Michel Enock Jean-Marie Schwartz
. s·p·r inger-Verlag
Anthropomorphic carving representing Duality (Totonac culture)
Michel Enock Jean-Marie Schwartz
Kac Algebras and Duality of Locally Compact Groups Preface by Alain Cannes Postface by Adrian Ocneanu
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest
Michel Enock Jean-Marie Schwartz CNRS, Laboratoire de Mathematiques Fondamentales Universite Pierre et Marie Curie 4 place Jussieu F-75252 Paris Cedex 05, France
The sculpture reproduced on cover and frontispiece is exhibited at the
Museo de antropo/ogia de Ia Universidad Veracruzana, Jalapa, E. U. de Mexico
Mathematics Subject Classification (1980): 22025,22035, 43A30, 43A65
iSBN 3-540-54745-2 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-54745-2 Springer-Verlag NewYork Berlin Heidelberg
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To Professor Jacques Dixmier "What's the matterY" Macbeth (11,2)
The question is the story itself, and whether or not it means something is not for the story to tell. Paul Auster (City of glass)
Preface
This book deals with the theory of Kac algebras and their duality, elaborated independently by M. Enock and J .-M. Schwartz, and by G.I. Kac and L.I. Vajnermann in the seventies. The subject has now reached a state of maturity which fully justifies the publication of this book. Also, in recent times, the topic of "quantum groups" has become very fashionable and attracted the attention of more and more mathematicians and theoretical physicists. One is still missing a good characterization of quantum groups among Hopf algebras, similar to the characterization of Lie groups among locally compact groups. It is thus extremely valuable to develop the general theory, as this book does, with emphasis on the analytical aspects of the subject instead of the purely algebraic ones. The original motivation of M. Enock and J .-M. Schwartz can be formulated as follows: while in the Pontrjagin duality theory of locally compact abelian groups a perfect symmetry exists between a group and its dual, this is no longer true in the various duality theorems ofT. Tannaka, M.G. Krein, W.F. Stinespring ... dealing with non abelian locally compact groups. The aim is then, in the line proposed by G.I. Kac in 1961 and M. Takesaki in 1972, to find a good category of Hopf algebras, containing the category of locally compact groups and fulfilling a perfect duality. It is natural to look for this category as a category of Hopf-von Neumann algebras since, first, by a known result of A. Weil, a locally compact group G is fully specified by the underlying abstract group with a measure class (the class of the Haar measure), and, second, by a result of M. Takesaki, locally compact abelian groups correspond exactly to co-involutive Hopf-von Neumann algebras which are both commutative and cocommutative. A co-involutive Hopf-von Neumann algebra is given by a morphism r : M-+ M ® M of a von Neumann algebra M
VIII
Preface
in its tensor square M ® M and a co-involution "' which together turn the predual M* into an involutive Banach algebra. A Kac algebra is a co-involutive Hopf-von Neumann algebra with a Haar weight, i.e. a semi-finite faithful normal weight on M which is left-invariant in a suitable way. In this book, the theory of Kac algebras and their duality is brought to a quite mature state, relying a lot on the modular theory of weights developped also in the seventies. The resulting category of Kac algebras fully answers the original duality problem, but is not yet sufficiently non-unimodular to include quantum groups. This of course opens a very interesting direction of research, undertaken recently by S. Baaj and G. Skandalis.
Paris
Alain Connes
Table of Contents
Introduction
1
Chapter 1. Co-lnvolutive Hopf-Von Neumann Algebras
7
1.1 1.f
1.9
1..4 1. 5 1.6
Von Neumann Algebras and Locally Compact Groups . . . . . . Co-Involutive Hopf- Von Neumann Algebras . Positive Definite Elements in a Co-Involutive Hopf- Von Neumann Algebra Kronecker Product of Representations Representations with Generator Fourier-Stieltjes Algebra
8 13
19 23 30
36
Chapter 2. Kac Algebras . .
44
JU An Overview of Weight Theory f.f Definitions . . . . . . . . f.9 Towards the Fourier Representation f.4 The Fundamental Operator W f.5 Haar Weights Are Left-Invariant . f. 6 The Fundamental Operator W Is Unitary f. 7 Unicity of the Haar Weight . . . . .
45 55
58
60 66
71 76
Chapter 3. Representations of a Kac Algebra; Dual Kac Algebra . . . . . . . . . .
9.1 9.2 9.9
The Generator of a Representation . . The Essential Property of the Representation The Dual Co-Involutive Hopf- Von Neumann Algebra 9.4 Eymard Algebra . . . . 9.5 Construction of the Dual Weight 9.6 Connection Relations and Consequences 9. 7 The Dual K ac Algebra . . . . . . .
83 84
>.
89 92
97 101 104 111
X
Table of Contents
Chapter 4. Duality Theorems for Kac Algebras and Locally Compact Groups . . . . . . .
4.1
Duality of Kac Algebras . . . . . . . Takesaki's Theorem on Symmetric Kac Algebras Eymard's Duality Theorem for Locally Compact Groups . . . . 4-4 The Kac Algebra K 8 (G) . . . . 4.5 Characterisation of the Representations and Wendel's Theorem . . . . . . 4.6 Heisenberg's Pairing Operator . . . . 4. 7 A Tatsuuma Type Theorem for Kac Algebra 4.2 4.9
124 125 130 136 140 144 152 158
Chapter 5. The Category of Kac Algebras
161
5.1 5.2 5.9 5.4 5.5 5. 6
162 166 172 174 176 184
Kac Algebra Morphisms l!ll-Morphisms of Kac Algebras . . . Strict l!ll-Morphisms . . . . . . . Preliminaries About Jordan Homomorphisms Isometries of the Preduals of K ac Algebras Isometries of Fourier-Stieltjes Algebras
Chapter 6. Special Cases: Unimodular, Compact, Discrete and Finite-Dimensional Kac Algebras
192
6.1 Unimodular Kac Algebras 6.2 Compact Type Kac Algebras 6.9 Discrete Type Kac Algebras 6.4 Krdn's Duality Theorem . 6.5 Characterisation of Compact Type Kac Algebras 6.6 Finite Dimensional Kac Algebras
193 197 208 213 219 232
Postface
243
Bibliography
245
Index
255
Introduction
In the early nineteen thirties, L.S. Pontrjagin ([121]) established his famous duality theorem for abelian locally compact groups: he showed that the set of characters of an abelian locally compact group G is an abelian group, called the dual group of G, and noted G, which is locally compact for a suitable topology. Moreover, the bidual GAA is canonically isomorphic and homeomorphic to G; the Fourier transform carries the convolution algebra L 1 ( G) into the multiplication algebra L 00 ( G); conversely, the Fourier inverse transform carries the convolution algebra L 1 ( G) into the multiplication algebra L00 (G). These two transformations are transposed of each other in the following scheme: Fourier Fourier inverse
Since this fundamental result, which can be considered as one of the founding theorems of abstract harmonic analysis, a series of duality theorems for nonabelian locally compact groups has been gradually set up, but these dualities were not formulated within the category of locally compact groups. In 1938, T. Tannaka ([166]) proved a duality theorem for compact groups, involving the irreducible representations of a compact group G (one should bear in mind, that, in the case of an abelian locally compact group, the characters are the irreducible representations); although no group-like structure is to be put on that class (in particular, the irreducible representations being finite-dimensional, the tensor product of two irreducible representations is still finite-dimensional, but may no longer be irreducible), Tannaka showed that it is possible to recover the compact group G from the set of its irreducible representations. In 1941, M.G. Krein ([82]) obtained an equivalent result, and, in 1949, ([83], (84]), he took the system of matrix algebras generated by these representations as the dual of the compact group G, and, without invoking the group G, gave an intrinsic description ("matrix block algebras") of these objects.
2
Introduction
The next step was done in 1959 by W.F. Stinespring's duality theorem for unimodular locally compact groups in [148]. There, Stinespring emphasizes the crucial role played by the unitary operator Won L 2 (G X G) defined by:
(Wf)(s,t)
= f(s,st)
(s,t E G;
f
E
L 2 (G x G))
and by the left regular representation >.a of G on L 2 (G), which are linked by: (s E G) W*(>.a(s) ® 1)W = >.a(s) ® >.a(s) The operator W allows us then to define a Hop£ structure on the von Neumann algebra .C(G) generated by the representation >.a. Stinespring showed how it is possible to recover the group G from this Hop£ structure. One must notice that this unitary W defines, too, a coproduct on L 00 (G) by the formula: (W(1 ® f)W*)(s, t) = f(st) (s, t E G; f E L 00 (G)) Then, in 1961, G.l. Kac ([66], [70]) introduced the notion of "ring-group", closely related to Hop£ algebras; he gave a satisfactory abstract framework to Stinespring's results, and was again in a situation similar to Pontrjagin's theorem, where the initial objects and their duals are both of the same kind. These "ring-groups" generalize the algebra L 00 (G), for a unimodular locally compact group G, as well as the von Neumann algebra .C(G), these two objects, for a given G, being dual to each other. In 1964, P. Eymard ([46]), at last, gave a duality theorem valid for every locally compact group, even non-unimodular: taking an arbitrary pair f, g in the Hilbert space L 2(G), the convolution product f * gP (where gb(s) is equal to g(s- 1 ) for alls in G) belongs to L 00 (G), and the functions ofthat type form an involutive algebra A( G), called the Fourier algebra of G. With a suitable norm, it becomes an involutive Banach algebra, whose dual is the von Neumann algebra .C(G). Moreover, the spectrum of A( G) is equal to G. In order to explain the importance of that theorem, let us assume again that G is abelian. Then, the Fourier-Plancherel transform carries respectively L 2 (G) onto L 2 (G), the left regular representation of L 1 (G) onto the Fourier tranform, and the von Neumann algebra .C( G) onto the von Neumann algebra L 00 (G), acting by multiplication on L 2 (G). So, the predual A( G) is isomorphic to the convolution algebra L 1 (G), whose spectrum is, by definition, the bidual a~~. By dealing with this particular case, Eymard's theorem allows us to recover Pontrjagin's duality theorem. In the general case, Eymard's theorem brings us to write down the following scheme, which is a generalization of the abelian case: .Xa
L 1 (G) - - - - - - + .C(G)
L00 (G)
Gelfand transform ~---------
A( G)
Introduction
3
In 1967, N. Tatsuuma ([167], [168]) gave another duality theorem for arbitrary locally compact groups, recovering G, not as Eymard did, only from the left regular representation of G, but, in the spirit of Tannaka, from all the unitary representations of G. In [44], J. Ernest showed how Tatsuuma's theorem was closely related to the Hop£ structure of the algebra £(G) introduced by Stinespring, and to the Hop£ structure of the enveloping algebra W*(G) that he introduced in [43] and [44]. In 1968, M. Ta.kesa.ki, using both Kac's work and these recent duality theorems, gave a characterization of group algebras: every co-involutive commutative von Neumann algebra with a left invariant measure corresponds to a locally compact group ([157]). This crucial result, making the link between the abstract constructions of Kac and the duality theorems for locally compact groups, is actually the keystone of the theory. Afterwards, Takesa.ki endeavoured, in 1972 ([159]), to complete Kac's programme for non-unimodular groups, i.e. to establish a general duality theory which would work inside a wider category than the one of locally compact groups, and would restrict itself to the Eymard-Tatsuuma-Emest theorem for locally compact groups. Due to the incompleteness of non-commutative integration theory at that moment, he did not reach a perfect symmetry. This was done in 1973, independently by L.I. Va.lnermann and G.I. Kac ([170], [179], [180]), and the authors ([34], [35], [36]) who named "Kac algebras" this wider category, because of Kac's 1961 basic work in that direction. The duality obtained may be sketched in the following scheme:
where M and M are von Neumann algebras, their preduals M* and M* being involutive Banach algebras, and .X and Xnon-degenerate faithful representations. Many improvements were made, by E. Kirchberg ( [78], [79]), J. De Canniere ([18]), the authors ([135], [136], [38], [40]), J. De Canniere and the authors ([21], [22]). In particular, was done in [22] the generalization of M.E. Walter's work, who had noticed ([193], [194]) in 1970, that the Banach algebra A( G) characterizes the group G (although A( G) is always abelian, even when G has not the same property), exactly like L 1 (G) does ([198]), and had strongly indicated these two results should be just two particular cases of a stronger one (see also Akemann-Walter [2]). As of now, the theory of duality of Kac algebras, and the links with duality of locally compact groups, seem to have reached a state of maturity, which is described in the present book. Surveys of Kac algebras theory may be found in [45], [20], [139] and [175]. For a historical view of abstract harmonic analysis, we refer to J.-P. Pier ([117]).
