. And there is a variant of A  B that is universal for “almost commuting” pairs of algebra homomorphisms. We define almost multiplicative, close, and almost commuting maps in Chapters 4 and 5

A.4 Fedosov type products

315

using classes of generalised nilpotent algebras. Here we recall some prerequisites for this from [25, §1]. We write down most of the following constructions in terms of elements for simplicity. These computations extend to algebras in symmetric monoidal categories as explained in §A.2.1.

A.4.1 The Fedosov product on a differential graded algebra Definition A.61. The Fedosov product ˇ on a differential Z=2-graded algebra .; d / is defined by ! ˇ  WD !    .1/k d.!/  d./ for all ! 2 k ;  2 l : Lemma A.62. The Fedosov product is associative. Proof. We check associativity on k ˝ l ˝ m . The graded Leibniz rule and d 2 D 0 yield .! ˇ / ˇ  D !  .1/k d.!/d./  .1/kCl d.!/d./ D !  .1/k d.!/d./  .1/kCl d.!/d./  .1/l !d./d./; ! ˇ . ˇ / D !  .1/l !d./d./  .1/k d.!/d./ D !  .1/l !d./d./  .1/k d.!/d./  .1/kCl d.!/d./I both products are equal. The algebra .; ˇ/ is still Z=2-graded, so that the even part is a subalgebra for the Fedosov product. We are interested in the Fedosov product on .A/ WD L

1 M

n .A/:

nD0

We have .A/ ˝ .A/ Š m;n2N m .A/ ˝ n .A/ because direct sums in C are assumed to be compatible with ˝. Hence the multiplication of differential forms turns .A/ into a differential graded algebra. We first consider .even .A/; ˇ/, then ..A/; ˇ/.

A.4.2 The tensor algebra The restriction of ˇ to even .A/ simplifies to ! ˇ  D !    d.!/  d./: We let A W A Š 0 .A/ ! even .A/ be the obvious linear map.

316

Appendix. Algebraic preliminaries

Theorem A.63. Let f W A ! B be a linear map between two algebras. Then there is a unique algebra homomorphism hf i W .even A; ˇ/ ! B with hf i ı A D f , namely, hf i.x0 dx1 : : : dx2n / WD fC .x0 /  !f .x1 ; x2 /    !f .x2n1 ; x2n / nC1 or hf i D mB ı .fC ˝ !f˝n / on 2n .A/. Thus ..A/even ; ˇ/ Š TA.

Proof. The curvature of A is ! .x; y/ D A .xy/  A .x/ ˇ A .y/ D dx dy. Hence hf i is the only possible extension of A to an algebra homomorphism. It remains to check that hf i is multiplicative. Since A generates .even A; ˇ/, it suffices to check multiplicativity on A ˝ even .A/. In terms of elements, this follows from the computation f .x/  fC .x0 /  !f .x1 ; x2 /    !f .x2n1 ; x2n /   D f .x  x0 /  !f .x; x0 /  !f .x1 ; x2 /    !f .x2n1 ; x2n / D f .x  x0 dx1 : : : dx2n /  f .dx dx0 dx1 : : : dx2n / D f .x ˇ x0 dx1 : : : dx2n /: An object V 22 C becomes an algebra for the zero multiplication. Up to signs, ..V /even ; ˇ/ amounts to the usual construction of TV in this case. Notation A.64. Let A WD hidA i W TA ! A; this is the projection even .A/ ! 0 .A/ that annihilates 2n .A/ for n  1. We let JA WD ker.A / D

1 M

2n .A/  TA:

nD1

Since A is a section for A , we get a semi-split algebra extension JA  TA  A called the tensor algebra extension of A. These extensions play a crucial role for the bivariant K-theory of Joachim Cuntz ([17]). Example A.65. Let C D Cborn, so that 1 D C. Then TC C Š CŒt , and TC is y isomorphic to the ideal t CŒt  in CŒt  because C ˝n Š C for all n  1. The map C W TC ! C is given by evaluation at 1. Thus JC Š .1  t /t CŒt . Once dim V  2, the tensor algebra TV becomes non-commutative. x even .A/; ˇ/ has the following universal property: If A is a unital algebra, then . even x .A/; ˇ/ ! B correspond to unital maps A ! B, unital algebra homomorphisms . that is, maps f W A ! B with f ı uA D uB .

A.4.3 The free double Next we consider the Z=2-graded algebra ..A/; ˇ/. The computation .x ˙ dx/ ˇ .y ˙ dy/ D xy  dx dy ˙ .dx/y ˙ x.dy/ C dx dy D xy ˙ d.xy/ shows that the maps ˙ W x 7! x˙dx are algebra homomorphisms ˙ W A ! ..A/; ˇ/.

A.4 Fedosov type products

317

Theorem A.66. Let B be a Z=2-graded algebra and let f W A ! B be an algebra homomorphism .not compatible with the grading/. There is a unique algebra homomorphism hf i W ..A/; ˇ/ ! B with hf i ı C D f that preserves the Z=2-grading. If we decompose f into its even and odd parts f0 and f1 , then we get hf i.x0 dx1 : : : dxn / D f0 .x0 /  f1 .x1 /    f1 .xn /: Proof. The even and odd part of C .x/ are x and dx, respectively. Hence the only possible way to extend f to a grading preserving algebra homomorphism is hf i. It is clear that hf i preserves the gradings. Since elements of the form x and dx generate .A/, it suffices to check multiplicativity on A ˝ .A/ and dA ˝ .A/. This follows easily from f0 .xy/ D f0 .x/f0 .y/ C f1 .x/f1 .y/;

f1 .xy/ D f1 .x/f0 .y/ C f0 .x/f1 .y/:

Thus ..A/; ˇ/ is the Z=2-graded algebra generated by A. Any Z=2-graded algebra carries a canonical grading automorphism, which leaves the even part invariant and changes sign on the odd part. If C is ZŒ1=2-linear, then we can recover the grading from the grading automorphism. In this case, we can reformulate the universal property of ..A/; ˇ/ as follows: Theorem A.67. Let C be ZŒ1=2-linear. Then any pair of algebra homomorphisms fC ; f W A ! B induces a unique algebra homomorphism fC  f W ..A/; ˇ/ ! B with .fC  f / ı C D fC and .fC  f / ı  D f ; it maps x0 dx1 : : : dxn 7!

n fC .x0 / C f .x0 / Y fC .xj /  f .xj /  : 2 2 j D1

Thus .A; ˇ/ Š A  A is the free product A  A of two copies of A. If C fails to be ZŒ1=2-linear, then we can still describe the free product A  A as a deformation of .A/; this time, we replace the Fedosov product by !   WD !    odd.m/ !  d./ for all ! 2 m .A/;  2 n .A/; where odd.m/ is 1 for odd and 0 for even m. We omit the proof that this product is associative. It is easy to check that x 7! x and x 7! xCdx are algebra homomorphisms and x0 dx1 : : : dxn D x0  dx1      dxn : A pair of algebra homomorphisms fC ; f W A ! B induces an algebra homomorphism fC  f W ..A/; / ! B;

x0 dx1 : : : dxn 7! fC .x0 / 

n Y 

 f .xj /  fC .xj / :

j D1

x x If A is a unital algebra, then ..A/; ˇ/ and ..A/; / have similar universal properties: any unital algebra homomorphism into a Z=2-graded algebra extends uniquely

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x to a unital grading preserving algebra homomorphism on ..A/; ˇ/; pairs of unital algebra homomorphisms A  B correspond to unital algebra homomorphisms x x ..A/; / ! B and to unital algebra homomorphisms ..A/; ˇ/ ! B if C is ZŒ1=2linear. Notation A.68. Let QA WD A  A and let qA be the kernel of the natural algebra homomorphism A W A  A ! A with A ı i1 D A ı i2 D idA . Since the two canonical algebra homomorphisms i1 ; i2 W A ! QA are sections for A , we get an algebra extension qA  QA  A that splits in two different ways. When we describe QA as .A; ˇ/ or .A; /, then qA corresponds to 1 .A/ in both cases.

A.4.4 Free products versus tensor products Now first consider the unital free product of two different unital algebras A and B; let uA W 1 ! A and uB W 1 ! B be the unit maps. We define x .A; B/ WD

1 M

x n .B/I x n .A/ ˝  

nD0

x x this is a subalgebra of .A/ ˝ .B/, but not closed under the differential. We define a Fedosov-like product as in [25, §1] by . 0 ˝ 0 / ~ . 1 ˝ 1 / WD 0 1 ˝ 0 1  .1/j0 j 0 d. 1 / ˝ .d0 /1 : We omit the computation that this product is associative. This works for any pair of differential graded algebras, by the way. x Since d ı1 D 0 in .A/, the embeddings idA ˝uB W A ! A˝B and uA ˝idB W B ! A ˝ B become unital algebra homomorphisms A

B

x A ! ..A; B/; ~/  B: Theorem A.69. Let fA W A ! D and fB W B ! D be two unital algebra homomorx phisms. There is a unique unital algebra homomorphism fA fB W ..A; B/; ~/ ! D with .fA  fB / ı A D fA and .fA  fB / ı B D fB . It maps a0 da1 : : : dan ˝ b0 db1 : : : dbn 7! fA .a0 /fB .b0 /ŒfA .a1 /; fB .b1 /    ŒfA .an /; fB .bn /: x Thus ..A; B/; ~/ is canonically isomorphic to the coproduct A C B of A and B in the category of unital algebras.