4
Introduction
In the theory of Kac algebras, the crucial idea is the role played by the coproduct on L 00 ( G); it is well known that, for a locally compact semi-group, the product (s, t) --.. st is completely given by all the functions on G X G defined by r(f): (s, t)--.. f(st), for all fin a suitable space of functions on G (which separates the points of G, as, for example, C0 (G) or L 00 (G) for a suitable measure on G); if G is a group, the inverses --.. s- 1 will be given by all the functions ~(f) : s--.. f(s- 1 ), for all fin C0 (G) or L 00 (G). Moreover, it is known, thanks to a A. Weil's result ((197]), that the topology of the locally compact group G is completely given by its Haar measure. These simple remarks show how, on L 00 ( G), the coproduct, the co-involution, and the Haar measure give back the group G, both algebraically and topologically. Many important results have been obtained about actions of Kac algebras on von Neumann algebras and crossed-products (i.e. non-commutative dynamical systems) ((33], (37], (39], (103], (61], (106]) and are to be linked to the numerous works which have been made about duality and crossed products ((161], (156], (151], (102], (58], (88], (89], (188], (132], (190], (104], (153], (187], (77], (134], (139], (90], (60]). In (37], the authors have proved a duality theorem for Kac algebras crossed-products, which generalized Takesaki's theorem ((161]) about crossed-products by group actions. In (107], (108], (60], (61], (106] is developped, by A. Ocneanu and others, a Galois theory for inclusions of von Neumann algebras, in which Kac algebras play a crucial role; are obtained intrinsic characterizations of crossed-products by a Kac algebra in term of position of the initial algebra in the crossed-product. More precisely ([106]), if Mo C M1 is an inclusion offactors, with a faithful semi-finite normal operator valued weight from M1 to Mo, then M1 is the crossed product of Mo by an outer action of some Kac algebra, eventually twisted by a cocycle, if and only if the relative commutant Mb n M1 is C and Mb n M3 is a factor (where Mo C M1 C M2 C M3 C ... is the canonical tower of von Neumann algebras constructed from the initial inclusion); these results open a wide range of opportunities for further study of Kac algebras. Other directions have been studied: amenability of Kac algebras ((112], (192], (41], (115], (116])); C*-Kac algebras ((111], (96], (187], (4], (42]), and actions of C*-Kac algebras ((187], (3], (4]). Precise examples of Kac algebras which are neither abelian, nor symmetric (i.e. which are neither groups, nor group duals) have been found and studied in (73], (74], (75], (19], (97], [4]. To exhibit and classify these examples, even in the finite-dimensional case, is a difficult problem, which is far from being solved. We refer to (4] for recent researches on that question. This theory, which remained in a kind of shade for some years, is nowadays valuable to more and more mathematicians, first of all those who want to give an even more general framework for duality theory, for instance Yu. M. Berezanskil ([6], [7], [8], [9]), L.l. Valnerman ([182], [171], [172], (173], (174], (175], (181], [176], (178]), S.L. Woronowicz ([200], (201], (202], (203], (120], (204], [205], (206]), A. Ocneanu ((107], (108], (109]), S. Baaj and G. Skandalis
Introduction
5
([3], [4]), M.E. Walter ([195], [196]) or S. Doplicher and J. Roberts ([26], [27]) (we apologize to the others when we may have ommitted them). In the early eighties, many theoretical physicists from the Leningrad school introduced the notion of quantum group, as it appeared in the quantum inverse scatterring transform method ([144], [140], [141], [143], [165], [85], [87], [137]). A quantum group is generally considered to be mean a Hopf algebra obtained from a Lie group by deforming the envelopping algebra of its Lie algebra, in a way to obtain a non-commutative algebra. Historically, the first example of such a construction is to be found in [73], where Kac and Paljutkin gave a non-trivial infinite dimensional example of a Kac algebra. These ideas were developped in ([86], [142], (30], [63], (64]), and the link was made with Lie bialgebras ([10], [5], [28], [29], [51]), mostly by V.G. Drinfel'd. This theory has become very fashionable, thanks to V.G. Drinfel'd's lecture at the ICM-86 ([31]), and his Fields medal at the ICM-90, and many mathematicians are nowadays involved in quantum groups, from Yang-Baxter equation ([52], [53], (54], [55]), Poisson-Lie groups ([91], [138]), studying representations of quantum groups ([92], [93], [100], [101], [128], [80], [81]), or making the link with a geometrical point of view ([99], [129], [169], [164], [123], [17]). The non-commutative function algebras on quantum groups are studied in ([183], [184], [185], [145], [186], [146]). Other recent works are (32], [98], [122], [14 7], [119]. Constructing such objects in a C* -algebra setting has been done, mostly by S.L. Woronowicz ([200], [201], [202], [203], [204], [205]); related works has been made by M.A. Rieffel ([125], [126]), A. Van Daele ([189]), P. Podle5 ([118]), P. Podle5 and S.L. Woronowicz ([120]), S.L. Woronowicz and S. Zakrzewski ([206]), I. Szymczak and S. Zakrzewski ([155]). M. Rosso has shown ([127], [130]) the link between Drinfel'd's work and Woronowicz's "pseudo-groups" (see also J. Grabowski [50]). For a survey of quantum group theory, we refer to [31], [191], [13], [131]. Let us now describe briefly the link between Kac algebras and Woronowicz's point of view. Roughly speaking, in Woronowicz's "compact matrix pseudo-groups" ([201], [205]), the co-involution K satisfies weaker axioms: in Kac algebras, it is involutive (i.e. K( x*) = K( x )* for all x in M) and is an involution (i.e. K 2 (x) = x for all x in M); in Woronowicz's algebras, K is defined only on a dense sub-involutive algebra A and satisfies K(K(x)*)* = x for all x in A. Woronowicz then obtains a left-invariant state, i.e. satisfying (i ®
6
Introduction
More generally, if a coproduct ron a von Neumann algebra Misgiven by a unitary W by the formula:
T(x)
= W(l ® x)W*
as it is the case for L 00 ( G), then, the co-associativity of r leads to a pentagonal relation for W, which occurs both in Kac algebras and in pseudo-groups. This relation is the starting point ofS. Baaj and G. Ska.ndalis ([4]), where such unitaries are called "multiplicative". They define supplementary conditions to get "Kac systems", where duality theorems are proved. Both Kac algebras and Woronowicz's pseudo-groups are Kac systems, and, within Kac systems, pseudo-groups are characterized. So, the close relationship between Kac algebras, Woronowicz's work and quantum groups appears more and more often. We do hope our work will be useful for researches in that direction. We would like to express our profound gratitude to Jacques Dixmier, who oriented us to this subject and who always provided us with true help and support. Special thanks are due to Alain Connes, who kindly accepted to write the preface of this book. We are indebted to Masamichi Takesaki, whose work is the founding stone of the present one, and who gave us permanent encouragement. Thanks also to Adrian Ocneanu, who contributed most to bring back Kac algebras into bright light, and wrote a very encouraging postface. We are thankful to many other colleagues for fruitful exchanges, specially to Franf$ois Combes for so many substantial conversations, and to the C.N.R.S. to which we both belong and which made this research possible. Last but not least, we are obliged to Ms. C. Auchet, who processed the typescript.
Chapter 1 Co-Involutive Hopf-Von Neumann Algebras
This chapter is devoted to the structure of co-involutive Hopf-von Neumann algebras, which has been introduced by Ernest ([44]), and mostly studied by Kirchberg ([79]), and de Canniere and the authors ([21]). The paradigm, from which the whole theory comes, is the algebra L 00 (G) of all the (classes of) essentially bounded measurable (with respect to a left Haar measure) complex valued functions on a locally compact group G, equipped with a coproduct and a co-involution, which are nothing but the duals of the usual product and involution of the involutive Banach algebra £ 1 (G) of all (classes of) integrable (with respect to a left Haar measure) complex valued functions on G (let us recall that L 1 (G) is the predual of L 00 (G)). Other examples will be found later: the von Neumann algebra of G (1.6.8), the von Neumann algebra generated by the left regular representation of G (9.9.6), the dual of the involutive Banach algebra M 1 (G) of bounded measures on G (4.4.1 ). A co-involutive Hopf-von Neumann algebra (1.2.5) is a von Neumann algebra M, equipped with a coproduct, i.e. a normal injective unital morphism r from M to the von Neumann tensor product M ® M, which is co-associative, i.e. which satisfies: and a co-involution "' satisfying:
where ~(a® b)= b ®a for any elements a, b of M. The predual M* is then, in a natural way, an involutive Banach algebra, and so we may study the theory of continuous positive forms on it (called positive definite elements in M ( 1. 9.1 ) ) and the theory of representations of M •. In the case of L 00 (G), we recover the usual positive definite functions on G (1.9.11) and the unitary representations of G (1.1.6). Convenient morphisms (11-morphisms) for co-involutive Hopf-von Neumann algebras are defined and allow us to define a category (1.2.6).
8
1. Co-Involutive Hopf-Von Neumann Algebras
A crucial result is the construction of the Kronecker product of two representations, due to Kirchberg (1.4.2), which, in the case of L 00 (G), is the tensor product of unitary representations of G (1.4. 7) and may be considered as an important and useful tool for the general theory as well. The Kronecker product enables us to put, on the enveloping von Neumann algebra of M., almost a structure of co-involutive Hopf-von Neumann algebra (eventually, we only miss the coproduct being non-degenerate; (1.6.5) and ( 1. 6. 6) ). So, its predual (which is, too, the dual of the enveloping C* -algebra of M.), is therefore an involutive Banach algebra, called the Fourier-Stieltjes algebra (1.6.9); it has a faithful representation in M, called its FourierStieltjes representation. In the case of L 00 ( G), we get B( G), the FourierStieltjes algebra of G, studied by Eymard ((46]), and the canonical imbedding of B(G) into L 00 (G). It is well known that non-degenerate representations of L 1 ( G) are given by unitary representations of G. A similar situation, for a general co-involutive Hopf-von Neumann algebra, is given by the "representations with generator", which, after Kirchberg, are studied in 1.5. The particular case of finite-dimensional representations is especially studied, and will be useful in Chap. 6.