A.4 Fedosov type products

319

Proof. Any algebra homomorphism with f  A D fA and f  B D fB must map da ˝ db to ŒfA .A/; fB .b/ because ŒA .a/; B .b/ D .a ˝ 1/ ~ .1 ˝ b/  .1 ˝ b/ ~ .a ˝ 1/ D da ˝ db: This leads to the formula in the theorem because a0 da1 : : : dan ˝ b0 db1 : : : dbn D .a0 ˝ a1 / ~ .da1 ˝ db1 / ~    ~ .dan ˝ dbn /: It remains to check that the map so defined is multiplicative. Since A ˝ 1 and 1 ˝ B x x generate ..A; B/; ~/, it suffices to check multiplicativity on A ˝ .A; B/ and B ˝ x .A; B/. The first is trivial, the second is a simple computation based on fB .b/  fA .a0 /  fB .b0 / D fA .a0 /  fB .b  b0 /  ŒfA .a0 /; fB .b/fB .b0 / D fA .a0 /  fB .b  b0 /  ŒfA .a0 /; fB .b/fB .b0 / C fB .b/ŒfA .a0 /; fB .b0 /: x x 0 .A/ ˝  x 0 .B/ D A ˝ B is an algebra The canonical projection .A; B/ !  x homomorphism. Thus weL get a semi-split algebra extension N .A; B/  .A; B/  1 n n x x A ˝ B with .A; N B/ WD nD1  .A/ ˝  .B/ with multiplication ~. Now we consider the free products in the category of non-unital algebras. We let x C ; BC /. Explicitly, .A; B/ for A; B 22 Alg.C/ be the augmentation ideal of .A 1    M  .A; B/ Š A  B ˚ n .A/ ˝ n .B/; ~ D A  B ˚ N .AC ; BC / I nD1

we have defined A  B in §A.2.4. Putting .A; B/ WD N .AC ; BC /, we get a semi-split algebra extension .A; B/  .A; B/  A ˚ B:

A.4.5 Modules and derivations for free algebras and free products We describe the categories of modules and the bimodule of non-commutative 1-forms over free algebras and free products of algebras. We only use the universal properties, the concrete realisation of the algebras using the Fedosov product is irrelevant here. Lemma A.70. The TV -module structures m W TV ˝ M ! M on M 22 C correspond bijectively to maps V ˝ M ! M via m 7! m ı .V ˝ idM /. Proof. Instead of proving this in general (which is not hard), we explain the elegant proof in the case where C is closed. Then a TV -module structure on M is equivalent to an algebra homomorphism TV ! Hom.M; M /, which is equivalent to a map V ! Hom.M; M /, which is, in turn, equivalent to a map V ˝ M ! M .

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Lemma A.71. The A  B-module structures m W .A  B/ ˝ M ! M on M 22 C correspond bijectively to pairs .mA ; mB / consisting of an A- and a B-module structure without any compatibility condition. Proof. The proof is similar to that of Lemma A.70. Proposition A.72. The map d ı V W V ! 1 .TV / induces bimodule isomorphisms x 1 .TC V / TC V ˝ V ˝ TC V Š 1 .TV / Š  x 1 .TC V / is the for any V 22 C. Thus 1 .TV / is the free TV -bimodule on V and  free unital TC V -bimodule on V . Proof. By the defining property of free modules, it suffices to check that composition with d ı V yields a bijection HomA;A .1 .TV /; M / Š Hom.V; M /. First, Proposition A.55 identifies HomA;A .1 .TV /; M / Š Deriv.TV; M /. Theorem A.53 yields a bijection to the set of algebra homomorphisms TV ! TV Ë M that lift the identity map on TV . By the universal property of TV , we can identify these with liftings V ! TV Ë M of V W V ! TV . Since only the second component of a lifting V ! M is relevant and can be arbitrary, we get HomA;A .1 .TV /; M / Š Hom.V; M / as desired. Proposition A.73. Let A and B be algebras in C. Then there is a natural isomorphism     1 .A  B/ Š .iA /Š 1 .A/ ˚ .iB /Š 1 .B/ : Here we use the non-unital bimodule versions of .iA /Š and .iB /Š , .iA /Š .M / WD .AB/C ˝A M ˝A .AB/C ;

.iB /Š .M / WD .AB/C ˝B M ˝B .AB/C :

Proof. We check that both sides represent the same functor on Mod.A  B; A  B/. We get HomAB;AB .1 .A  B/; M / Š Deriv.A  B; M / by Proposition A.55 and   HomAB;AB iAŠ .1 A/ ˚ iBŠ .1 B/; M     Š HomAB;AB iAŠ .1 A/; M ˚ HomAB;AB iBŠ .1 B/; M Š HomA .1 .A/; iA M / ˚ HomB .1 .B/; iB M / Š Deriv.A; iA M / ˚ Deriv.B; iB M / because iAŠ and iBŠ are left adjoint to iA and iB , respectively (see §A.2.8); here iA M and iB M denote M viewed as an A- or B-bimodule via iA or iB , respectively. Derivations A  B ! M correspond to algebra homomorphisms A  B ! A  B Ë M that lift the identity map on A  B. By the defining property of free products, such algebra homomorphisms correspond to pairs of algebra homomorphisms A ! A  B Ë M and B ! A  B Ë M that lift iA W A ! A  B and iB W B ! A  B, respectively. These, in turn correspond to pairs of derivations A ! iA M and B ! iB M , as desired. Proposition A.73 applies to both Q.A/ Š A  A and .A; B/ Š A  B.

A.5 Homological algebra for modules

321

A.5 Homological algebra for modules A.5.1 The basic setup Let A be a unital algebra in C. When we do homological algebra in the category ModC .A/ of unital A-modules, then we do not care about the internal homological algebra of C. That is, we work relative to the forgetful functor ModC .A/ ! C. This ensures that we get a trivial theory for A D 1. The right way to ignore homological contributions from C is to restrict the class of extensions. An additive category together with a class of admissible extensions satisfying some assumptions is called an exact category. We have already met this notion in §2.3.6 in connection with local homotopy equivalences of chain complexes. The relevant class of extensions for our purposes are the semi-split ones, that is, those extensions in ModC .A/ that split in the underlying category C. Theorem A.74. The additive category ModC .A/ with semi-split extensions as admissible extensions is an exact category. Moreover, it is bicomplete if C is. A simplified form of the axioms of an exact category can be found in [60]. It is easy to verify them in our situation. Hence the general machinery of homological algebra, including derived categories and functors, applies to ModC .A/. We will only use the following basic notions explicitly: Definition A.75. • A chain complex is semi-split exact if its image in C is contractible. • A resolution of M 22 ModC .A/ is a chain complex .Cn ; dn /n0 in ModC .A/ with an augmentation morphism C0 ! M such that dn

dn1

   ! Cn ! Cn1 !    ! C0 ! M ! 0 !    is a semi-split exact chain complex. • A module P is projective if the functor HomA .P; / is exact on split-exact extensions or, equivalently, on semi-split exact chain complexes. • A projective resolution of a module is a resolution by a chain complex of projective objects. We will not use the dual notions of injective object and (co)resolution. For topological algebras and, in particular, for Banach algebras, this reproduces familiar notions that have been used by many authors for a long time (see [40]). We will soon see that any module has a canonical projective resolution, the bar resolution. The same argument as for Abelian categories shows that any two projective

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resolutions of the same module are chain homotopy equivalent and that the passage to projective resolutions is functorial up to chain homotopy, that is, it is a functor ModC .A/ ! HoKom.ModC .A//: Finding convenient resolutions is the main method to compute Hochschild homology. In addition, it is a crucial step in our proof of the Excision Theorems for periodic, analytic, and local cyclic homology. It follows immediately from the defining universal property that free modules are projective because HomA .A ˝ V; / Š Hom.V; / is exact on split-exact sequences. Most projective resolutions that we shall use are actually free resolutions.

A.5.2 The bar resolution Let A be a unital algebra in C and let M be a unital A-module. The bar resolution provides a natural free (and hence projective) resolution of M . Its basic ingredient is the unital A-bimodule extension m x 1 .A/  A ˝ A  A; 

(A.76)

which splits as an extension of left or right A-modules via idA ˝ u or u ˝ idA , respectively; but it does not split as a bimodule extension. x 1 .A/ ˝A M  .A ˝ A/ ˝A M  A ˝A M is again a semi-split The diagram  extension in ModC .A/ because (A.76) splits as a sequence of right modules. Recall that x 1 .M / WD  x 1 .A/ ˝A M for M 22 ModC .A/ and that A ˝A M Š M . we abbreviate  A Hence we get a semi-split extension m x A1 .M /  A ˝ M   M

(A.77)

in ModC .A/, where m is the multiplication map and A˝M carries the free left module x 1 .M / Š AN ˝ M structure. Since 1 .A/ Š AN ˝ A as a right A-module, we have  A 1 x in C. The left module structure on A .M / translates to N N / ! A˝M; m˝idM idA ˝m W A˝.A˝M

x0 ˝xS1 ˝x2 7! x0 xS1 ˝x2 x0 ˝xS1 x2 I (A.78) more precisely, the map mA ˝ idM  idA ˝ mAM W A ˝ A ! M ! A ˝ M descends to a map as claimed because ˝ is compatible with cokernels. x 1 .A/ by 1 .A/ and get a semi-split extension In the non-unital case, we replace  m

A1 .M /  AC ˝ M  M I

(A.79)

we have A1 .M / Š A ˝ M with a module structure as in (A.78). Since A ˝ M (or AC ˝ M in the non-unital case) is free, (A.77) is the beginning of x n .M / for n 2 N, we get semi-split a free resolution of M . When we apply (A.77) to  A extensions x nC1 .M /  A ˝  x An .M /   x An .M /  (A.80) A

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x n .M / by A ˝ AN˝n ˝ M . The in ModC .A/ for all n 2 N; we may replace A ˝  A Yoneda product (concatenation) of these semi-split extensions is a semi-split exact chain complex  " x 3 .M /  A

/ A ˝ AN˝2 ˝ M 6 (

b20

/ A ˝ AN ˝ M 7

x 2 .M /  A

'

b10

/ A˝M 9

b00

/M

/0

x 1 .M /  A

or, equivalently, a free resolution of M in ModC .A/; it is called the reduced bar resolution of M and denoted by Bar  .A; M /. Similarly, we have the (unreduced) bar resolution Bar  .A; M / WD Bar  .AC ; M / in Mod.A/ for M 22 Mod.A/ and A 22 Alg.C/. Explicitly, we have Bar n .A; M / Š A ˝ AN˝n ˝ M;

Bar n .A; M / Š AC ˝ A˝n ˝ M;

for n 2 N with boundary maps bn0 D

n X

.1/j mj;j C1 W AC ˝ A˝n ˝ M ! AC ˝ A˝n1 ˝ M;

(A.81)

j D0

where mj;j C1 multiplies the j th and j C 1th tensor factors and acts identically on the others. In terms of elements, this reads bn0 .x0 ˝    ˝ xnC1 / WD

n X

.1/j x0 ˝    ˝ xj 1 ˝ .xj  xj C1 / ˝ xj C2 ˝    ˝ xnC1 :

j D0

In the reduced case, essentially the same formula works. But the individual summands mj;j C1 do not descend to the quotient A ˝ AN˝n ˝ M ; only their sum does so because of cancellation between neighbouring terms. We are particularly interested in the chain complexes Bar  .A/ WD Bar  .A; A/ for unital A and Bar  .A/ WD Bar  .A; AC / for general A, which are called the (reduced) bar resolution of A. Notice that they are free bimodule resolutions of A and AC , respectively.