1.1 Von Neumann Algebras and Locally Compact Groups 1.1.1 Notations and Definitions. We refer to the usual text books about operator algebras ((105], (12], (24], (25], (133], (124], (113], (152], (162]); here follow some basic facts about C* -algebras and W* -algebras: (i) A C*-algebra A is a complex Banach algebra, equipped with an antilinear involution x --+ x* such that (xy)* = y*x* and llx*xll = llxll 2 for all x, y in A. Let H be a Hilbert space; we shall denote by C(H) the involutive Banach algebra of all bounded linear operators on H equipped with the norm: llxll the involution X
--+
= sup{!lxell,
eE H,
11e11 S 1}
x* being defined by, for all
(x*e I7J)
e. 7J in H:
= ce IX7J)
Then C(H) and all its norm-closed involutive subalgebras are C*-algebras; conversely, by Gel'fand-Na.lmark's theorem, for every C* -algebra, there exists a Hilbert space H such that A is isomorphic to a C*-subalgebra of C(H). A W* -algebra M is a C* -algebra which is the dual of some Banach space; then, the above Banach space is unique, called the predual of M, and denoted
1.1 Von Neumann Algebras and Locally Compact Groups
9
M •. The algebra .C(H) is a W*-algebra, its predual .C(H). being the space of all trace-class operators, that is the set of elements of .C( H) such that ltl = (t*t) 112 satisfies tra(ltl) < +oo (where trn is the canonical trace on .C(H)), equipped with the norm t--+ trn(ltl), whose dual is then the set of all the linear forms t--+ tra(xt), for all x in .C(H), and may be identified, as a Banach space, with .C(H). The topology u(.C(H),.C(H).) is called the ultraweak topology. A von Neumann algebra M on H is an involutive subalgebra M of .C(H), ultra-weakly closed, containing 1. It is easy to show that it is then a W*-algebra, the predual of M being then the Banach space of all the ultra-weakly continuous linear functionals on M. Conversely, by Sakai's theorem, for every W* -algebra, there exists a Hilbert space H such that M is isomorphic to a von Neumann algebra on H. (ii) In all what follows, M will denote a von Neumann algebra on a Hilbert space H, and M. its predual, Z(M) its centre, M+ (resp. M't) the positive part of M (resp. M.), M+l (resp. M't 1 ) the subset of M+ (resp. M't) composed of the elements of norm one. For all w in M. and x in M, to emphasize the fact that M is the dual of M., we shall often write {x,w) instead of w(x). H y is in M and win M., we define w, y ·wand w ·yin M., for x in M, by: w(x) = w(x*)-
{x, y · w) = {xy,w) (w · y)(x) = w(yx)
He,"' are in H, we shall denote by ile,TJ the linear form on .C(H) defined by: {x, ne.TJ) = (xe I'll) and ile instead of ile,e. H M is a von Neumann algebra on H, ~e shall still write ne.TJ instead of ne.TJ IM, if there is no confusion. We have ne.TJ = nTJ.e· For any subset S of .C(H), we define the commutant S' by:
S'
= {x,
x E .C(H), xy
= yx for ally inS}
which is a von Neumann algebra on H. Then, by von Neumann's theorem, S" is the von Neumann algebra generated by S, that is the intersection of all von Neumann algebras on H containing S. H Tis a closed operator on H, we say that T is affiliated to M if uTu* = T for all unit aries in M'. So, thanks to von Neumann's theorem, if Tis bounded and affiliated to M, it belongs to M. If P is a projection in H, belonging to M, the reduced von Neumann algebra Mp is the set of all x in M such that Px = xP = x, and is a von Neumann algebra on the Hilbert space PH; its commutant will be the induced von Neumann algebra (M 1 )p, whose elements are all the restrictions to PH of elements of M 1• H Mi (i = 1,2) is a von Neumann algebra on a Hilbert space Hi, the von Neumann tensor product M1 ®M2 is the von Neumann algebra on the tensor
10
1. Co-lnvolutive Hopf-Von Neumann Algebras
product Hilbert space H1 ® H2 generated by the algebraic tensor product M1 0 M2. The algebraic tensor product (Ml)* 0 (M2)* is then norm dense in the predual (M1 ® M2) •. Moreover, by Tomita's theorem, we have:
(M1 ® M2)' = Mf ® M~ (iii) Let M be a von Neumann algebra on H; M is said to be standard in H ((57]) if there exists an antilinear bijective isometry J : H -+ H and an autopolar convex cone 'P in H such that:
JMJ=M 1
J
e= efor all ein 'P
xJxJ'P C 'P for all x in M
We shall say, eventually, that the triple (H, J, 'P) is standard forM, or that (H, J) is standard for M. For example, the von Neumann algebra £(H), identified to £(H)® Cfl, is standard in H ® fi (where fi is the conjugate Hilbert space of H), thanks to the antilinear isometry J from H ® fi to H ® fi defined by J ( ® Tj) = rJ ® { ( rJ E H) and the closed convex cone 'P generated by {e ® {;e E H}. In an obvious way, the von Neumann algebra ®ieiC(Hi) is standard in ®iei(Hi®fii)· Thanks to Haagerup's result, every von Neumann algebra can be represented on a Hilbert space in which it is standard, and, then, two technically very useful events occur: first, for all w in M., there exist rJ in H such that w = il{,,.,l M. We shall write then we, 11 instead of ile, 11 1 M, if there is no confusion. Secondly, every automorphism a of M is actually implemented by a unique unitary u of £(H) (that is a(x) = uxu* for all x in M) such that uJ = Ju and u'P C 'P. Then u is called the standard implementation of a.
e
e,
e,
1.1.2 Notations and Definitions. In all the chapters, G will denote a locally compact group; we refer to the usual text books about locally compact groups ((197], (11], (62], (12], (25]). We shall denote by Cb(G) (resp. Co(G), resp. K( G)) the algebra of bounded continous complex valued functions on G (resp. vanishing at the infinity, resp. with compact support). Let ds be a left Haar measure on G. We shall write L 2 (G) for the Hilbert space of all (classes of) square-integrable measurable (with respect to ds) complex-valued functions on G. Let L 00 ( G) denote the (abelian) W* -algebra of all (classes of) essentially bounded measurable (with respect to ds) complex-valued functions on G. Let us recall that the predual of L 00 ( G) is isomorphic to the Banach space L 1 (G) of all (classes of) ds-integrable functions over G. The essential results we shall need are recalled thereafter: (i) The von Neumann tensor product L 00 (G) ® L 00 (G) may and will be identified with L 00 (G x G). For every fin L 00 (G), we define a two variables function Fa(!): Fa(f)(s, t) = f(st) (s, t E G)
1.1 Von Neumann Algebras and Locally Compact Groups
11
Thanks to the above identification, Fa(!) belongs to L=(G)®L=(G). Along the same way, we define Ka(/) by the equality:
(s E G) Evidently, Fa and Ka are linear, multiplicative and preserve the adjoint. (ii) The Banach space L 1 ( G) is a Banach involutive algebra, when equipped with the convolution product * and the involution ° respectively defined by,
f*g(t) = fa!(s)g(s- 1t)ds
res)= Lla(s- )/(s1
(f,g E L 1 (G), s,t E G)
1)
(! E L 1 (G), s E G)
where Lla is the modular function on G. It is known that L 1 (G) has a bounded approximate unit, and has a unit if and only if G is discrete; the unit of L 1 (G) is then the characteristic function of {e}, where e is the unit of G. (iii) Let M 1 (G) be the Banach space of bounded measures on G, which is dual to the Banach space Co( G). Then, M 1 (G) is a Banach involutive algebra, when equipped with the convolution product * and involution ° defined by, for m, n in M 1 (G), f in Co( G), where s, t belong to G: m
* n(f) = f
lax a
m 0 (/)
f(st)dm(s)dn(t)
= (faJCs- 1 )dm(s))-
It is known that L 1 (G) may be considered as a closed subspace of M 1 (G), a function f of L 1 (G) being identified with the bounded measure f(s)ds. Under that identification, L 1 (G) is a two-sided ideal of M 1 (G), and is equal to M 1 (G) if and only if G is discrete. It is known that L 1 (G) and M 1 (G) are abelian if and only if G is abelian.
1.1.3 Proposition. The morphisms Fa and Ka defined in 1.1.2 verify, for any
f in L=(G), h, k in L 1 (G): (i) (ii)
(Fa(/), h ® k)
=
Ia
f(t)(h
* k)(t)dt
{KaCf),h)- = fa!(s)h 0 (s)ds
So, Fa and Ka are normal.
12
1. Co-Involutive Hopf-Von Neumann AlgebrBB
Proof. Let have:
f
be an element of L 00 (G), hand k be two elements of L 1 (G). We
fa f(t)(h
* k)(t)dt =
fa f(t) (fa h(s )k(s- 1 t)ds) dt
= [
f(t)h(s)k(s- 1t)dsdt
=
f(st)h(s)h(t)dsdt
lax a [
lax a =
[
lax a
Fa(f)(s, t)(h ® k)(s, t)dsdt
= (Fa(/), h ® k)
Since the algebraic tensor product L 1 (G) 0 L 1 (G) is weakly dense in the predual of L 00 (G)®L 00 (G), this equation ensures the normality of Fa. With the same assumptions, we have also:
which brings the normality of "-a and completes the proof. 1.1.4 Definition. Let P.a be a continuous unitary representation of G on some Hilbert space 'H. Then it is well known ([25] 13.3) that the equality:
p.(f)
= fa!(s)p.a(s)ds
defines a non-degenerate representation of L 1 (G) on 'H; we establish by this way a one-to-one correspondance between continuous unitary representations ofG on 'Hand non-degenerate representations of L 1 (G) on 'H. More generally, it is true as well that there is a one-to-one correspondance between continuous unitary representations of G and representations of M 1 (G), whose restriction to L 1 (G) is non-degenerate, given by:
1.1.5 Definition. For all s in G, we define a unitary operator ..\a( s) on L 2 ( G), by, for any fin L 2 (G), tin G:
(..\a(s)J)(t) = f(s- 1 t) It is well known that >.a is a continuous unitary representation of G, called the left regular representation of G.
1.2 Co-lnvolutive Hopf-Von Neumann Algebras
13
The corresponding representation of M 1 ( G), called the left regular representation of M 1 (G), is given, for all m in M 1 (G), f in L 2 (G), t in G, by: (.Aa(m)f)(t) =
Ia
f(s- 1t)dm
The corresponding non-degenerate representation of L 1 (G), called the left regular representation of L 1 ( G), is given, for all f in L 1 (G), g in L 2 ( G), t in G, by: (.Aa(f)g)(t) =
Ia
f(s)g(s- 1 t)ds
We shall write .C(G) for the von Neumann algebra generated by the representation .Aa of L 1 (G). Iff is in L 1 (G), and gin L 1 (G) n L 2 (G), we have: .Aa(f)g
= f *g
1.2 Co-Involutive Hopf- Von Neumann Algebras 1.2.1 Definition. A couple (M, r) is called a Hopf-von Neumann algebra if: (i) M is a W*-algebra; (ii) r is an injective normal morphism from M to the W*-algebra M®M, such that F(1) = 1®1, and which has the co-associativity property, i.e. which makes the following diagram commute:
r is then called a co-associative co-product. If (M,r) is a Hopf-von Neumann algebra, so is (M,c;F) where c;(a ®b) is equal to b ® a for any elements a, b of M. The co-product r will be called symmetric if c;F = r. For every x in M and every w,w' in M., let us write:
(x,w*w'} = (F(x),w®w 1 } This formula defines a product* on M •. As equipped, M* becomes a Banach algebra; it is abelian if and only if r is symmetric.
14
1. Co-Involutive Hopf-Von Neumann Algebras
We have then, for all w, w 1 in M*:
(w * w1 ) - =
w* w'
1.2.2 Definition ([44]). Let (M, F) be a Hopf-von Neumann algebra. The set of all invertible elements x of M such that:
F(x)
=X
0
X
is clearly a subgroup of the group of invertible elements in M, called the intrin8ic group of ( M, F).
1.2.3 Proposition ([44]). The element8 of the intrin8ic group of a Hopf-von Neumann algebra are unitarie8. Proof. If xis an element of the intrinsic group, so is x*x. Therefore, we have:
which implies llx*xll = 1. Since (x*x )- 1 belongs also to the intrinsic group, we have ll(x*x)- 1 11 = 1 as well; x*x being a positive operator, the spectral theorem then implies x*x = 1. The same arguments work for xx* = 1, which completes the proof.
1.2.4 Remark. Since the weak, ultraweak, strong and ultrastrong topologies coincide on the unitary group of a von Neumann algebra (cf. [24)), the intrinsic group of a Hopf-von Neumann algebra is a topological group for this topology. 1.2.5 Definition. A triple llli = (M, r, Yi) is called a co-involutive H opf-von Neumann algebra if: (i) (M,F) is a Hopf-von Neumann algebra (ii) Yi is an involutive anti-automorphism of M, i.e. a linear mapping from M toM such that, for every x, yin M:
ri( xy) ri(x*) ri(ri(x)) Yi
is then called a co-involution.
= ri(y )ri( x) = ri(x)*
=x
1.2 Co-Involutive Hopf-Von Neumann Algebras
15
(iii) the co-product and the co-involution are such that the following diagram commute:
It is easy to check that (ii) implies ~~:(1) = 1. If x belongs to the intrinsic group of (M,r), it is clear it is the same for ~~:(x). Let JH[ = (M,F,~~:) be a co-involutive Hopf-von Neumann algebra; then JH[' = (M,c;F,~~:) is also a co-involutive Hopf-von Neumann algebra, called the symmetric of JH[. When M is commutative (resp. r is symmetric), JH[ is said to be abelian (resp. symmetric). For every x in M and win M., let us define w 0 = w o ~~:,that is:
this formula defines an involution ° on the Banach algebra M* (1.2.1); as equipped, M* becomes an involutive Banach algebra, and we have, for all w in M.: W=W 0 0K=(wo~~:) 0
If x, yare in M, win M., we have, with the notations of 1.1.1 (ii):
and so:
1.2.6 Definition. Let JH[l = (MI.H,~~:I) and JH[2 = (M2,r2,11:2) be two coinvolutive Hopf-von Neumann algebras; a morphism from JH[l to JH[2 is a normal morphism u from M1 to M2 such that:
(i) (ii) (iii)
u(1) = 1 r2u = (u ® u)rl
Such morphisms will be called .IH[-morphisms. The class of all co-involutive Hopf-von Neumann algebras, equipped wi_th JH[-morphisms, is a category.
16
1. Co-Involutive Hopf-Von Neumann Algebras
1.2.7 Proposition. With the notations of 1.2.6, let Pu be the greatest projector of the ideal Keru, Ru = 1- Pu be the support of u; then Pu and Ru belong to the centre of Mt. We have: rt(Ru) ~ Ru ® Ru lq(Ru) = Ru .
(i) (ii}
Proof. We have: (u ® u)H(Ru)
= rtu(Ru) = 1
which proves (i). We have:
so: As
11:1
is involutive, we get (ii).
1.2.8 Notations. Let H be a Hilbert space such that M is standard in H ( cf. 1.1.1 (iii)); the co-involution"' is then implemented by an anti-linear isometric involution of H, noted .:1; that is, for every x in M, we have:
K(x) = .:Jx* .:1 Moreover, for any x in M, we have:
(x,w 3 e,311 )
= (x.:re I.:fry) = (ryi.:Jx.:re) = (K(X)'fll e) = {K(x),w 11 ,e)
= ("-(x),we, 11 o "-) =
by 1.2.5
(x,we, 11 )
which implies:
1.2.9 Example. Let G be a locally compact group. It follows from 1.1.3 that the triple (L 00 (G),ra,"-a) is an abelian co-involutive Neumann algebra. We shall denote it lH!a(G). Furthermore we have the involutive Banach algebra structure induced on the predual which is isomorphic to L 1 (G), is the usual one.
1.1.2 and Hopf-von seen that L 00 (G)*,
1.2 Co-lnvolutive Hopf-Von Neumann Algebras
17
Let G0 PP stand for the opposite group of G. As the left Haar measure on G 0 PP is nothing but the right Haar measure L1a( s )ds on G, the algebras L 00 (G0 PP) and V)O(G) are identical, and it is immediate that lHia(G0 PP) is equal to lHia (G)'. So lHia( G) is symmetric if and only if G is abelian. 1.2.10 Proposition. Let lHI = (M, r, ~~:) be a co-involutive Hopf von Neumann algebra. Let H be a Hilbert space, and J be an anti-linear bijective isometry on H such that (H, J) is standard forM {1.1.1 (iii)); then M' = J M J; let us now define, for all x in M': r'(x)
= (J ® J)F(JxJ)(J ® J) ~~: 1 (x) = J~~:(JxJ)J
then: (i) JHI' = (M', F', ~~:') is a co-involutive Hopf von Neumann algebra. (ii) the application u: x - t J~~:(x)* J from M toM' is an lHI-isomorphism from lHI' to JHI'. Proof. A straightforward calculation left to the reader gives (i); further on, we have: F'u(x) = (J ® J)T(~~:(x*))(J ® J) = ~(J ® J)(~~: ® ~~:)F(x)*(J ® J) = (u ® u)~F(x)
and:
~~: 1 u(x)
= J~~:(Ju(x)J)J = J~~:(~~:(x*))J = Jx*J
= u(~~:(x)) which proves (ii). 1.2.11 Lemma. Let I be a set, and, for all i in I, let di be in N, Hi be a Hilbert space of dimension di, {e}h$j$d; an orthonormal basis of Hi, e~,k the matrix units associated to these basis. We shall write H = tiJiEIHi, D the discrete von Neumann algebra (f}iEI.C(Hi), Pi the projection on Hi, which is in the centre of D. Let us suppose that there are a coproduct r and a coinvolution K such that ( D, r, K) is a co-involutive H opf-von Neumann algebra. Then: (i) for all i and j in I, there exist k1, ... , kp in I and mi,j ,k1 , •.• , mi,j ,kp inN, such that Ef=l mi,j,kldkl = didj, and a unitary ui,j from Hi® Hj to tiJt(Cm;,;,kl ® Hk 1 ) such that, for all x = tfJkEJXk in D, we have: F(x)(Pi ®Pj)
= Ui,j(tiJt(l.C(Cm;,;,k 1 )
®xk 1 ))Ui~j
18
1. Co-lnvolutive Hopf-Von Neumann Algebras
(ii) for all i in I, there exist i 1 in I such that K.(.C(Hi)) = .C(Hi' ), and di = di'· Moreover, there exists a unique unitary V; from Hi to Hi' such that, for all x in .C(Hi), we have: K.(x)
= V;xtl/i*
where xt is the element of .C(Hi) whose matrix with respect to the basis {ejh~j~d; is the transposed matrix of x. The operator Pi defined by: P.i
= dt-1 . I: e.},i k ® "' <eki ,}.) j,k
belongs to .C(Hi) ® K.(.C(Hi)) and is the projection on the one-dimensional subspace of Hi® Hi' generated by the vector I:i ej ® Viej. Proof. The support of the homomorphism X ---+ r(x)(Pi ® Pj) from D to .C(Hi ® Hj) is a finite-dimensional projection in the centre of D; therefore, there exist k1. ... , kp in I such that this support is I:f= 1 Pkr Moreover, the homomorphism xk 1 ---+ F(xk 1 )(Pi®Pj)r(Pk,) from .C(Hk,) to .C(Hi®Hj)r(Pk,) is the composition of an ampliation and a spatial isomorphism; so, there exist mi,j,k, in N, and a unitary Ui,j,k, from F(pk 1 )(Hi ® Hj) to cm;,j,k' ® Hk, such that, for any xk, in .C(Hk1), we have:
= Ui,j,k 1(1C(Cm;,j,k') ®xk 1 )U~j,k, and then, for all X = ffikeJXk in D, with ui,j = ffilUi,j,kp we have: r(x)(Pi ® Pj) = ffilF(xk,)(Pi ® Pj)r(pk,) = ui,j(ffil(lc(Cm;,j,kl) ® Xk,))Ui~j F(xk 1 )(Pi ®Pj)r(Pk,)
which gives (i). For all i in I, K.(Pi) is a minimal projection in the centre of D; therefore, it is equal to some Pi' and we have then K.(.C(Hi)) = .C(Hi' ); by dimension arguments, we get di = di'· As the mapping x ---+ K.(xt) is an isomorphism from .C(Hi) to .C(Hi' ), we get the existence (and unicity) of V;. Moreover, we have:
(E oj,,
2
® •({;)) =
},k=1
L
e},kef,m ®
j,k,l,m
e3i. kef m ® K.( e~ lei 3.)
= """" ~
j,k,l,m
=
L
K.(eL)K.(e~,l)
''
e~,m ® K.(e~,j)
j,k,m d;
= di
L j,k=1
e~,k ® K.(eL)
''
1.3 Positive Definite Elements in a Co-Involutive Hopf-Von Neumann Algebra
which shows that pi is a projection. Moreover, let in Hi ® Hi'; we have:
Pie= di 1
L
j,lc,l,m
19
e = El,m al,mef ® Vie:n
al,me~,lcef ® K(e1.J)Vie!n
-a.·(~"···) (~ei ® V;ej)
by (i)
from which we get (ii).
1.3 Positive Definite Elements in a Co-Involutive Hopf- Von Neumann Algebra In the following paragraph llll = (M,T,K) will be a co-involutive Hopf-von Neumann algebra. 1.3.1 Deftnition. An element x in M is told to be po8itive definite if it induces a positive form on the involutive Banach algebra M., i.e. if for all w in M. the number {x,w 0 * w) is positive. The set of all positive definite elements will be denoted by P(llll). 1.3.2 Lemma. Let H be a Hilbert 8pace and 3 an anti-linear i8ometric involution on H. The clo8ed convex cone c(.:J) generated in H ® H by the 8et {.:Je ® e; e E H} i8 autopolar. Proof. Let it be the conjugate Hilbert space of H. The closed convex cone generated in it ®H by {e ®e; e E H} is the autopolar cone associated to the von Neumann algebra Cil ®C(H) which is standard in it ®H (cf. 1.1.1 (iii)). It is clear, by hypothesis, that the mapping 3 ® 1 is an isomorphism from H ® H which sends the above mentioned autopolar cone onto c(.:J); that completes the proof. 1.3.3 Theorem. (i) The product of two po8itive definite element8 i8 po8itive definite. (ii) Let x in P(lll). Then K(x) = x*, and thi8 element 8till belong8 to P(lll). In other word8, every x in P(lll) i8 an hermitian form on M •. Proof. We assume that M is standard on a Hilbert space H. Then, for all x in M and all w = we,IJ in M. (withe,"' in H), we have, by 1.2.8:
{x,w0 *W) = {T(x),w.:re,.7'1 ®we,q) = (r(x)(.:Je ®e) 1.:1"1 ® "')
20
1. Co-Involutive Hopf-Von Neumann Algebras
Thus, x belongs to P(lll[) if and only if that last quantity is positive for all
e, 1J in H. Using the above lemma, by linearity and continuity, it is equivalent to suppose that:
T(x)c(..J) C c(..J) As
r is a morphism, (i) follows. Since the preceeding computation may be prolonged by:
it is clear, by virtue of the same reasoning as above, that x belongs to P(lll[) if and only if x* also does. Moreover, for x in P(lll[) we get:
0 ~ (x*,w 0 *w) = (x,w 0 *W) = (T(x),w o "'Q9w 0 o "') = (("' ® ,)r(x),w ® w0 ) = (r(,(x)),w 0 ®w)
by 1.2.5 by 1.2.5
By polarization, it implies that for all w,w' in M*, we have:
(T(x*),w ® w1) By density, we deduce that T(x*) the proof of (ii).
= (T(,(x)),w ® w1)
= r(,(x));
r being injective, it completes
1.3.4 Definitions and Notations. Let 1-' be a representation of M* over a Hilbert space 'Hil; we shall note All the von Neumann algebra generated by fl(M*) in C('Hil). If 1-'I.I-'2 are two representations of M*, we shall write Hom(fll, !-'2) the set of all intertwining operators between Ill and 1-'2· We define a linear mapping ~-'* from the predual of AI' to M by writing, for all 0 in (AI'h and w in M*:
(!-'*( 0), w) = (!-'( w ), 0) It is obvious that~-'* is norm-continuous and that 111-'*11 ~ 111-'11· It is clear that fl is non-degenerate if and only if fl(M*) is weakly dense in All and if and only if ~-'* is injective.