A.5.3 Cohomological dimension and connections We will use the following notions of dimension for modules and algebras: Definition A.82. A module M 22 ModC .A/ has .projective/ dimension k if it has a projective resolution of length k, that is, of the form 0 ! Pk ! Pk1 !    ! P0 ! M , and none of shorter length. We denote the dimension by dim M .

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Thus dim M D 0 if and only if M itself is projective. We write dim M D 1 if M has no projective resolution of finite length. Definition A.83. The bidimension dim A of a unital algebra A is the projective dimension of A as a unital A-bimodule. For non-unital A, we define dim A WD dim AC . Projective modules and connections. We relate the dimension to the existence of maps with certain properties called connections, following [25, §8]. These differ from connections in classical differential geometry because we use non-commutative differential forms even for commutative algebras. Nevertheless, they share many of their formal properties. Definition A.84. A .reduced/ connection on a unital left A-module M is a morphism x 1 .M / in C that satisfies the Leibniz rule rW M !  A x1 r ı mA;M D mA;

x 1 .M / ı .idA ˝ r/ C d ˝ idM W A ˝ M ! A .M /; A

x 1 .A/ is the universal where mA;‹ W A˝‹ !‹ is the module structure on ‹ and d W A !  derivation. In terms of elements, this becomes r.a  x/ D a  r.x/ C d.a/ ˝A x

for all a 2 A; x 2 M:

The difference between connections and module homomorphisms is the additional x 1 .M / in the Leibniz rule. If r0 is any connection, summand d ˝ idM W A ˝ M !  A   x 1 .M / . then the set of connections is equal to the affine space r0 C HomA M;  A x 1 .A/ and the x 1 .M / Š  Examples A.85. Let A be a unital algebra. If M D A, then  A 1 x n .A/ ! x derivation d W A !  .A/ is a reduced connection. The differentials d W  nC1 x  .A/ are reduced connections as well because of the (graded) Leibniz rule. x 1 .M / D  x 1 .A/ ˝ V , and d ˝ idV is a reduced If M D A ˝ V is free, then  A connection. Retracts evidently inherit the existence of a connection. Since free modules admit connections, so do projective modules. The following proposition yields the converse assertion: Proposition A.86. Let M 22 ModC .A/. The following assertions are equivalent: (1) M is projective; (2) the extension (A.77) splits by an A-module homomorphism M ! A ˝ M ; x 1 .M /. (3) there is a connection r W M !  A x1 Similar assertions hold if A and M are non-unital, replacing A by AC in .2/ and  A 1 by A in .3/.

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Proof. (1) , (2): if M is projective, then HomA .M; / maps (A.77) again to an extension. Hence mAM induces a surjective map HomA .M; A˝M / ! HomA .M; M /, so that idM 2 HomA .M; M / lifts to s 2 HomA .M; A ˝ M /; this is the desired section. Conversely, if there is such a section, then M is a retract of a free module, and hence projective. u˝idM

(2) , (3): The extension (A.77) has an obvious section s0 W M Š 1 ˝ M ! x 1 .M /. We A ˝ M . Any other section is of the form s0 C r for a map r W M !  A claim that s0 C r is a module homomorphism if and only if r is a connection. We need r as a map into A ˝ M . Since d.a/ D a ˝ 1  1 ˝ a, we have d.a/  x D a ˝ x  1 ˝ ax D a  s0 .x/  s0 .a  x/: Hence r.a  x/ D a  r.x/ C d.a/  x is equivalent to .s0 C r/.a  x/ D a  .s0 C r/.x/. x1 xn Now we generalise the  passage  from d W A !  .A/ to the maps d W  .A/ ! n x .M / x  .A/. The sequence  in C is a graded unital left module over .A/ A n2N using the obvious multiplication maps x nC1

x m .A/ ˝  x An .M /   x m .A/ ˝A  x mCn .M / x An .M / Š   A for m; n  0; these are associative in the obvious sense. Recall that  a differential  graded module (briefly DG-module) over a differential graded algebra .n /n2N ; d such as x .A/ is a graded -module .Mn /n2Z together with differentials r W Mn ! MnC1 for n 2 Z that satisfy the graded Leibniz rule r.  x/ D .d /  x C .1/n  r.x/

for all 2 n ; x 2 Mk :

x  .M / that yields a differential graded  x  .A/-modLemma A.87. A differential on  A ule structure restricts to a connection on M , and this defines a bijection between x  .M /. connections on M and DG-module structures on  A Proof. On A˝M , the graded Leibniz rule required for a DG-module structure restricts to the ungraded Leibniz rule required for a connection. Hence DG-module structures restrict to connections. Conversely, we may extend a connection r on M by id

˝n ˝r

N x An .M / Š AN˝n ˝ M A x A1 .M / Š  x nC1 .M /  ! AN˝n ˝  A

or r.da1 : : : dan ˝ m/ WD da1 : : : dan  r.m/. A straightforward computation shows that these maps satisfy the graded Leibniz rule because r does so on M and d 2 D 0. Extend r as above. The computation   r 2 .  x/ D r .d /  x C .1/n r.x/ D .1/nC1 .d /  r.x/ C .1/n .d /r.x/ C .1/2n  r 2 .x/ x homomorphism; it is called the curvature of the shows that r 2 is a .A/-module connection r.

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Appendix. Algebraic preliminaries

Some elementary properties of the dimension. Here we recall some facts from homological algebra with only sketches of proofs because we do not want to develop the whole extension theory only for this purpose. Lemma A.88. dim.M1 ˚ M2 / D maxfdim M1 ; dim M2 g for M1 ; M2 22 ModC .A/. Proof. It is clear that dim.M1 ˚ M2 /  maxfdim M1 ; dim M2 g because two resolutions of length k for M1 and M2 combine to one for M1 ˚ M2 . For the converse implication, we characterise modules of projective dimension k by the vanishing of ExtAn .M; / for all n > k. Since ExtAn is additive, this yields the result. We will only use the following consequence, which is more elementary and can be proved without using extension theory. Lemma A.89. Let M1 ; M2 22 ModC .A/ be stably equivalent, that is, there exist projective modules P1 ; P2 22 ModC .A/ with M1 ˚ P1 Š M2 ˚ P2 . Then dim M1 D dim M2 . Lemma A.90. If K  P  M is a semi-split extension with projective P , then dim K D maxfdim M  1; 0g: x n .M / has dimension dim.M /  n for n < dim.M / and 0 for n  dim.M /. Hence  A Proof. Clearly, if K has a projective resolution of length k, then M has one of length k C 1 by concatenation. Conversely, suppose that M has a projective resolution of length k C 1  1, which begins with some augmentation map ı W P 0 ! M , and let K 0 WD ker ı 0 ; then K 0  P 0  M is another semi-split extension with projective P 0 , and our resolution of M provides one for K 0 of length k. Schanuel’s Lemma – which holds in any exact category – yields a bimodule isomorphism K 0 ˚ P Š K ˚ P 0 . Hence Lemma A.89 yields dim K D dim K 0  k. x 1 .M / D maxfdim M  1; 0g. Now we use the resolution (A.77) and get dim  A n x Iterating this, we get the formula for dim A .M /. Characterisations of k-dimensional modules Proposition A.91. Let A 22 AlgC .C/, M 22 ModC .A/, and k 2 N. The following are equivalent: (1) dim M  k; (2) M has a projective resolution of length k; x k .M / is projective; (3)  A x k .M / !  x kC1 .M / on the module  x k .M /; (4) there is a connection r W  A A A x l .M / !  x lC1 .M / for l  k that turn  x k .M / into a (5) there are maps rl W  A A A x  .A/-module. differential graded 

A.5 Homological algebra for modules

327

Proof. We have (1) , (2) by the definition of the dimension. Lemma A.90 yields (1) , (3) because the dimension vanishes exactly for projective modules. The equivalences (3) , (4) and (4) , (5) follow from Proposition A.86 and Lemma A.87. Now we consider the bidimension of algebras. Although this is, by definition, a special case of the dimension of modules, we proceed differently and use the free bimodule resolution Bar  .A/, which exploits the fact that A is already free as a onesided module. Proposition A.92. Let A 22 AlgC .C/. Then the following are equivalent: (1) dim A  k; (2) A has a projective A-bimodule resolution of length k; x k .A/ is a projective A-bimodule; (3)  x k .A/ !  x kC1 .A/ that is a (4) there is a right A-module homomorphism r W  connection for the left module structure; we call r a left connection on the x k .A/; bimodule  x l .A/ !  x lC1 .A/ for k  l that turn (5) there are left A-module maps rl W  k x  .A/ into a differential graded right A-module. The last two conditions are equivalent to symmetric ones with left and right exchanged. Connections on right modules are defined in essentially the same way; the graded Leibniz rule for a connection on a right module M contain the signs x Ak .M /; y 2  x l .A/: r.x  y/ D r.x/  y C .1/k x  d.y/ for all x 2  Proof. Arguments as in the proof of Proposition A.91 yield (1) , (2) , (3) and (4) , (5). It remains to prove (3) , (4). x n .A/ is free as an A-bix n .A/ is free as a one-sided unital module, A ˝  Since  x n .A/ is projecmodule. Copying the argument of Proposition A.86, we see that  x n .A/ !  x n .A/ ˝ A tive as a bimodule if and only if there is a bimodule section  for the multiplication map. Again we write this section as s0 C r for some map x k .A/ !  x kC1 .A/ and the tautological section s0 WD id x n ˝ u. Since s0 is a rW 

.A/ left module homomorphism, s0 C r is a left module homomorphism if and only if r is. As in Proposition A.86, s0 C r is a right module homomorphism if and only if r is a right module connection. Both Proposition A.91 and A.92 have versions for non-unital algebras and modules, x n .A/ by n .A/ everywhere. where we simply replace A by AC and  Example A.93. Let V 22 C. Proposition A.72 shows that 1 .TV / is a free nonunital TV -bimodule. Hence TV has bidimension 1 and 1 .TV / supports bimodule connections. More precisely, Proposition A.72 yields a bijection between left or right

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Appendix. Algebraic preliminaries

bimodule connections r W 1 .TV / ! 2 .TV / and maps r0 W V ! 2 .TV / in C. Given r0 , its unique extension to a right bimodule connection is given by r.!0 .dx/!1 / D !0 r.dx  !1 / D !0 r0 .dx/!1  !0 dx d!1 for any !0 ; !1 2 TC V , x 2 V . The standard choice is r0 D 0, which leads to the standard right connection r.!0 .dx/!1 / WD !0 dx d!1 on 1 .TV /.

A.6 Hochschild homology and cyclic homology A.6.1 Definition of Hochschild homology The normalised Hochschild complex Definition A.94. The normalised Hochschild chain complex for a unital algebra A is HH.A/ WD Bar  .A/=Œ ;  Š Bar  .A/ ˝A˝Aop A;

viewed as an object of HoKom.C/. It is clear that HH is a functor from the category of unital algebras to Kom.C/. We will use an appropriate variant of this chain complex for non-unital algebras to define Hochschild homology and cohomology. Since Bar.A/ is a free bimodule resolution, we can easily compute the commutator quotients: we get x n .A/: Bar.A/ Š .A ˝ AN˝n ˝ A/=Œ ;  Š A ˝ AN˝n Š  The boundary map

bn W A ˝ AN˝n ! A ˝ AN˝n1

is induced by b 0 in (A.81). Since the identification .A ˝ AN˝n ˝ A/=Œ ;  Š A ˝ AN˝n involves the map mnC1;0 that multiplies the last tensor factor onto the first one, we get bn D

n1 X

.1/j mj;j C1 C mn;0

(A.95)

j D0

or, in terms of elements, bn .x0 ˝    ˝ xn / WD

n1 X

.1/j x0 ˝    ˝ xj 1 ˝ .xj  xj C1 / ˝ xj C2 ˝    ˝ xn

j D0

C .1/n xn  x0 ˝ x1 ˝    ˝ xn1 : (A.96) We have b ı b D 0 by construction; this can easily be checked by hand as well.

A.6 Hochschild homology and cyclic homology

329

x n .A/. In this The formula for the boundary map simplifies if we view HHn .A/ Š  description, we can rewrite (A.96) as bn .x0 dx1 : : : dxn / D .1/n1 x0 dx1 : : : dxn1  xn C .1/n xn x0 dx1 : : : dxn1 D .1/n Œxn ; x0 dx1 : : : dxn1  (A.97) D .1/n1 Œx0 dx1 : : : dxn1 ; xn : Example A.98. HH.1/ Š 1 because 1N D 0. Recall that any two projective resolutions of the same (bi)module are chain homotopy equivalent as chain complexes of (bi)modules and hence have chain homotopy equivalent commutator quotient complexes. Therefore, if P ! A is any projective resolution of A in ModC .A ˝ Aop /, then P =Œ ;  is chain homotopy equivalent to HH.A/. This is the most important method to compute HH.A/. We use it to prove excision for periodic cyclic homology theories in §4.3.4. Extension to non-unital algebras Definition A.99. The Hochschild chain complex for an algebra A (with or without unit) is HH.A/ WD ker.HH.AC /  HH.1/ Š 1/: Its homology and cohomology       HHn .A/ WD Hn HH.A/ ; HHn .A/ WD Hn HH.A/ D Hn Hom.HH.A/; 1/ are the Hochschild homology and Hochschild cohomology of A. Lemma A.100. If A is unital, then HH.A/ and HH.A/ are chain homotopy equivalent. For any algebra A, we have a canonical chain homotopy equivalence HH.AC / HH.A/ ˚ 1;

where we view 1 as a chain complex supported in degree 0. Proof. Let A be unital. Then AC Š A ˚ 1 as algebras and as A-bimodules. Therefore, free unital A-bimodules remain projective as unital AC -bimodules. Hence we get HH.A/ and HH.AC / by applying the commutator quotient functor to certain projective AC -bimodule resolutions Bar.A/ and Bar.AC / of A and AC , respectively. Since AC Š A ˚ 1 and 1 is projective as an AC -bimodule, Bar.A/ ˚ 1 is another projective AC -bimodule resolution of AC . Since any two projective AC -bimodule resolutions of AC are chain homotopy equivalent, we get Bar.AC / Bar.A/ ˚ 1 and hence HH.AC / HH.A/ ˚ 1. The passage to HH.A/ removes the additional summand 1, that is, HH.AC / HH.A/ ˚ 1. Now we allow A to be non-unital. Since the algebra extension A ! AC ! 1 splits, so does the extension of chain complexes HH.A/  HH.AC / ! 1; here we use HH.1/ Š 1. Thus HH.AC / Š HH.AC / HH.A/ ˚ 1.

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Appendix. Algebraic preliminaries

Therefore, it makes no difference whether we use HH.A/ or HH.A/ for unital A. But the chain maps that implement this homotopy equivalence are not obvious; we will describe them more explicitly below. Next we describe HH.A/ in terms of tensor products. Recall that HH.AC / Š x C /; this induces an isomorphism HHn .A/ Š n .A/ for all n 2 N, including .A n D 0. If n  1, we can split n .A/ Š AC ˝ A˝n Š A˝nC1 ˚ A˝n : In terms of elements, this corresponds to the splitting into differential forms of the form x0 dx1 : : : dxn and dx1 : : : dxn . We write the boundary map b W n .A/ ! n1 .A/ as a block matrix with respect to this decomposition. A computation yields   b 1 W A˝n ˚ A˝n1 ! A˝n1 ˚ A˝n2 ; bD 0 b 0 where we write b 0 and b for the maps b 0 ; b W A˝k ! A˝k1 given by the familiar formulas 0

b WD

k2 X

j

.1/ mj;j C1 ;

b WD

j D0

k2 X

.1/j mj;j C1 C .1/k1 mk1;0 ;

j D0

as in (A.81) and (A.95), and  W A˝k ! A˝k is the signed cyclic rotation operator which, in terms of elements, reads .x0 ˝    ˝ xk1 / WD .1/k1 xk1 ˝ x0 ˝    ˝ xk2 : Hence HH.A/ is isomorphic to the total complex of the bicomplex :: :

:: :  A˝3 o 

1

b

A˝2 o

1



b 0

(A.101)

A˝2 b 0

b

 Ao

 A˝3

1D0

 A.

This is exactly the truncation of the cyclic bicomplex in Figure 2.2 to the first two columns. The fact that (A.101) is a bicomplex amounts to the formulas b ı b D 0;

b 0 ı b 0 D 0;

which are easy to check by hand.

b ı .1  / D .1  / ı b 0 ;

A.6 Hochschild homology and cyclic homology

331

This also says that .A˝n ; b 0 /n2N and .A˝n ; b/n2N are chain complexes and that the maps 1   W A˝n ! A˝n form a chain map between them; its mapping cone is HH.A/. Lemma A.102. If A is unital, then the chain complex b0

b0

b0

   ! A˝nC1  ! A˝n  ! A˝n1 !    ! A˝2  !A!0 is semi-split exact .that is, contractible/. Proof. The maps s0 WD u ˝ idA˝n W A˝n ! A˝nC1 provide a natural contracting homotopy, that is, b 0 ı s0 C s0 ı b 0 D idA˝n for all n 2 N. Hence the canonical embedding .A˝n ; b/ ! HH.A/ is a chain homotopy equivalence for unital A because it is part of a semi-split extension .A˝n ; b/  HH.A/  .A˝n ; b 0 / whose quotient is contractible. Let s0 W A˝n ! A˝nC1 denote the contracting homotopy constructed above. Then         id 0 0 0 b 1 id .1  / ı s0 :  ; D 0 0 0 id 0 s0 0 b 0 Thus the pair of maps .id; 1   ı s0 / defines a deformation retraction HH.A/ ! .A˝n ; b/. It is evident that the canonical projections A˝n ! A ˝ AN˝n1 define a chain map ˝n .A ; b/ ! HH.A/. This map is a chain homotopy equivalence if A is unital. Thus we get explicit chain homotopy equivalences HH.A/ ! .A˝n ; b/ ! HH.A/:

Definition A.103. An algebra A is H-unital if the chain complex .A˝n ˝ X; b 0 ˝ idX / is exact for all X 22 C, and strongly H-unital if .A˝n ; b 0 / is contractible. The notion of H-unital (or homologically unital) rings is due to Mariusz Wodzicki (see [110]). For algebras in C, it requires an additional exact category structure on C besides the trivial one that we implicitly use in Theorem A.74. We need strongly H-unital algebras to study the chain complex HH.A/ up to chain homotopy. Almost by definition, A is strongly H-unital if and only if the embedding of .A˝n ; b/ in HH.A/ is a chain homotopy equivalence. Many formal properties that we may want Hochschild homology to have – like excision, matrix-stability (or Morita invariance), additivity and a formula for tensor products – only hold for H-unital algebras. Corresponding statements for HH.A/ require strong H-unitality instead. We state these results without proof:

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Appendix. Algebraic preliminaries

Theorem A.104. Let A and B be strongly H-unital algebras. Then there are natural chain homotopy equivalences HH.A ˝ B/ HH.A/ ˝ HH.B/;

HH.A ˚ B/ HH.A/ ˚ HH.B/:

Moreover, the corner embedding A  ! Mn .A/ induces a natural chain homotopy  equivalence HH.A/ HH Mn .A/ for all n 2 N [ f1g. p

If I  E  Q is a semi-split algebra extension and I is strongly H-unital, then the canonical map HH.I / ! HHrel .p/ is a chain homotopy equivalence.

A.6.2 Cyclic homology via the cyclic bicomplex Operators on non-commutative differential forms. The following computations are due to Joachim Cuntz and Daniel Quillen ([24]). They work for algebras in additive symmetric monoidal categories. The operators b and d generate the Karoubi operator  WD 1  Œd; b D 1  .db C bd /:

(A.105)

In terms of elements, we have .! dx/ D ! dx  .1/n d.Œx; !/  .1/nC1 Œx; d! D ! dx  .1/n Œdx; ! D .1/n1 dx  !

for ! 2 n1 .A/, x 2 A.

Splitting n .A/ Š A˝n ˚ A˝nC1 as above, this yields the block matrix description    .1/n1 mn;0 D W A˝n ˚ A˝nC1 ! A˝n ˚ A˝nC1 : 0  We also define an operator B W .A/ ! .A/ of degree C1 by n X

B WD

j ı d

on n .A/

(A.106)

j D0

or, in terms of elements, B.x0 dx1 : : : dxn / D

n X

.1/j n dxj : : : dxn dx0 : : : dxj 1 :

j D0

In the decomposition n .A/ Š A˝n ˚ A˝nC1 , this corresponds to the block matrix 

 0 0 BD W A˝nC1 ˚ A˝n ! A˝nC2 ˚ A˝nC1 ; Q 0

QD

n X j D0

j

on A˝nC1 .