1.3.5 Proposition. Let 1-' be a representation of M*. For any positive element il of (AI'h, ~-'*(il) belongs to P(lll[). Proof. For all win M*, we have:
(!-'*(il),w 0 *W} = (fl(W 0 *W),il} = {1-'(w)*!-'(w),il} 2::0 which completes the proof.
by 1.3.4 by hypothesis
1.3 Positive Definite Elements in a
C~rlnvolutive
Hop£-Von Neumann Algebra
21
1.3.6 Definition. Let x in P(IEll). If there exists a representation 1-L of M. and a positive element Q in the predual of AIJ, such that x = 1-L•(Q), we shall say that x is representable. The subset of P(IEll) made of all representable elements will be denoted by P R(IEll). 1.3.7 Example. Since F(1) = 1 ® 1 and ~~::(1) = 1, it is obvious that 1 induces a one-dimensional representation on M* which shall still be noted 1; using that representation, it is trivial to show that 1 (as an element of M) belongs to PR(IEll).
1.3.8 Proposition. {i) Let x be in P(IEll); x belongs to P R(IEll) if and only if it exists a positive number p such that, for all w in M.: l(x,w}l 2 ~ p(x,w 0
* w}
More precisely, in that case, there exists a Hilbert space H, an involutive non-degenerate representation 1-L of M. on H, and a vector in H with:
e
(ii) Whenever M. has an approximate unit, we have P R(IEll) = P(IEll).
Proof. Let us assume that x belongs to P R(IEll), therefore, there exists a nondegenerate representation 1-L and Q in (AIJ)* such that x = 1-L•(Q) and for all win M., we have then: l(x,w}l 2 = I(IL•(il),w}l 2 = 1(1-L(w), il}l 2 ~ llilii(IL(w)*IL(w),il} = llilii(IL•(il),w 0 = llilll(x,wo * w}
by 1.3.4 for Q is positive
* w}
Conversely, let x be an element of P(IEll) satisfying such an inequality. Taking 1.3.3 (ii) into account, x induces a positive form on the Banach algebra obtained by adding a unit to M •. Therefore by ([124], 4.5.11 and 4.5.5), there is a Hilbert space H, a vector in H with llell 2 =panda representation 1-L of M. on H such that for every win M., we have:
e
Then, xis equal to 1-L•(ile) and so belongs to PR(IEll), and we have:
which completes the proof of (i).
22
1. Co-lnvolutive Hopf-Von Neumann Algebras
H M* has an approximate unit, it results from ([25], 2.1.5(i)) that every element in P(llll) satisfies the inequality of (i); the proof of (ii) follows.
1.3.9 Proposition. Let Jll and Jl2 be two representation, of M •. Let w be in Hom(Jll, Jl2)· For all vectors 6 in 'HIJ1 and '72 in 'HIJ2 , we have:
Proof. Let win M •. We have:
(Jl2•(nw6,1'/2),w)
= (Jl2(w), nwet,l'/2) = (Jl2(w)w61 '72) = (wJll(w)61'72) = (Jll(w)61 w*'72)
by hypothesis
= (Jll(w), ne1,w•112)
= (Jlh(nel,w•l12),w) which completes the proof. 1.3.10 Example. Let G be a locally compact group and Jl be a non-degenerate representation of L 1 (G); let JlG the corresponding continuous unitary representation of G. We have, for fin L 1 (G) and n in the predual (AIJ)*:
(Jl(f), n)
=
fa
f( s )(Jla( s ), n)ds
And so the element Jl•(n) in L 00 (G) is almost everywhere equal to the continuous function s -+ (Jla ( s)' n). 1.3.11 Theorem. Let G be a locally compact group, and f a function in L 00 ( G). Then, the following three assertion, are equivalent: (i) The function f is positive definite, i.e.
Ia
f(t)(h 0
* h)(t)dt ~ 0
for all h in L1 (G)
(ii) There exists a Hilbert space H, a vector~ in H, a continuous unitary representation JlG of G on H such that, for almost all t in G: f(t)
= (Jla(t)~ I~)
(iii) The function f is positive definite and almost everywhere equal to a continuous function.
1.4 Kronecker Product of Representations
23
Proof. Let us suppose {i); then f is in P{lla{G)); as L 1 (G) has an approximate unit, f is in PR(llla(G)), which gives {ii), thanks to 1.3.8(ii) and 1.1.4. The implications (ii) => (iii) => (i) are trivial, which ends the proof.
1.4
Kronecker Product of Representations
r,
1.4.1 Lemma. Let 111 = (M, K) be a co-involutive Hopf-von Neumann algebra. Let I be a set and, for all i,j in I, Xi,j be elements in M such that:
(i)
K(xi,j) = Xj,i
(ii)
r(xi,j) = L Xi,k ® Xk,j kEI
the sum being convergent for the ultra-weak topology. Then, for all finite subset I 0 C I, and all family (ai)ielo of complex numbers, the element L:i,j iiiajXi,j belongs to P R(lll). More precisely, there exists a representation 1-' of M. and n in (A,)t such that: and
LiiiajXi,j i,j
= 1-'•(fJ)
which implies:
L aiajXi,j ~ L i,j i
lail2.
Proof. Let w in M •. We have: (LaiajXi,j 1 W 0 iJ
* w) = Liiia;{r(xi,j),w
0
®w)
i,j
=
Liiiaj{Xi,k®Xk,j 1 W 0 ®w)
by (ii)
i,j,k
=
L
iiia;{xi,k 1 W0 ){xk,j 1 W)
i,j,k
= L
iiiaj{K(xi,k),w)-{xkJ,w)
i,j,k
= L i,j,k
iiia;{xk,i,w)- {xk,;,w)
by (i)
24
1. Co-Involutive Hopf-Von Neumann Algebras
Therefore I:i,j ii:iajXi,j belongs to P(llll). Moreover, we have: 2
(I: ii:iO!jXi,j,W)
2
= LQ:iLO!j(Xi,j,W}
i,j
j
~ (~ lail ~ ~aj(Xi,j,w) 2)
I
2
J
I
by using the Cauchy-Schwartz inequality
= (~ lail 2 )
(~ O:iO!jXi,j W * w) 1
0
1 1J
I
according to the above calculation. Thanks to 1.3.8 (i), it completes the proof.
1.4.2 Proposition ([79]). Let llli = (M, F, K.) be a co-involutive H opf-von N eumann algebra, I be a 8et, for all i, j in I, Xi,j be element8 in M and {eii i E I} the canonical ba8i8 of the Hilbert 8pace £2 (I). The following a88ertion8 are equivalent: {i) There exi8t8 a repre8entation /J of M* on £2 (I) 8uch that:
'Vi,j E I
{ii) The element8 Xi,j Mti8jy, for all i,j in I: K.(xi,j)
= Xj,i
F(x 1· J·) = ~ L...J x a,· 1c ® x1c ,J· 1
IcE I
{the 8um being ultra-weakly convergent). (The author8 are indebted to Prof8. E. G. Effro8 and Zong-jin Ruan who pointed out a mi8take in the preprint ver8ion of the following proof.) Proof. Let us assume (i). Let w be in M.; we have:
{K.(xi',j),w)
and then:
= (IJ•(fle;,eJ,w
0 }-
= (IJ(w 0 )ej lei)= (IJ(w )ei Iej) = (IJ•(fle;,e; ),w} = (xj,i,w)
1.4 Kronecker Product of Representations
25
Now, let WI,W2 be in M•. We have:
{F(xi,;),wl ®w2}
= {JL•(il€;,€i),wt *w2} = (JL(wt)JL(w2)e; Iei) = (JL(w2)e; IJL{wt)*ei)
= ~)JL(w2)ej IeA:)(eA: IJL(wl)*ei)
kEI thanks to {eii i E I} being an orthogonal basis
= L {JL.(
n€j ,€/o ),
w2} (JL.( n€J: ,ei ), Wt}
kEI =
L{xi,k ® Xk,j,Wl ®w2} kEI
Let us assume now that M is standard in a Hilbert space H, and let ~ be implemented by an antilinear isometric involution 3 of H ( cf. 1.2.8). We have then, for all in H:
e,"'
(F(xi,i)(:re ®e) I:r., ® "!) = (r(xi,i),w.1e,.1, ®we,,} =
L(xi,k ® Xk,i,w.1e,.1, ®we,,} kEI
= L(xi,k•we,,Hxk,i,we,,} kei
by 1.2.8
= L(~(xi,k)*,we,,}-(xk,i,we, 71 }
by 1.2.5
kEI = L l(xk,i,we, 71 }1 2 kEI
by the result above
The same calculation proves that, for every finite subset J if I, we have:
( ( L Xi,k ® Xk,i) (:re ®e) I:r., ® .,) keJ
=
L l(xk,i,we,,}l 2 keJ
~ L l(xk,i,we,,}l2 kEI = (F(xi,i)(:re ®e) I37] ® TJ) and, by linearity and continuity, we have, for every E 1, E2 in the closed convex cone c(:J) introduced in 1.3.2:
26
1. Co-lnvolutive Hopf-Von Neumann Algebras
As every element E in H ® H may be written as E = E1 - E2 + iEs - iE4, with En in c(:J) (n = 1, ... ,4), and E!=1 11Enll 2 = 115'11 2 , we then easily get that II E~ceJXi,lc ® x~c,ill $llxi,ill· The algebraic tensor product M. 0 M. being dense in (M ® M)., we can conclude that the sum EA:ei xi,lc ® Xk,i is ultra-weakly convergent, and is equal to r(xi,i), which, by polarization, implies (ii). Let us assume (ii). Let E be an element of c(I), which can be considered as a dense subset of £2 (1). Therefore there exists a finite subset 10 of I, and complex numbers O:i (i E 10 ) such that:
L
E=
O:iei
iEio Thanks to the calculation of lemma 1.4.1, we know that: 2
L L
o:i(xlc,i,w)
lcEl iElo
is finite and equal to:
(:?; aiO:jXi,j,W
0
* w) $
l,j
~ lo:il llw 2
0
* wll $11~ O:iei
I
llwll 2
= IIEII 2 11wll
I
Since:
L L
2
2
O:i(xlc,i,w)
=
lcEI iEio
L L o:i(XA:,i,w)elc
2
lcEI iEio
we have, by this way, defined a linear mapping ~t(w) from c(I) to £2 (1) such that:
~t(w)E =
LL
o:i(xlc,i,w)elc
lcEI iEio This mapping is continuous and such that
~t(w)ei
ll~t(w)ll
$ llwll, and we have:
= L(xlc,i,w)elc leE I
which implies: (~t(w)ei
lej) = (xj,i,w)
Thank to its continuity, it is possible to extend ~t(w) to l 2(I). For all wbw2 in M., we have:
1.4 Kronecker Product of Representations
27
(J.L(W1 *W2)ei lej) = {xj,i,W1 *W2) = {F(x;,i),w10w2)
= (L Xj,k 0
xk,i,w1 0 w2)
by hypothesis
kEI
= L{xj,k,w1){xk,i,w2) kei
=
L(J.L(wl)ek le;)(J.L(w2)ei lek) kEI
= L(ek IJ.L(wi)*e;)(J.L(w2)ei Iek) kEI = (J.L(w2)ei IJ.L(w1)*e;)
= (J.L(wl)J.L(w2)ei Ie;) Therefore: We have:
(J.L(W 0 )ei le;) = {xj,i,w 0 )
= {~~:(x;,i)*,w) = {xi,;,w)= (J.L(w)ej lei)-
by hypothesis
= (J.L(w)*ei le;) therefore J.L(w 0 ) = J.L(w)* and J.L is an involutive representation of M. on £2(1). The remaining of (i) follows immediately, which completes the proof.