333

A.6 Hochschild homology and cyclic homology

Theorem A.107. The operators b; d;  satisfy the following relations on n .A/. ıd Dd ı ıb Dbı d ı  nC1 D  nC1 ı d b ı n D n ı b ıB DB ı n

.  1/. n

(A.108) (A.109)

Dd Db DB D 1 C b ı n ı d

(A.110) (A.111) (A.112) (A.113)

b ı d D  nC1  

(A.114)

d ıb D1

(A.115)

nC1

nC1

 1/ D 0

(A.116)

.1   /  .1   / 1 The right hand side in (A.117) is not 0 but  plugged into the polynomial b ı B D B ı b D

.1  x n /  .1  x nC1 /=.1  x/ D

n

n1 X

nC1

xj 

j D0

2n X

(A.117)

xj :

j DnC1

Proof. Equations (A.108) and (A.109) follow easily from d 2 D 0 and b 2 D 0. Since  acts like  on closed forms, we get (A.110); this together with (A.108) also implies (A.112). On the one hand, we have 1   n .x0 dx1 : : : dxn / D Œx0 ; dx1 : : : dxn  because  n .x0 dx1 : : : dxn / D .1/n1  n1 . dxn x0 dx1 : : : dxn1 / D    D dx1 : : : dxn x0 : on the other hand, (A.97) yields 2

b ı  n ı d.x0 dx1 : : : dxn / D .1/n b. dx1 : : : dxn dx0 / D Œx0 ; dx1 : : : dxn : Both computations together yield (A.113). Hence  nC1 D  C b nC1 d D  C bd by (A.109) and (A.110). This is (A.114). We also get (A.115) because db D 1    bd by (A.105). Equation (A.113) also implies b ı  n D b because b 2 D 0. Together with (A.109) this yields (A.111). Equations (A.113) and (A.115) imply (A.116) because . n  1/ ı . nC1  1/ D n b d ı .db/ D 0. Now we use (A.106) (on n1 .A/) and (A.115) to compute Bb D

n1 X

 j db D

j D0

n1 X j D0

 j .1   nC1 / D

.1   n /  .1   nC1 / : 1

Combining this with (A.106), (A.114), and (A.112), we get bB D

n X j D0

bd  j D

. nC1  /.1   nC1 / D Bb D Bb: 1

334

Appendix. Algebraic preliminaries

Hence we get (A.117); this finishes the proof of the theorem. Let D be the ring of operators on .A/ generated by b and d . The homogeneous operators in D are linear combinations of .db/n and .bd /n for n 2 N because d 2 D 0 and b 2 D 0. Equations (A.114) and (A.115) show that this ring is generated by . Equations (A.108) and (A.109) show that it is central in D. When we restrict to n .A/, this central subring is, for generic A, isomorphic to ZŒ=. n  1/. nC1  1/, that is, (A.116) usually describes the minimal polynomial of j n .A/ . The cyclic bicomplex. The relations b 2 D 0, B 2 D 0, and Bb C bB D 0 mean that the diagram in Figure A.1 is a bicomplex. Identifying n .A/ Š A˝n ˚ A˝nC1 and

 4 .A/ o

B

 3 .A/ o

B

 2 .A/ o

B

 1 .A/ o

B

 0 .A/

b

 3 .A/ o

B

 1 .A/ o

B

 0 .A/

B

 1 .A/ o

B

 0 .A/

b

b

b

 1 .A/ o

 2 .A/ o

b

b

 2 .A/ o

B

B

 0 .A/

b

b

b

b

 0 .A/ Figure A.1. The cyclic bicomplex via differential forms.

using our block matrix descriptions of b and B above, we see that this chain complex is isomorphic to the homological cyclic bicomplex in Figure 2.2 (up to rearranging the entries). Definition A.118. We let HC.A/ be the total complex of the bicomplex in Figure A.1. The cyclic homology and the cyclic cohomology of A are defined by       HCn .A/ WD Hn HC.A/ ; HCn .A/ WD Hn HC.A/ D Hn Hom.HC.A/; 1/ : The first column of the cyclic bicomplex is the Hochschild chain complex HH.A/. If we remove the first column, we retain a shifted copy of the original complex HC.A/. Hence we get a semi-split extension of chain complexes I

S

HH.A/  HC.A/  HC.A/Œ2:

A.6 Hochschild homology and cyclic homology

335

This generates an exact triangle in HoKom.C/, whose boundary map HC.A/Œ1 ! HH.A/ is closely related to B. The projection S W HC.A/  HC.A/Œ2 is called the periodicity operator. Taking (co)homology, we get Connes’ SBI exact sequences I

S

B

I

 HCn .A/  ! HCn2 .A/  ! HHn1 .A/ !   ;    ! HHn .A/ ! B

S

I

B

   ! HHn1 .A/  ! HCn2 .A/  ! HCn .A/ !  HHn .A/  !  : They often allow to compute the cyclic homology from the Hochschild homology. In addition, both theories are related by a spectral sequence because the columns in the cyclic bicomplex are shifted copies of the Hochschild complex. Q   Definition A.119. We let .A/ 22 C be the product 1 nD0 .A/ in the category   of projective systems C . Let HP.A/ be the Z=2-graded chain complex in C with  underlying pro-object .A/ and differential b C B. The periodic cyclic homology and the periodic cyclic cohomology of A are defined by       HPn .A/ WD Hn HP.A/ ; HPn .A/ WD Hn HP.A/ D Hn Hom.HP.A/; 1/ : The relationship between cyclic and periodic cyclic homology is as follows. The periodicity operator generates a projective system of chain complexes SŒ2

SŒ4

SŒ6

SŒ8

HC.A/   HC.A/Œ2  HC.A/Œ4  HC.A/Œ6     :

 This projective system is isomorphic in Kom. C I Z/ to the 2-periodic chain complex bCB  even bCB  odd bCB  even bCB  odd    !  .A/ !  .A/ !  .A/ !  .A/ !    :

  Disregarding the distinction between Kom. C I Z/ and Kom. C I Z=2/, we get   HP.A/ Š lim HC.A/Œ2n; S :   Since projective limits in C are purely formal, this projective limit is also a homotopy  projective limit in the triangulated category HoKom. C /. This follows because we have a semi-split extension of chain complexes HP.A/ 

1 Y nD0

HC.A/Œ2n 

1 Y

HC.A/Œ2n;

nD0

where the first map is the canonical embedding of the projective limit in the product and the second map is induced by id  S. This yields an exact sequence lim1 HCC2nC1 .A/  HP .A/  lim HCC2n .A/  

336

Appendix. Algebraic preliminaries

and an isomorphism

HP .A/ D lim HCC2n .A/ ! for  D 0; 1. These formulas often allow to compute the periodic cyclic (co)homology of an algebra by first computing its Hochschild (co)homology and then its cyclic (co)homology. This approach is feasible for algebras such as group rings whose Hochschild homology is accessible.

A.6.3 The X-complex and its relatives Definition A.120 ([26, §2]). The Hodge  filtration on the chain complex HP.A/ is the decreasing filtration by subcomplexes Fn HP.A/ n2N defined by 1 Y   Fn HP.A/ WD b nC1 .A/  k .A/: kDnC1

Warning A.121. Michael Puschnigg indexes the Hodge filtration differently in [86], [89]: our Fn is FnC1 for him. Definition A.122. We let X .k/ .A/ 22 Kom.CI Z=2/ for k 2 N be the quotient complex .HP.A/=Fk HP.A/; B C b/; that is, its underlying object of C is X .k/ .A/ WD

k1 Y

k1 Y k .A/ Š j .A/   kC1 j .A/  k .A/=Œ ; I b  .A/ j D0 j D0

these two agree by (A.97). For k D 1, we get the X-complex X.A/ WD X .1/ .A/ of A, which is of the form Ao

\ıd bN

/

1 .A/=Π; ;

where \ W 1 .A/  1 .A/=Œ ;  is the quotient mapping and bN W 1 .A/=Œ ;  ! A is induced by b W 1 .A/ ! A. The following theorem is essentially due to Christian Kassel ([58]); but our formulas seem slightly different. Theorem A.123. For a quasi-free algebra A, the canonical chain maps HP.A/  X .k/ .A/  X.A/ are chain homotopy equivalences for all k  2. More generally, if the bidimension of A is at most m, then the maps HP.A/  X .k/ .A/  X .m/ .A/ are chain homotopy equivalences for all k  m. Proof. Recall that A is quasi-free if and only if dim A  1 by Theorem 4.16. Hence it suffices to prove the second statement. We use the method of §A.2.1 to write down proofs using element notation.

A.6 Hochschild homology and cyclic homology

337

If A has bidimension at most m, then m .A/ is a projective A-bimodule. Hence there is a sequence of left A-module maps r W k .A/ ! kC1 .A/ for k  m that define a differential graded .A/-module structure on m .A/. We put r D 0 on k .A/ for k < m. Using the graded Leibniz rule and (A.97), we compute Œr; b D id on >m .A/: r ı b.x0 dx1 : : : dxn / C b ı r.x0 dx1 : : : dxn /   D .1/n r Œxn ; x0 dx1 : : : dxn1  C .1/nC1 Œxn ; r.x0 dx1 : : : dxn1 / D .1/n r.xn  x0 dx1 : : : dxn1 /  .1/n xn  r.x0 dx1 : : : dxn1 /  .1/n r.x0 dx1 : : : dxn1  xn / C .1/n r.x0 dx1 : : : dxn1 /  xn D x0 dx1 : : : dxn :   On m .A/, we have r ıb D 0, so that the range of Œr; b is contained in b mC1 .A/ . Since Œr; b is a chain map by definition and Œr; b D id on mC1 .A/, it follows that  mC1 Œr; bj m .A/ is a projection onto b  .A/ . As a result, Œr; b is a retraction HP.A/  Fm HP.A/, so that the chain complex .Fm HP.A/; b/ is contractible. This shows that HH.A/ is chain homotopy equivalent to the truncated complex b

b

b

b

0 ! m .A/=Œ ;  !  m1 .A/ !   !  1 .A/ !  0 .A/ ! 0: For the boundary map b C B, r yields a chain homotopy id id  Œb C B; r. The latter restricts to ŒB; r on Fm HP.A/. Now we use the operator h WD

1 X

.1/ r ı ŒB; r D j

j

j D0

1 X

  on Fm HP.A/. .1/j r ı .Br/j C .rB/j

j D0

(A.124)  This infinite sum exists on .A/ because ŒB; r has degree 2, so that the projection to k .A/ for any fixed k annihilates all but finitely many terms in the series. It follows from ŒB; r D Œb C B; r  id that ŒB; r commutes with B C b, so that Œb C B; h D

1 X



.1/ b C B; r ı ŒB; r D j

j

j D0

1 X

.1/j .ŒB; rj C ŒB; rj C1 / D id;

j D0

that is, h is a contracting homotopy for Fm HP.A/. We have seen above that the extension of chain complexes Fm HP.A/  HP.A/  X .m/ .A/ is semi-split via b ı r on m .A/. Thus the contractibility of Fm HP.A/ implies that the projection HP.A/  X .m/ .A/ is a chain homotopy equivalence. In the above situation, we also get an explicit chain homotopy inverse for the projection HP.A/  X .m/ .A/, namely, ˛ WD lim .id  P  Œb C B; r/n : n!1

338

Appendix. Algebraic preliminaries

We show that this limit exists and describe it more concretely. Recall that P WD Œb; r is a projection onto Fm HP.A/, so that id  P D id on k .A/ for k  m  1. The operator ŒB; r has degree 2 and vanishes on k .A/ for k  m  2. Thus its range is contained in Fm HP.A/ and thus annihilated by id  P . Hence induction shows that ˛ D lim .id  P  ŒB; r/n n!1   D lim 1  ŒB; r ˙    C .1/n1 ŒB; rn1 ı .id  P / C .1/n ŒB; rn D

n!1 1 X

.1/j ŒB; rj ı .id  P /:

j D0

Notice that ˛ vanishes on Fm HP.A/ and that ˛j k .A/ D id for k  m  2 because r vanishes on l .A/ for l  m  1. We are particularly interested in the resulting chain homotopy inverse ˛ W X.A/ ! X .2/ .A/ for the canonical projection X .2/ .A/  X.A/ for quasi-free A. Disregarding terms in F2 HP.A/, we get ( id  r ı d on X0 .A/ D A, ˛D (A.125) id  P D id  b ı r on X1 .A/ D 1 .A/=Œ ; . Example A.126. We compute the chain map ˛ W X.A/ ! X .2/ .A/ more concretely for a tensor algebra TV for some V 22 C and the standard connection on 1 .TV / described in Example A.93. We have .id  b ı r/.!0 .da/!1 / D !0 .da/!1 C b.!0 da d!1 / D !0 .da/!1 C !0 a d!1  !0 d.a!1 / C !1 !0 da D !1 !0 da: In particular, the range of the idempotent map id  b ı r is the subspace TC V dV of all 1-forms ! dx with ! 2 TC V , x 2 V . Write an element of TV as a formal product x1    xn with x1 ; : : : ; xn 2 V . The Leibniz rule yields r ı d.x1    xn / D r

n X

x1    xj 1 .dxj /xj C1    xn



j D1

D

n X

x1    xj 1 dxj d.xj C1    xn /:

j D1

A.7 Biprojective algebras Biprojective Banach algebras and topological algebras were introduced by Alexander Ya. Helemski˘ı (see [39], [40], [41]). Other authors who have worked on biprojective

A.7 Biprojective algebras

339

algebras are Yuri˘ı V. Selivanov ([94], [95], [96], [97]), Alexei Yu. Pirkovskii ([81]), and Hanno Baehr ([1]). We need some basic facts about them for the computations in §2.4.2. Definition A.127. Let A be an algebra in an additive symmetric monoidal category. We call A biprojective if the multiplication map A ˝ A ! A splits by an A-bimodule homomorphism  W A ! A ˝ A. If A is biprojective, then the multiplication map A ˝ A ! A is an epimorphism (even split). Iterating  , we get a bimodule section  .2/ W A ! A ˝ A ˝ A  AC ˝ A ˝ AC for the multiplication map AC ˝A˝AC ! A, so that A is projective as an A-bimodule. The converse also holds, that is, an algebra A is biprojective if and only if it is projective as a bimodule and the multiplication map A ˝ A ! A is epic. This explains why such algebras are called biprojective. Most biprojective algebras are not unital. In the unital case, biprojective algebras have a very simple structure. We note the following result: Lemma A.128. An algebra A is biprojective and unital if and only if it has cohomological dimension 0. Proof. By definition, A has cohomological dimension 0 if and only if AC is a projective A-bimodule. If A is unital, then we have an algebra homomorphism AC Š A˚1. This shows that A is a projective A-bimodule if and only if AC is one. Thus a unital algebra is biprojective if and only if it has cohomological dimension 0. Conversely, let A have cohomological dimension 0. We must show that A is unital. We use that any extension in Mod.A/ or Mod.Aop / splits. In particular, we get a left and a right module section for the extension A  AC  1. Taking the differences of the unit map 1 ! AC with such sections, we get a left and a right unit for A. In such a situation, both units coincide and A is unital. Theorem A.129. Let A be biprojective. Then dim A  2, and we have dim A  1 if and only if the diagonal embedding A ˝ A ! .AC ˝ A/ ˚ .A ˝ AC / splits by a bimodule homomorphism. Proof. Equip 1 with the zero bimodule structure. We have a semi-split bimodule extension A  AC  1. Therefore, it suffices to construct a projective bimodule resolution of length 2 for 1; this resolution can then be patched together with the length0 projective bimodule resolution A ! A. We get the desired bimodule resolution of 1 by taking the tensor product of two copies of the extension A  AC  1, that is, A ˝ A  .A ˝ AC / ˚ .AC ˝ A/ ! AC ˝ AC  1:

(A.130)

This is a (semi-split) resolution because it is a tensor product of two semi-split extensions. It is projective because A and AC are projective as left and right modules. The cohomological dimension of the bimodule AC is 1 if and only if this holds for 1, if and only if the resolution (A.130) splits. This yields the last assertion.

340

Appendix. Algebraic preliminaries

Proposition A.131. If A is biprojective, then HH.A/ Š HC.A/ Š HP.A/ Š A=Œ ; :

Thus HHn .A/ D 0 and HHn .A/ D 0 for n  1, and HP0 .A/ Š HC2n .A/ Š HH0 .A/ Š A=Œ ; ;

HP1 .A/ Š HC2nC1 .A/ D 0;

HP .A/ Š HC .A/ Š HH .A/ Š Hom.A=Œ ; ; 1/;

HP1 .A/ Š HC2nC1 .A/ D 0;

0

2n

0

for all n 2 N, and the periodicity operators S W HC2nC2 .A/ ! HC2n .A/;

S W HC2n .A/ ! HC2nC2 .A/

are invertible for all n 2 N. Proof. The canonical map .A ˝ A/=Œ ;  ! A, x ˝ y 7! y  x, is an isomorphism because A is biprojective. The commutator quotients for A ˝ AC and AC ˝ A are isomorphic to A for any algebra A. Now we plug this into the explicit projective resolutions for 1 and AC constructed in Theorem A.129; we see that A is strongly H-unital and that HH.A/ A=Œ ;  (supported in degree 0). It follows that HH.A/ Š HC.A/ Š HP.A/.

A.7.1 Some examples of biprojective algebras The following general construction of biprojective algebras is related to spaces of nuclear operators in functional analysis. Let W; V 22 C and let f W W ˝ V ! 1 be a map. Then V ˝ W becomes an associative algebra with respect to the product idV ˝ f ˝ idW W V ˝ W ˝ V ˝ W ! V ˝ W: f

The map  WD f ı ˆV;W W V ˝ W Š W ˝ V  ! 1 is a trace for this multiplication:    .v1 ˝ w1 /  .v2 ˝ w2 / D .f .w1 ; v2 /  v1 ˝ w2 / D f .w1 ; v2 /  f .w2 ; v1 /     yields  .v1 ˝ w1 /  .v2 ˝ w2 / D  .v2 ˝ w2 /  .v1 ˝ w1 / because 1 is commutative. We assume that there are maps W 1 ! W ,  W 1 ! V , such that f ı . ˝ / D id1 ; that is, and  are elements of W and V with f . ; / D 1 (see §A.2.1). Then the map idV ˝ ˝  ˝ idW W V ˝ W Š V ˝ 1 ˝ 1 ˝ W ! V ˝ W ˝ V ˝ W is a bimodule homomorphism and a section for the multiplication map. Thus V ˝ W is biprojective. Viewing V ˝ W as the range of the idempotent bimodule homomorphism p WD .idV ˝ ˝  ˝ idW / ı m on V ˝ W ˝ V ˝ W , we can easily compute the commutator quotient of V ˝ W because .V ˝ W ˝ V ˝ W /=Œ ;  Š V ˝ W via v1 ˝ w1 ˝ v2 ˝ w2 7! v2  v1 ˝ w2  w1 I

A.7 Biprojective algebras

341

it turns out that p induces the idempotent map v ˝ w 7! f .w; v/   ˝ on V ˝ W ; thus the trace f W V ˝ W ! 1 induces an isomorphism V ˝ W =Œ ;  Š 1. Thus V ˝ W has the same Hochschild, cyclic, and periodic cyclic (co)homology as 1. This is not surprising because V ˝ W is Morita equivalent to 1 and strongly H-unital. Examples A.132. Let V be a complete bornological vector space that has the global approximation property ([67]). Let V 0 WD Hom.V; C/ be its bornological dual space y V ! C be the canonical bilinear map. Then V ˝ y V 0 is the algebra and let f W V 0 ˝ of nuclear operators on V . If we drop the assumption on the approximation property, y V 0 ! Hom.V; V / need not be faithful any more, so that then the representation V ˝ 0 y V differs from the algebra of nuclear operators. V ˝ y V 0 Š Mn .C/. If V D C n , then V ˝ y V 0 is the algebra of trace-class operators on V . If V is a Hilbert space, then V ˝ y W ! C in Cborn is equivalent to a bounded linear A bounded bilinear map V ˝ map W ! Hom.V; C/ D V 0 by adjoint associativity. Therefore, any algebra of the y W comes equipped with an algebra homomorphism V ˝ y W !V ˝ y V 0. form V ˝ L y If V D W D n2N C and f W V ˝ W ! C is the obvious bilinear map, then y W is isomorphic as a bornological algebra to M1 .C/. V ˝ The cohomological dimension for algebras of the form V ˝ W can take all possible values 0; 1; 2. Although the criterion in Theorem A.129 may look simple enough, it is quite subtle to decide whether a biprojective algebra has cohomological dimension 1 or 2, that is, whether it is quasi-free or not. This is also illustrated by the Examples A.133. Examples A.133. We consider the convolution algebras C 1 .K/ and L1 .K/ of smooth and measurable functions and the group C  -algebra C  .K/ for a compact Lie group K. These examples play a role in §2.4. The Banach algebras L1 .K/ and C  .K/ are biprojective and have cohomological dimension 2 by [39, Theorem 9]. It is shown in [81, Corollary 5.4] that C 1 .K/ is biprojective of cohomological dimension 1, that is, C 1 .K/ is quasi-free. These y assertions are shown in the symmetric monoidal category of Fréchet algebras with ˝ as tensor product functor. Since the precompact bornology functor from Fréchet spaces ! to Cborn and the dissection functor Cborn ! Ban are symmetric monoidal, this ! remains the case when we view them as bornological algebras or algebras in Ban.