r,
1.4.3 Theorem ([79)). Let J8I = (M, ~~:)be a co-involutive Hopf-von Neumann algebra. Let J.L1 and J.L2 be two representations of M*. There exists a unique representation v of M., .mch that Av C A 111 0 A 112 and that, for every il1 in (A111 )* and il2 in (A112 )., we have:
We shall write v = J.Ll X J.L2 and we shall call it the Kronecker product of J.L1 by J.L2. This operation is obviously associative. If M is abelian, this operation is commutative. Proof. Let 11 and 12 such that 'H111 and 'HI-' 2 be respectively isomorphic to £2(11) and £2(12)· Let {e~; i E h} and {e~; m E 12} be respectively the canonical basis of £2(1}) and £2(12)· Let us put, for i,j in h, and m, n in 12:
x~,3•
= J.Lt.(ilt:~J'•t:!)
and
x~ 'n
= J.l2•(ilt:2"'m t:2 )
28
1. Co-lnvolutive Hopf-Von Neumann Algebras
The Hilbert space e2(h X /2) can be identified with £2(11) 0 £2(h), its canonical basis being then {c:~ 0c:~J. Thus if we put x(i,m),(j,n) = x~,ix~,n• we have:
by 1.4.2 = x(j,n),(i,m) and: r(x(i,m),(j,n)) =
=
r(x~,j)r(x~,n)
(I:
x},k 0 xl,j)
=
(I: x~,l
0 xr,n)
lEh
kElt
"" 1 2 L.J xi,kxm,l
1 2 0 xk,jxl,n
k,l
L
=
X(i,m),(k,l) 0 X(k,l),(j,n)
(k,l)Eh x/2
We are then faced with the conditions 1.4.2 (ii) applied to £2 (h xh); therefore it exists a representation v of M* on £2(h x /2) such that: X(i m) (in) '
'
'
= v*(QE:~.o.E:2 E:~®E:2m ) J'
Using the natural identification of QE:~.o.E:2 E:~.o.E:2 with QE:~ E:~ 0 ilE:2 E:2 , we J'
Jlh(ile1,'1l)JL2*(n6,T/2)
= v*(ne1,T/1
0 n6,T/2)
Let a 1 be in A~ and a2 in A2. For all 6, 7J1 in 1iJJ1 and 6, 7]2 in 1iJJ 2 , and w in M*, we have: (v(w)(a1 0 a2)(6 0 6) 1711 0 '72) = {v(w),ila 1{t,T/1 0 ila26,712}
= {v*(ila16,T/1 0 ila26,712),w) = {JLh(Qa16 ,T/JJL2*(Qa26,T/2 ),w} = {JLh(n6,ai771 )JL2*(n{2,a2772),w) = {v*(ilet,ai711
= (v(w)(6 0
by 1.3.9
0 n6,a2712),w)
711) I (al ® a2)*(6 0 772))
from what follows that v( w) belongs to AJJ 1 0 AJJ 2 , which completes the proof.
1.4 Kronecker Product of Representations
29
1.4.4 Corollary. Let JH[ be a co-involutive Hopf-von Neumann algebra; the set P R(JH[) is stable by multiplication. Proof. With the notations of 1.3.6, let x = J.Lh(ill) and x2 = J.L2*(il2) be in P R(JH[); !11 and !12 are therefore positive. We have: X1X2 = J.Lh(.f11)!-'2*(.f12)
by 1.4.3
= (1-'1 x J.L2)*(n1 ® n2) As !11 ® !12 is positive, it completes the proof.
1.4.5 Proposition. Let JH[ = (M, r, "') be a co-involutive H opf-von Neumann algebra. Let J.L, J.Ll. 1-'2 be three representations of M*. {i} Let w be in Hom(J.LI. !-'2)· Then 1 ® w belongs to Hom(J.L X J.Ll. J.L x J.L2)· (ii) Let ~ be a normal morphism from Al-' 1 to Al-' 2 such that ~(1) = 1 and ~ o J.L1 = fL2· Then, we have:
(i
i8) ~) 0
Proof. Let w in M*, e, 7J in 1£1-',
(!-'X !-'1) = J.L X J.L2 .
6
in 1£1-'1' 7J2 in 1£1-' 2 • We have:
((1 ® w)(J.L X J.L1)(w)(e ® 6) I7J ® 7J2) = ((J.L X J.L1)(w)(e ® 6) I7J ® w*7J2) ={(!-'X J.L1)(w), ile,TJ ® il6,w•172 )
= {J.L*(ne, 11 )J.Lh(n6,w• 112 ),w) = {J.L*(ne, 11 )J.L2*(ilw6, 112 ),w)
by 1.3.9
= ((J.L X J.L2)(w), ile, 11
by 1.4.3
i8)
by 1.4.3
ilw6,T/2)
= ((J.L x J.L2)(w)(1 ®w)(e®6)!7J®7J2) which by linearity, density and continuity completes the proof of (i). We have also, for all il in (AJ.'h and !12 in (AI-' 2 )*:
{(i i8) ~) o (!-'X !-Ll)(w), .f1 i8) il2) = ((J.L x J.Ll)(w), n ® n2 o ~)
= (J.L*( il)!-'h( il2 o ~), w) = (!-'*(il)!-'2*(il2),w) = {(J.L x J.L2)(w), il ® il2)
by hypothesis
which by linearity, density and continuity completes the proof of (ii).
30
1. Co-lnvolutive Hop£-Von Neumann Algebras
1.4.6 Remark. Let 111 = {M, r, ~t) be a co-involutive Hopf-von Neumann algebra. It is straightforward from what is above that the operation which associates 1-'1 x 1-'2 to the couple (!-'1, JJ2) is a functor from Rep M. x Rep M* to RepM•. It is easy to check that the representation 1 {1.3.7) is a unit for that product. Since the Kronecker product is also associative, it is clear that (Rep M., x, 1) is a strict monoidal category in the sense of [95]. 1.4. 7 Example. Let G be a locally compact group and JJ, v non-degenerate representations of L 1(G). Let JJG, va be the corresponding continuous unitary representations of G. We have seen, in 1.3.10, that, for {}in (Ap)., JJ•({}) is a.e. equal to the functions-+ (JJa(s ), n). So, using 1.4.3, we get that, for {} in (Ap)., n' in (A11 ) . , the element (JJ X v).({} ® {}1) is a.e. equal to the function:
s-+ (JJa(s), n}(va(s), n'}
= (JJa(s) ® va(s), {} ® n'}
It is easy to conclude that 1-' x vis the non-degenerate representation of L 1 ( G) associated to the tensor product JJG ® liG.
1. 5 Representations with Generator 1.5.1 Proposition. Let {M, F) be a Hopf-von Neumann algebra, A be a von Neumann algebra on a Hilbert space 'H., V be in A®M. We define a bounded linear mapping JJ: M* -+A by, for all w in M.: JJ(w)
= (i ®w)(V)
Then we have: (i} The two following assertions are equivalent: (a) JJ is multiplicative (b} V satisfies the formula: (i ® F)(V)
= (V ® 1){1 ® u)(V ® 1)(1 ® u)
where u is the flip operator from H ® 1i to 1i ® H, or from 1i ® H to H ®'H. (ii) If K is a co-involution on M such that (M, r, ~t) is a co-involutive Hopfvon Neumann algebra, then the three following assertions are equivalent: (c) JJ is involutive . (d) for all w in M., we have:
(i ®w o ~t)(V) = (i ®w)(V*)
1.5 Representations with Generator
(e)
for all fl in A., we have: K((fl ® i)(V))
Proof. Let
31
WI,W2
= (fl ® i)(V*) .
in M •. We have, on the one hand: fJ(WI
* w2) = (i ® (wl * w2))(V) = (i ®wl ® W2)((i ® r)(V))
and, on the other hand: fJ(WI)fJ(W2)
= (i ® WI)(V)(i ® w2)(V) = (i ®wi)(V((i ®w2)(V) ® 1)) = (i ® WI)(V((i ® w2 ® i)(V ® 1)) = (i ® WI)(V((i ® i ® w2)(1 ® u)(V ® 1)(1 ® u)) = (i ® wi)((i ® i ® w2)((V ® 1)(1 ® u)(V ® 1)(1 ® u)) = (i ®w1 ®w2)((V ® 1)(1 ® u)(V ® 1)(1 ® u))
For any {lin A., the equality:
is therefore equivalent to:
which implies (i) by linearity and density. For every win M., {lin A., we have, on the one hand: {fl, fJ(w 0 )} = {fl ® w 0 , V}
= (ii ®woK, V*}= {ii,(i®woK)(V*)}On the other hand: {il,fJ(w)*} = {ii,fJ(w)}= {ii,(i ®w)(V)}-
Therefore we get that (d) is equivalent to (c). Moreover, (d) can also be written: (fl ® w o K)(V) = (fl ® w)(V*) (flEA.)
32
1. Co-lnvolutive Hopf-Von Neumann Algebras
which is trivially equivalent to:
w 0 K((n ® i)(V)) = w((n ® i)(V*)) and therefore to (e), which completes the proof.
r,
1.5.2 Definition. Let (M, K) be a co-involutive Hopf-von Neumann algebra; let p. be a representation of the Banach algebra M. on a Hilbert space 1£,.,. Let U be a partial isometry in £(1£,.,) ® M, whose initial and final supports are equal to P ® 1, where Pis a projector in £(1£,.,). We shall say that U is a generator for p. if it satisfies:
p.(w)
= (i ®w)(U)
for win M.
By linearity and density, U, if it exists, is unique; for any n in £(1£,.,)., we have p..(n) = (il®i)(U); by the bicommutant theorem, it is easy to see that U belongs to A,., ® M, and, thanks to 1.5.1 (i) and (ii), U satisfies:
(i ® r)(U) = (U ® 1)(1 ® u)(U ® 1)(1 ® u) (i ® w o K)(U) = (i ® w)(U*)(w EM.) 1.5.3 Lemma. The projection P in 1.5.2 is the projection on the essential space of p.. So U is unitary if and only if p. is non-degenerate.
Proof. Let us represent M on some Hilbert space H. Let a in 'HI-' such that p.(w)a = 0 for all win M •. This is equivalent to (U(a ® !1) I1® 6) = 0 for all /1,6 in H, 1 in 1£,.,, or to U(a ® !1) = 0 for all f1 in H, i.e. to Pa ® f1 = 0 for all f1 in H, and then to Pa = 0; this leads to the result. 1.5.4 Lemma. With the hypothesis of 1.5.2, let p., (P.i)iei be representations of the Banach algebra M., with generators u,.,, (UIJ;)iei· Then: (i} If p.1 is quasi-equivalent to p., then p.1 has a generator. (ii} If p. 11 is a sub-representation of p., then p. 11 has a generator. (iii) The representation $ieiP.i has a generator.
Proof. Let iJ be the morphism from A,., to AI-'' such that iJ o p. = p.1• It is easy to check that (iJ ® i)(U,.,) is a generator for p. 1, which gives (i). There exists a projection Pin (A,.,)' such that p. 11 = p.p. It is then easy to check that (U,.,)(P®l) is a generator for p.11 , which gives (ii). Finally, $ieiUIJ; is a generator for $ieiP.i· 1.5.5 Proposition. With the hypothesis of 1.5.2, let ILl and IL2 be two representations of M* having generators, respectively V1 and V2. Then the element
1.5 Representations with Generator
33
(u ® 1)(1 ® V1)(u ® 1)(1 ® V2) is the generator of 1'1 X Jl-2, and the essential space of 1'1 X 1'2 is the tensor product of the essential spaces of 1'1 and 1'2· So, if 1'1 and 1'2 are non-degenerate, 1'1 x 1'2 is non-degenerate too.