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Notation and Symbols

}

Fedosov product for differential forms over tensor algebras, p. 217

F

action of E by left multipliers on T L, p. 217



chain homotopy or chain homotopy equivalence, p. 292

k k1

operator norm on B.V /

1

unit object of a symmetric monoidal category; in categories such as ! Cborn or Ban, this is either R or C

AB

free product of two algebras

AB

augmentation ideal of AC ˝ BC , p. 302

A B

A is a uniformly dense bornological subalgebra of B, p. 131

A 22 C

A is an object of the category C

AËM

crossed product algebra of an algebra A with a bimodule M , p. 310

f 2C

f is a morphism in the category C

x 2} T , S } T means x 2 T } , S  T } x~ T , S ~ T

means x 2 T ~ , S  T ~

AC

unital algebra generated by an algebra

AN

coker.u W 1 ! A/

Aop

opposite algebra, p. 300

Alg.C/

category of algebras in C

AlgC .C/

category of unital algebras in C

Alg fp .C/

!  finitely presented objects of Alg. C /

A.Œ0; 1; V /

space of continuous, bounded variation functions Œ0; 1 ! V , p. 41

B}

disked hull of B, p. 15

B~

complete disked hull of B, p. 15

B1C .V; F; /

operators on V with ŒF;  D 0 and ŒF; V  2 `1 .V /, p. 274

350

Notation and Symbols

B1 .V; F /

operators on V with ŒF; V  2 `1 .V /, p. 274

 B1 .V; F /

operators on V with ŒF; V  2 K.V /, p. 281

C B1 .V; F; /

operators on V with ŒF;  D 0 and ŒF; V  2 K.V /, p. 281

b

Hochschild boundary operator (both homological and cohomological (both homological and cohomological)

b0

bar boundary operator (both homological and cohomological)

B

bootstrap category of C  -algebras

BV.Œ0; 1; V /

space of bounded variation functions Œ0; 1 ! V , p. 41

B.V /

algebra of bounded operators on V , p. 273

C Œn

translation functor for chain complexes, p. 291

ch

Chern–Connes character

C k .M; V /

space of C k -functions M ! V for k 2 N [ f1; !g, p. 22

C..t//

algebra of analytic power series, p. 116

C..t//0

algebra of analytic power series without constant term, p. 116

!  C

category of inductive systems in C, p. 53

 C

category of projective systems in C, p. 145

D

differential in .T A/

d

x differential in .A/ or .A/, p. 308

F .V /

algebra of finite-rank operators on V , p. 273

hf i

algebra homomorphism .even A; ˇ/ ! B induced by a map fWA!B

H .C /

cohomology of C , defined as H .C; 1/, p. 292

H .C /

homology of C , defined as H .1; C /, p. 292

H .C; D/

homology of the chain complex of maps C ! D or, equivalently, morphisms C ! DΠin the homotopy category of chain complexes, p. 292

HA .A/

analytic cyclic cohomology of A

HA .A/

analytic cyclic homology of A

Notation and Symbols

HL .A/

351

local cyclic homology of A

HoKom.CI Z=p/ homotopy category of Z=p-graded chain complexes in C, p. 293 Hom

generic notation for spaces of morphisms, especially in Born1=2 and ! Ban

Hom.n/

space of multi-linear maps

HA.A/

chain complex that computes the analytic cyclic theory of A

HArel

relative version of HA, p. 212

HL.A/

chain complex that computes the local cyclic theory of A

HLrel

relative version of HL, p. 212

HP.A/

chain complex that computes the periodic cyclic homology and cohomology of a pro-algebra A

Hom

generic notation for internal Hom especially for categories of bornological vector spaces

Inv

algebra of Laurent polynomials in one variable

Inv0

augmentation ideal in Inv

Iso

unital algebra generated by two elements v; w and the relation wv D 1

Iso0

augmentation ideal in Iso

JV

kernel of the canonical algebra homomorphism A W TA ! A

 JA

 kernel of A W TA ! A

JA

kernel of the canonical algebra homomorphism A W T A ! A

KK

Kasparov’s bivariant K-theory, p. 260

Kom.CI Z=p/

category of Z=p-graded chain complexes in C, p. 291

K.V /

algebra of compact operators on V

 L

 left ideal in TE generated by K, p. 163

`1 .V /

y V 0 , p. 273 algebra V ˝

L

left ideal in T E generated by K, p. 214



cyclic rotation operator, p. 76

352

Notation and Symbols

Mod.A/

category of A-modules in C

ModC .A/

category of unital A-modules in C

Mod.A; B/

category of A; B-bimodules in C

ModC .A; B/

category of unital A; B-bimodules in C

M.B/

left multiplier algebra of B, p. 217

B

gauge semi-norm of B, p. 13

O.X/

algebra of germs of holomorphic functions near X  C, p. 109

an A

completion of .A/ with respect to the analytic bornology

n .A/

bimodule of non-commutative differential forms over A, p. 308

n .A/

bimodule of non-commutative differential forms over A, p. 312

x n .A/ 

bimodule of reduced non-commutative differential forms over A, p. 308

x n .A/ 

bimodule of reduced non-commutative differential forms over A, p. 312

!f

curvature of f , mapping .x; y/ 7! f .xy/  f .x/f .y/

Piso

unital algebra generated by two elements v; w and the relations vwv D v, wvw D w

Piso0

augmentation ideal in Piso

A

canonical algebra homomorphism T A ! A

Q

averaging map based on the cyclic rotation operator (both homological and cohomological)

QA

free product of two copies of A

qA

kernel of the natural map QA ! A ı Calkin algebra B.V / K.V / S1 n nD1 S

Q.V / S1 hS i

behaves like S or nothing; thus hS1 iS2 WD S1 S2 [S2 and S1 hS2 i WD S1 S2 [ S1

hhS ii

hSi.dS/1  A for S 2 S.A/

S 1 .G/

rapidly decreasing functions on a discrete group G, p. 129

Notation and Symbols

353

S k .G/

functions on a discrete group G of polynomial decay of order k, p. 129

S ! .G/

functions on a discrete group G of subexponential decay, p. 129

S.V /

directed set of bounded subsets in V

Sc .V /

directed set of complete bounded disks in V

Scm .A/

directed set of complete submultiplicative bounded disks in a bornological algebra A

Sd .V /

directed set of bounded disks in V

S" .V /

bornology on B.V /, p. 79

Sm .A/

directed set of submultiplicative bounded disks in a bornological algebra A

sC

category of separable C  -algebras

†.xI A/

spectrum of x in A

A

canonical lanilcur A ! T A

TV

tensor algebra on V

TC V

unital tensor algebra on V

TA

analytic tensor algebra of A

TC A

.T A/C

 TC A

 . TA/C

k

Chern–Connes character of a finitely summable Fredholm module, see (7.24) and (7.32)

U.B; %/

algebra generated by B with relations , p. 242

VB

semi-normed subspace of V generated by the disk B

Z.M /

centre of bimodule M

Index Page numbers in italics refer to pages where a concept is defined

a-nilpotent, see analytically nilpotent Abelian category, 36 absolutely convex, 13 absolutely summable series, 22, 23, 26 absorb, 15, 67 absorbing, 13 additive, 35, 158, 159, 207, 208, 209, 332 additive category, 35 adjoint functor, 19, 27, 31, 32, 47, 49, 52, 56, 57, 59, 64, 241, 291, 298, 305, 306, 314, 320 algebra, 45, 46, 47, 58, 145, 270, 298–315 unital, 46, 47, 300, 304–308, 311, 312, 314, 321 almost commuting, 185, 186 amenable, 265, 270 analytic group action, 125 subspace for a group action, 125, 126, 254 analytic bornology, 181, 186 analytic cyclic cohomology, 79, 81, 102, 103, 251, 273–290 analytic cyclic homology, 79–81, 81, 102, 192, 193, 196–214, 251, 266, 270–273, 286 analytic power series, 116, 182, 188 analytic tensor algebra, 154, 180, 181–197, 247, 271, 272 analytically nilpotent, 116, 117, 119, 135, 141, 151, 180–193, 198, 202 analytically nilpotent curvature, see lanilcur analytically quasi-free, 103, 152, 187, 190, 191, 198–204, 214, 272 approximably dense, 73, 100, 141, 250, 251, 281 approximate local homotopy category, 255

approximate local homotopy equivalence, 247, 247–251 approximation property, 71, 71–73, 84, 251, 252, 254, 258, 274, 279–281 asymptotic morphism, 256 augmentation, 270, 301 Banach algebra, 76–79, 103, 108, 115, 118, 128–130, 134, 247, 281 Banach space, 13, 14, 19, 41, 42, 78, 85, 86, 98, 102, 146, 253, 279 bar resolution, 322, 323, 327 Baum–Connes Conjecture, 264–266, 270 bicomplete category, 52, 55, 303, 321 bidimension, 324, 336 bimodule, 154, 193, 273, 310, 311 biprojective, 100–104, 158, 273 bootstrap category, 100, 268–270 bornological algebra, 19, 44, 79–81, 246, 249, 251 bornological closure, 25, 26, 31, 33 bornological embedding, 34 bornological isomorphism, 18 bornological quotient map, 34 bornological subalgebra, 112, 116, 131 bornological topology, 25 bornological vector space, see bornology, 292 bornologically closed, 25 bornologically compact, 26 bornologically metrisable, 67, 67–71, 84 bornologically precompact, 26 bornologically relatively compact, 26 bornology, 14, 14–19, 86, 181, 281, 304, 306 von Neumann, 16, 17, 20–25, 28, 30, 37–39, 41, 66, 67, 71, 78, 79, 84, 103, 108, 109, 115, 118, 129, 130, 253, 285