(1'1
X
1'2).(/'h ® il2) = l'h(ill)l'2•(il2) =(ill® i)(VI)p.2*(il2)
by 1.4.3 by 1.5.2
=(ill® i)(V1(1 ® 1'2•(il2))) =(ill® i)(V1(i ® il2 ® i)(1 ® V2)) by 1.5.2 =(ill® i)((i ® il2 ® i)((u ® 1)(1 ® V1)(u ® 1)(1 ® V2))) because (i ® il2 ® i)((u ® 1)(1 ® VI)(u ® 1)) = V1 =(ill® il2 ® i)((u ® 1)(1 ® V1)(u ® 1)(1 ® V2)) Let V1,2 = (u ® 1)(1 ® V1)(u ® 1)(1 ® V2). We easily get:
vt 2V1 2 = V1 2vt 2 = P .. '
'
'
,
,..
l
® P r.. 2 ® 1
and then deduce that P,., 1 ® P,., 2 is the projection on the essential space of 1'1 X 1'2· 1.5.6 Lemma. With the hypothesis of 1.5.2, let 1'1 and 1'2 be two representations of M., having respectively V1 and V2 as generators. Let t be in Hom(p.1, 1'2), and !I : A,., 1 -+ A,., 2 a morphism such that !I o 1'1 = 1'2i then, we have:
(i)
(t ® 1)Vl = V2(t ® 1)
(ii)
(!I® i)(V1) =
v2 .
Proof. For all w in M., we have:
(i ® w )((t ® 1)Vl) = t(i ® w)(V1) = tp.1(w) = 1'2(w)t = (i ®w)(V2)t = (i ®w)(V2(t ® 1)) from what (i) is easily reached. We have also: (i ® w)((!l ® i)(V1)) = !J(i ® w)(V1) =!I o l'l(w) = 1'2(w) = (i ® w)(V2) which allows to complete the proof.
34
1. Co-Involutive Hopf-Von Neumann Algebras
1.5.7 Proposition. With the hypothe&i& of 1.5.2, let p. be a non-degenerate repre&entation of M* on the finite-dimen&ional Hilbert &pace C", {ei}1
~ x*k ,a·Xk ,J. = 6i ,3·1 L....., k=l n
~ x-kx":k L....., a, J, = Di ,3·1 k=l
where Di,j i8 the wual Kronecker &ymbol. The generator of p. i& then the matriz [xi,;h~i,j~n' which may be con&idered a& an element of .C(C") ® M. Proof. Let ei,j be the matrix units associated to the basis {ei}. Let then:
U = L....., ~ e1· J· ® x 1· J· E .C(C") ® M 1
1
i,j
We have for any win M.:
and so, by linearity, for all win M.:
p.(w) = (i ®w)(U) As:
U* =
L e;,i ®xi,; i,j
we easily get:
U*U
= Lei,j ® I:xk,ixk,j i,j
and:
UU* =
k
L ei,j ® L xi,kxJ,k i,j
k
and so the conditions (ii) are equivalent to U being unitary, and the result is proved.
1.5 Representations with Generator
35
1.5.8 Corollary. With the hypothesis of 1.5.2, let {ei} be the canonical basis of and, for every i,j, 15 i, j 5 n, Xi,j be elements of M. The following assertions are equivalent: (i) There exists a non-degenerate representation p. of M. on with a generator, such that, for every 1 5 i, j 5 n:
en,
en,
(ii) The elements Xi,j satisfy: ~~:(xi,;)= Xj,i
r(xi,j)
= LXi,k ®xk,j k
~ L..J xk* ,,·Xk ,J· = 6i 3·1 J
k
L
xi,kxj,k = Di,j1 .
k
Proof. It results from 1.4.2 and 1.5.7. 1.5.9 Corollary. With the hypothesis of 1.5.2, let p. be a non-degenerate representation of M. on the Hilbert space en with a generator, {Ci} the canonical basis of en; let us write down, for all i,j = 1, ... 'n:
Then, the representation jJ defined by jJ(w) = p.(w o ~~:)t (where t means the transposition of matrices), has a generator, which is the matrix [xi,;h:$i,j:$n· Proof. It is clear that jl is a non-degenerate representation of M., and that: xi,;= ~~:(x;,i) = jl.(ile;,eJ Moreover, we have:
Lxk,ixk,; = L~~:(xi,~c)~~:(x;,k) k
=II: (
~ x;,kxi,k)
= ~~:(6i,;1) = 8i,j1 a.nd
E1c xi kx 3• k = I
I
by 1.4.2
k
Di,j by the same calculation.
by 1.5.7
36
1. Co-Involutive Hopf-Von Neumann Algebras
So, using again 1.5. 7, we see that the matrix
[xi,;]
is the generator of jl.
1.5.10 Example. Let G be a locally compact group, JH[a( G) the abelian coinvolutive Hopf-von Neumann algebra associated by 1.2.9. Let fLG be a unitary representation of G, fL the non-degenerate representation of L 1(G) associated to fLG by the formula (cf. 1.1. 4):
p(f)
=
Ia
f(s)pa(s)ds
We then see that the generator of fL is nothing but the bounded continuous functions~ fla(s) from G to £(1£110 ), which may be considered as a unitary element of £(1£11 ) 0 L 00 (G). So, any non-degenerate representation of L 1 ( G) has a unitary generator, and any representation of L 1 (G) has a generator.
1. 6 Fourier-Stieltjes Algebra 1.6.1 Introduction. (i) Let us put that, being given JH[ = (M,F,K) a coinvolutive Hopf-von Neumann algebra, C*(JH[) denotes the enveloping C*algebra of M* ([12), I §6, def. 5). We shall denote by 1r the universal representation of M* in C*(JH[). For every representation fL (resp. antirepresentation) of M*, there exists a surjective morphism (resp. antimorphism) fl from C*(JH[) onto the C*-algebra C*(p) generated by p(M*), such that fL = flrr. It is thus possible to identify C*(p) with the quotient of C*(JH[) by Ker fl. Consequently, if we denote the dual space of C*(JH[) by B(JH[), the dual of C*(p) is isometrically identified with a closed subspace B(p) of B(JH[). More precisely, to each element ()of (C*(p))*, one associates the element () o fl of B(JH[) and an element of B(JH[) belongs to B(p) if and only if it vanishes on Ker fl. (ii) Let W*(JH[) denotes the enveloping von Neumann algebra of C*(JH[) ([133), th. 1.17.2) (i.e. the bidual of C*(JH[), or, with the notations introduced in (i), the dual of B(JH[)). Let us still denote by 1r the universal representation of M* in W*(JH[). For any representation (resp. antirepresentation) of M*, there exists a normal morphism (resp. antimorphism) s 11 from W*(JH[) to A 11 , such that fl is the restriction of s 11 to C*(JH[), and, so, such that fL = s 11 rr ([12), I §6, prop. 10 and [25), 12.1.5). Going over to the preduals, we get a continuous morphism:
such that rr*(s 11 )* =~'-*'Moreover rr* implements a one-to-one linear mapping from B(JH[) toM (cf. 1.3.4).
1.6 Fourier-Stieltjes Algebra
37
(iii) When p. is non-degenerate, s 11 is surjective; then, we get s 11 (1) = 1. Furthermore, for any n in (A 11 )t, we have:
Let us remark that C*(llll) and W*(llll) contain more than 0, for it exists at least one non-null representation of M* (viz. 1).
r,
1.6.2 Proposition. Let llll = (M, ~~:) be a co-involutive Hopf-von Neumann algebra. We have: (i) an element x of M belongs to 7r*(B(p.)) if and only if we have:
sup{l(x,w)l, wE M., llp.(w)ll ~ 1} < +oo Then, we have:
sup{l(x,w)l, wE M., llp.(w)ll ~ 1} = 117l'; 1 (x)ll (ii) the mapping (s 11 )* is an i8ometry from (A 11 )* to B(llll), the image of which is included in B(p. ). Proof. Let 6 be in C*(p.)*, that is, by 1.6.1 (i), an element such that 6 o ji. belongs to B(p.); we have, for win M.:
and, therefore: sup{l(7r•(6 o ji.),w)l, wE
M., IIJ.L(w)ll
~
1} = 11611
by Kaplansky's theorem = 116 0 Ji.ll
Conversely, let x in M and c > 0 such that l(x,w)l ~ cllp.(w)ll for all win M •. It exists some 6 in C*(p.)* such that, for all win M., we have: (x,w)
= (p.(w),6) = (7r(w),6oji.) = (7r.(6oji.),w)
which implies: . and completes the proof of (i). Now, for n in (A 11 ). and win M., we have:
therefore, by Kaplansky's theorem, we have:
38
1. Co-Involutive Hopf-Von Neumann Algebras
From (i), we get that (siJ).(!~) belongs to B(p.) and that which completes the proof.
ll(s#J).(il)ll = !Iilii,
1.6.3 Proposition. Let G be a locally compact group; the enveloping C* -algebra and W*-algebra of L 1 (G) are noted respectively C*(G) and W*(G), and will be called the enveloping C* -algebra and W* -algebra of G; let us write 7rG the continuous unitary representation of G associated to the universal representation 1r of L 1 (G); 7rG will be called the universal representation of G. (i) For a non-degenerate representation p. of L 1 ( G), if P.G is the corresponding continuous unitary representation of G, we have, for all t in G: s1J1ra(t)
= p.(t)
(ii) Every element fin 1r*(B(p.)) C L00 (G) is almost everywhere equal to a continuous function. (iii) ([46]) For every bounded continuous function f on G, and every representation p. {including p. = 1r) of G, let us put:
then 1r*(B(p.)) {resp. 7r*(B(lllia(G))) appears as being the subset of all functions f in Cb( G) such that 11/IIIJ (resp. ll/ll1r) is finite. It is a complete normed vector space, denoted BIJ(G) (resp. B(G)). It can thus be identified to the dual of the C* -algebra generated by p.( L 1 (G)) ( resp. the dual of C* (G) i.e. the predual of W*(G)). Proof. By a straightforward application of 1.3.10, we get (i). Let 0 be in
(C*(p.))*; it can be written as a linear combination of positive elements. It implies that 1r*(O o Jl) = p.*(O) is a linear combination of positive definite elements of L 00 (G), which, according to 1.3.11, completes the proof of (ii) and (iii) is just an application of 1.6.2. 1.6.4 Lemma. Let llli be a co-involutive Hopf-von Neumann algebra, p. 1 and p. 11 two representations of M*. We put s 1 = siJ' and s" = siJ"· We have:
= p.1 X p. 11 = SIJ'XIJ"
{i)
( s 1 ® i)( 7r
{ii)
(s 1 ® i)s?rXIJ" ( i ® s 11 )(p.1 X 7r) =
(iii)
(i ® s 11 )SJIIX1r ( s' ® s 11 )( 1r x 1r) (
SI
10. '01
X
p. 11 )
S" ) S1rX1r
p.'
X
p. 11
= SJI'XJI" = p.' x p.11 = SIJ'XJI"
1.6 Fourier-Stieltjes Algebra
39
Proof. Let w be in M., !J' in (A,,). and D" in (A,,) •. We have: {(s' ® i)(1r
X
Jt 11 )(w), !J1® !J11 ) = {1r ® Jt11 (w),s~(!J1 ) ® !J 11 ) = {(1r X Jt11 ).(s~(!J1 ) ® !J11 ),w) = {1r*s~(!J1 )Jt~(!J 11 ),w) = {JJ~(!J')JJ~(!J"),w) = {(Jt1 X Jt 11 ).(!J1® !J11 ),w) =
by 1.4.3 by 1.6.1 (ii) by 1.4.3
{(JJ' x JJ")(w), n' ® n")
from what, by linearity, continuity and density, (i) results; the proof of (ii) is identical and leads to (iii), thanks to the decomposition:
s1® s11 = (s 1® i)(i ® s11 ) • 1.6.5 Proposition. Keeping the notations used above, s71'X71' is a one-to-one normal morphism from W*(lll) to W*(lll) ® W*(lBl) such that:
If every representation has a generator, then s71'X71' is a co-product in the sense of 1.2.1.