356

Index

complete, 15, 16, 17, 31, 32 countably generated, 69, 70, 96, 98, 134, 135 direct product, 35 direct sum, 35 fine, 16, 19, 25, 30, 36, 98, 101, 134, 304 generated by a set of bounded subsets, 16, 20, 35, 36, 38, 39, 71 precompact, 17–30, 17, 26, 37, 40, 66, 67, 71, 84, 98, 109, 111, 114, 118, 128–130, 251–254, 266, 280, 281, 285 quotient, 30 separated, 15, 31, 32 subspace, 30 bounded, see bornology bounded map, 18–20, 25, 28, 29, 135 bounded variation, 41, 41–44, 61, 123, 124, 155, 189, 198, 244, 246, 256

compact Lie group, 98–104 compact operator, 127, 262 complete disked hull, 15, 17, 26 completion, 31, 32, 56, 57, 64–72, 93, 112, 116, 138, 140, 141, 281 cone, 50, 293, see mapping cone connection, 103, 157, 324, 324–328, 336, 338 Connes–Thom Isomorphism, 265 continuous, 14, 22, 23, 41–43, 61, 123–125, 127, 133, 180, 189, 226, 249, 253, 256, 287, 301 group action, 125 continuously differentiable function, 22, 23, 61, 68, 123, 124 contractible chain complex, 293 convergent sequence, 22, 23, 42, 61 convex, 13 coproduct, see direct sum, 51 corner embedding, 158, 205, 206 crossed product, 193, 264–266, 310, 311 C  -algebra, 98, 101, 122, 126, 127, 133, curvature, 148, 180, 181, 183, 247, 325 251–254, 258, 260–270, 272, 286 cut-off sequence, 279, 280–287 Cauchy sequence, 22, 23, 42, 61 cyclic bicomplex, 77, 80, 154, 196–197, centre, 101, 307 334–336 chain complex, 85, 86, 291–297, 321 cyclic cohomology, 334 chain homotopy, 155, 291, 292, 293 cyclic homology, 101, 232–239, 334 chain homotopy equivalence, 193, 213, cyclic rotation operator, 76 216, 293, 322, 336 cylinder functor, 189, 244 chain map, 85, 86, 297 Chern–Connes character, 99, 102, 260–290 close, 185, 192 closed symmetric monoidal category, 47, 52, 57, 58, 297, 298, 305–307 cofinal, 15, 54 cohomological, 294 cohomology, 292 cokernel, 32, 33, 35, 51, 298 commutator map, 185 commutator quotient, 101, 154, 192, 193, 199, 307 compact, see also bornologically compact, 122, 252, 279–281, 285

dense, 27, 27–29, 117–135, 141, 142 dense bornological subalgebra, 131 derivation, 154, 193, 275, 310, 311–313, 320 inner, 311, 312 derived category, 95 determined by homology, 267, 267–269 diagram, 50 differential form, 308–314, 309, 312, 320, 322 differential graded algebra, 314, 315 dimension, 323, 326–328 direct limit, 51, 50–56, 303

Index

direct product, 35, 51, 159 direct sum, 35, 51, 112, 116, 138, 140, 141, 159, 202, 210, 241 direct union, 62 directed set, 15, 53, 54 disk, 13, 13–15, 22 complete, 14 norming, 14 disked hull, 15 dissection functor, 59, 59–72, 82–84, 249, 251 dual, 69, 96–98, 292

357

finitely presented, 86, 86, 241–245 Fréchet space, 17, 20–30, 33, 34, 37–41, 67–72, 96, 98, 101, 253, 254, 256 Fredholm module, 274, 281, 285, 286 free product, 185, 186, 202, 314, 317–320 fully exact functor, 34, 63 functional calculus, 109, 182, 203 Goodwillie’s Theorem, 157, 198 graded commutator, 292 group algebra, 125, 129, 130

HA-equivalence, 81 Hahn–Banach Theorem, 96 element, 298 HL-equivalence, 94 embedded bornological subalgebra, 112 Hochschild cohomology, 334, 336 entire cyclic cohomology, 76–79, 78, 103, Hochschild homology, 101, 204, 322, 334, 253, 285, 286 336 entire growth condition, 78, 79 Hodge filtration, 336 epic, 32 holomorphic functional calculus, epimorphism, see epic see functional calculus equibounded, 18, 19 homological, 294 essential range, 62 homology, 292 exact category, 94–98, 321 homotopy, 151, 152, 153, 189–191, 253, exact chain complex, 95, 321, 323 261, 287 exact functor, 94–98, 321, 322 operator, 287 Excision Theorem, 159–173, 213 homotopy category, 189–191, 244, 293 extension, 33, 146, 180, 184, 187, 321 homotopy direct limit, 90–92, 96, 97, 216 analytically nilpotent, 117, 119 homotopy equivalence, 81 locally semi-split, 213, 216, 222 homotopy invariance, 152, 154, 155, 157, locally split, 94, 95–97 189, 198, 253, 261, 263 of bornological algebras, 113–127, homotopy inverse limit, 90 140–142 semi-split, 34, 152, 153, 157, 180, 184, idempotent, 182, 200, 201, 270, 287 187, 189, 191, 192, 198, 213, 261, index, 288, 289 inductive limit, 53–59, 65, 82–86, 91, 108, 296, 316, 321, 322 112, 116, 122, 135–142, 241, 251, split, 34, 187, 261, 309–311, 318, 324, 292 326, 327 inductive system, 53–66, 81–86, 96, 108, square-zero, 309 111, 135–142, 145, 241–244 exterior product, 173, 212, 262, 264 essentially reduced, 62, 63, 82 Fedosov product, 149, 150, 181, 194, 315, reduced, 62, 63, 85, 108, 243 315–318 integral, see Stieltjes integral filtered, 67 internal Hom, 18, 19, 30, 36, 47–50, 52, 57–59 finitely generated, 243

358

Index

inverse limit, 50–70, 51, 116, 140, 141, mapping complex, 291 303 mapping cone, 89, 212, 293, 293–297 invertible, 202, 270, 287 metrisable, see bornologically metrisable isometry, 202, 206 module, 46, 58, 193, 298–308, 319, 320 isoradial, 100, 112, 117, 117–135, 142, bimodule, 301–310 142, 250–254, 281 cofree, 306 isoradial hull, 131 free, 154, 193, 306, 312, 320, 322 projective, 321, 322, 324–326 K-homology, 277–290 stably equivalent, 326 K-theory, 120, 271, 272, 277, 278, 286, unital, 47, 304–308, 311, 321 288, 289 monic, 32 Kasparov theory, 260–265, 286–290 monomorphism, see monic kernel, 32, 33, 35, 51 lanilcur, 151, 180, 180–193, 198 Laurent polynomials, 202, 270 left multiplier, 217 Leibniz rule, 310 LF-space, 21, 22, 40 local, 89, 91, 92, 94–98, 267 local approximation property, see approximation property local Banach algebra, see locally multiplicative local cohomology, 93, 96–98, 212 local cyclic cohomology, 94, 99, 102, 103, 262–270, 286, 290 local cyclic homology, 81, 82, 94, 99, 102, 192–226, 247, 251–254, 262–272, local homology, 93 local homotopy category, 91–94, 155, 217, 255 local homotopy equivalence, 75, 87, 88, 89, 91, 94, 213, 216, 222, 244–247, 244, 246, 251, 267 localisation, 93, 95, 189–192, 199 locally contractible, 87, 88, 89, 91, 244 locally dense, 28, 29 locally multiplicative, 108–126, 109, 131, 137–142, 188, 212, 241, 247, 256 locally multiplicatively convex, 109 locally nilpotent, see pro-nilpotent locally separable, 70, 71, 84 locally split-exact functor, 95

nilpotent, 146, 152 norm, 107, 128, 129 differential, 128 nuclear, 40, 70, 122, 205, 252, 273, 274, 279–281 null-sequence, 17, 30, 37, 65, 83 one-point compactification, 301 operator norm, 14 opposite algebra, 200, 300 partial algebra, 242, 250 partial isometry, 202 periodic cyclic cohomology, 103, 154, 155, 160, 277, 278, 335 periodic cyclic homology, 101, 154, 155, 157–159, 160, 173, 270, 277, 278, 335 Pimsner–Voiculescu Sequence, 266 pointed category, 32, 33 power-bounded, 106, 113, 116, 136, 139, 141, 180, 181 precompact, see bornologically precompact, see bornology, precompact pro-algebra, 145, 145–153 pro-nilpotent, 146, 148–153, 157 pro-tensor algebra, 149, 151–154 projective limit, 53, 116, 146, 159 projective resolution, 321, 322–327

Index

359

projective system, 53, 145, 335 stable, 158, 206, 332 pronilcur, 148–152 Stieltjes integral, 41–44, 42 Puppe Exact Sequence, 210, 213, 216, 217, strong local approximation property, see 267, 295 approximation property strongly spectrally invariant, 128, 254 quasi-free, 103, 152, 153, 155, 157, 187, subalgebra, 146 202, 204, 214, 246, 336 subcomplete, 32, 65, 66, 70–72, 83 quasi-isomorphism, 95 submultiplicative, 106, 108, 111, 130, 131 quotient, 112, 116, 138, 140, 141, 146 summable, 274, 276 suspension, 291 real-analytic function, 22, 23, 61, 98, 101, symmetric monoidal category, 44, 44–50, 103, 122–125, 133, 204, 253, 341 57, 93, 145, 158, 206, 270, 297–308 reduced inductive system, 251 symmetric monoidal functor, 50, 58, 59 relatively compact, see bornologically relatively compact tensor algebra, 200, 241–243, 314, resolution, 321 315–320, 327, 338 analytic, 116 Schatten ideal, 158 tensor product, 36–52, 65, 66, 71, 112, Schwartz space, see bornology, 138, 140–142, 146, 151, 173, 188, precompact 212, 302, 304, 306, 318–319, 332 section, 34 balanced, 304, 305, 307–308, 322 semi-norm, 13, 14 complete projective bornological, 36, closed unit ball, 13 36–50, 65, 66, 71, 119, 280 gauge, 13 complete projective topological, 37–50, separable, 70 71 separated quotient, 31, 33, 56, 57, 64, 65 inductive topological, 39, 40, 48 separately continuous map, 21, 22 projective bornological, 36, 36–50, 112, sequentially dense, 27 116, 119 Silva algebra, 130 projective topological, 48 Silva space, 25, 69, 78, 97, 98, 134, 135, Toeplitz algebra, 127, 202–204, 254, 254 263–264 small curvature, 247, 250 smooth, 21, 22, 23, 40, 61, 68, 98, 101, topological vector space, 14, 16–22, 48, 66, 146 103, 120, 123–125, 127, 133, 180, 189, 204, 225, 226, 249, 253, 287, trace, 274–278, 307 translation, 291 288, 303, 341 triangle, 294 group action, 125 exact, 294 subspace for a group action, 125, 126, functor, 294 254 mapping cone, 294 spectral radius, 106–142, 106, 136, 180, 185, 247, 250 uniformly bounded, see equibounded spectral triple, 285, 286 uniformly continuous, 22, 23 spectrum, 109, 115, 117, 122, 182 square-zero, 152 uniformly dense, 28, 29, 73

360

Index

uniformly dense range, 141 unit, 46 unit object, 57, 93 Universal Coefficient Theorem, 102, 266–270

X-complex, 154, 193, 195–197, 246, 336, 336–338 zero morphism, 32 zero object, 32