Proof. Let w be in M •. We have:
(i ® 871'X71' )s71'X71'( 1r(w)) = (i ® 871'X71' )( 11" X 1r)(w) = (1r X (1r X 1r))(w) = ((1r X 1r) X 1r)(w) = (s71'X71' ® i)(1r X 1r)(w) = (s71'X71' ® i)s71'x71'(1r(w))
by 1.6.1 (ii) by 1.6.4 (ii) by 1.4.3 by 1.6.4(i) by 1.6.1 (ii)
which by continuity and density implies the equality looked for. Considering the unit representation 1, we have also: (s1 ® i)s71'x71'(1r(w))
= (s1 ® i)(1r X 1r)(w) = (1 x 1r)(w) =1r(w)
which implies, by continuity and density:
by 1.6.1 (ii) by 1.6.4(i) by 1.4.6
40
1. Co-lnvolutive Hopf-Von Neumann Algebras
from what it results that s1rx1r is one-to-one, which completes the first part of the proof. If every representation has a generator, then 1r x 1r is non-degenerate (1.5.5) and s7rx7r(1) = 1, which ends the proof. 1.6.6 Proposition. We define an antirepre8entation ir of M*, by putting, for every w in M*: ir(w) = 1r(w o ~~:)
Then s1r i8 an involutive normal antiautomorphi8m of W*(IHI) which 8ati8jie8:
{i) (ii)
7r*(s*)* = 11:7r* S1rX1rS1i" = ,( S7r 0 S7r )s7rX7r .
Proof. Let w be in M*. We have:
S7rS7r(1r(W)) = S7r(1r(w)) = s7r(1r(w o = ir(w o ~~:) = 7r(w)
~~:))
by 1.6.1 (ii) by definition by 1.6.1 (ii) by definition
by continuity and density, we get that s1r is involutive. Let Q be in (W*(IHI))*; we have: {1r*((s*)*(il)),w) = {7r(w),(s*)*(Q)) = {s7r(7r(w)),il) = {ir(w),il) = {1r(w o ~~:), Q) = {1r(il),w o ~~:)
by 1.6.1(ii)
= {~~:7r*(il),w)
which proves (i). Let il1, il2 be in (W*(IHI)k We have: {s7rx7rs7r(7r(w)), il1 0 il2) = {s7rx7rir(w), il1 0 il2) = {s7rx1r1r(w o ~~:), il1 0 il2) = {1r X 1r(w o ~~:), il1 0 il2) = {(1r x 1r)*(il1 0 il2),w o ~~:) = (~~:(7r*(QI)7r*(Q2)),w)
=
by 1.6.1 (ii) by 1.6.1 (ii) by 1.4.3
{~~:7r*(Q2)K7r*(Ql),w)
= (7r*(s*)*(Q2)7r*(s*)*(Ql),w)
by (i)
1.6 Fourier-Stieltjes Algebra
= ((11" X 11").((s?r)•(.G2) ® (s?r).(.Gt),w} = ((11"
X
41
by 1.4.3
11")(w), (s1i').(!12) ® (s?r).(.Gt)}
= (s7rx7r11"(w),(s?r)•(.G2)®(s?r)•(.Gt)} = (( S?r ® S?r )s71"X7r11"(W ), !12 ® !11} = (<;(s1r®s1r)s7rx7r11"(w),.Gt ®!12}
by 1.6.1(ii)
which, by continuity and density, completes the proof of (ii).
1.6. 7 Theorem. Let llll be a co-involutive H opf-von Neumann algebra; let us suppose that every representation of the Banach involutive algebra M* has a generator; then the triple (W*(llll), <;S7rx1r, s1r) with the definitions of 1.6.5 and 1.6.6, is a co-involutive Hopf-von Neumann algebra. Ifllll is abelian, then W*(llll) is symmetric; if llll is symmetric, then W*(llll) is abelian. Proof. The first part of the theorem is given by 1.6.5 and 1.6.6. Moreover, if M is abelian, we have <;71" x 71" = 71" x 71" (1.4.3), and then s7rx1r = <;s7rx1ri if llll is symmetric, M. is abelian (1.2.1) and then W*(llll) too.
1.6.8 Corollary ([44]). Let G be a locally compact group, 1l"G its universal r.:presentation, W*( G) its enveloping W* -algebra. There exists a unique morphism r from W*(G) to W*(G) ® W*(G), and a unique antiautomorphism K. in W* (G) such that, for all t in G: r(11"a(t))
= 11"a(t) ® 11"a(t) = '~~"a(t)*
~t('~~"a(t))
Moreover, (W*(G),F,~t) is a symmetric co-involutive Hopf-von Neumann algebra, which we shall call the Ernest algebra of G. Proof. Using 1.4.7 and 1.6.3(i), we have, for all tin G: s1rx1r'~~"G(t) = '~~"a(t) ® '~~"a(t)
s*'~~"a(t) = '~~"a(r 1 ) a.nd so, in particular s7rx7r(1)
= 1 ® 1, which ends the proof.
1.6.9 Theorem. Let llll be a co-involutive Hopf-von Neumann algebra; then, the space B(llll) = (W*(llll)). is canonically equipped with a structure of semisimple involutive Banach algebra with unit of norm one. It will be called the Fourier-Stieltjes algebra associated to llll. The mapping K.11"* is a faithful representation of this algebra in M; it will be called its Fourier-Stieltjes representation.
42
1. Co-lnvolutive Hopf-Von Neumann Algebras
Proof. By transposing the mappings c;s11'X11' and s1r, we get a product * and an involution ° on B(JBI). For !h and il2 in B(JB!) and win M., we have:
(/\:7r*(n1
* n2),w) = (7r(w 0 /\:), n1 * n2) = (c;s11'X11'7r(W 0 11:), f11 ® f12) = ((1r x 1r)(w o ~~:),
n2 ® n1)
= {1r.(!.?2)7r.(il1),w o ~~:) = (~~:?r.(il1)/\:7r*(n2),w)
by definition by 1.6.1 (ii) by 1.4.3
Moreover, we have: (~~:7r.(il 0 ),w) =
((7r(w 0 ~~:), il0 ) = (si'( 1r(w 0 II:)*), n)= (s1i'(1r(W 0 0 ~~:)), n)= (1T"(w 0 0 ~~:), n)= (7r(w 0 ),n)= (7r.(!.?),wo)-
by definition by 1.6.1 (ii)
= (~~:?r.(il)*,w) Since 1r is non-degenerate, 1r* is faithful and B(JB!) is semi-simple. It results from the decomposition J.t• = ?r.(sp)• that we have 7r.(B(JB!)+) = PR(JB!); we have seen in 1.3.7 that the unit of M belongs to PR(JB!), it follows that 1r; 1(1) is a unit for B(JB!); it is easy to check that 1111";\1)11 = 1, which completes the proof. 1.6.10 Remarks. Since it results from 1.6.2 (i) that 1r*( B(JB!)) is stable by ~~:, the fact used above that ?r.(B(JB!)+) = PR(JB!) implies that the image of B(JB!) by the Fourier representation is the set of linear combinations of elements of P R(JB!), that is, also, the set of elements which can be written J.t•(il), where J.t is a representation of M. and n is in (Ap)•· Since 1r* is faithful, we can deduce that B(JB!) is composed of those elements of the form (sp).(il). Thanks to 1.6.2(i), we observe, at last, that PR(JB!) is stable by 11:. 1.6.11 Theorem ([46]). Let G be a locally compact group, and f a function on G. The following assertions are equivalent: {i) The funCtion f belongs to B(G), i.e. {1.6.3 (iii)), f is a bounded continuous function on G, and:
sup{
I.Ia f(t)h(t)dtl,
h E £ 1( G), ll1r(h)ll ::::; 1}
where 1r is the universal representation of L 1 ( G).
< +oo
1.6 Fourier-Stieltjes Algebra
43
(ii) The function f iJJ a linear combination of continuouJJ poJJitive definite functionJJ on G. (iii) There exiJJtJJ a Hilbert 3pace H, vectorJJ {, '7 in H, and a continuous unitary repreJJentation P.G on H such that f(t) = (p.a(t)e I77) Moreover, B( G) is a subalgebra of the algebra Cb( G) of continuouJJ bounded functions on G. Equipped with the norm:
B(G) iJJ a Banach algebra, which can be identified with the dual of C*(G) (i.e. the predual of W*(G)), the duality being given, if f(t) = (p.a(t){ I71), and h iJJ in C*(G), by: (!,h) = (Jt(h)e 171 ) (where jl denotes here the representation ofC*(G) aJJsociated with P.a). Proof. Using 1.6.3(iii) and 1.6.10, we see that B(G) is the image of the involutive algebra B(lllla(G)) by its Fourier-Stieltjes representation. Then we have the implication (i) => (ii) by 1.6.10 and 1.3.11; (ii) => (iii) is a corollary of 1.3.11, and (iii) => (i) is a straightforward calculation. The end of the theorem is just a re-writing of 1.6.3 (iii).
Chapter 2 Kac Algebras
This chapter deals with technical results about Haar weights, as they have been studied by the authors in [36] and [136], and, independently, by Vainerman and Kac ([180]). On a co-involutive Hopf-von Neumann algebra (M,F,K), a Haar weight is a faithful, semi-finite, normal weight on M+, which is left-invariant with respect tor, i.e. such that:
(i ® cp)F(x) = cp(x)1 for all x in M+ (in 2.5, we show, after Kirchberg, that this axiom may be weakened), and, roughly speaking, satisfies two other axioms involving K. The quadruple (M' K, cp) is then called a Kac algebra. On L 00 (G), the integral defined by a left Haar measure is obviously a left-invariant weight, and it is straightforward to see that it is a Haar weight (£.£.£).Thus, L 00 (G), equipped with the convenient operations, is an abelian Kac algebra; we shall denote it Ka(G). Left-invariantness allows us to define the fundamental operator W, which is an isometry belonging toM® C(Hr.p), satisfying the so-called pentagonal relation (£.4.4):
r,
(1 ® W)(u ® 1)(1 ® W)(u ® 1)(W ® 1)
= (W ® 1)(1 ® W)
With the other axioms, we get that W is unitary (£.6.£), "implements" the co-product r, i.e. that we have, for all X in M (£.6.9): F(x)
= W(1 ® x)W*
Moreover, the mapping: ..\: w--+ (w ® i)(W*)
from M* to C(Hr.p) is a non-degenerate representation of the involutive Banach algebra M* (2.6.1 and 2.6.9); it will be called the Fourier representation of the (predual of the) Kac algebra.
2.1 An Overview of Weight Theory
45
Then, W and A appear both as generalizations to an abstract Kac algebra of the left regular representation Aa of a locally compact group G; more precisely, for the Kac algebra lKa(G), the fundamental operator is the bounded continuous function s --+ Aa( s )*, when considered as an element of L 00 (G) 0 .C(L 2 (G)), and the Fourier representation is nothing but the left regular representation of L 1 (G) (2.4. 7 and 2.5.4 ). Another result strengthens the analogy with the case of locally compact groups: on a co-involutive Hopf-von Neumann algebra, the Haar weight (whenever it exists) is unique up to a positive scalar (2.8.6). The situation of locally compact groups is generalized, too, with the definition of square-integrable elements of M*, i.e. elements w of M* such that there exists a (unique) vector a(w) in H'P such that, for all x in~ (2.1.6):
(x*,w)
= (a(w) IA'P(x))
If the weight cp is left-invariant, the set I'P of such elements is a left ideal of M* (2.4.5), and, if cp is a Haar weight, we have, for all w1 in M* and w2 in
I"' (2.4.6): a(w1
* w2) =
A(w1)a(w2)
In the case of lKa( G), we recover L 1 ( G)nL 2 (G), which is a left ideal of L 1 ( G), and the formula:
h *h = Aa(h)h which links the convolution and the left regular representation of G. If lK = (M,r,,,cp) is a Kac algebra, so is JK' = (M,~r,,,cp o K), and it is possible, too, to define a Kac algebra JK' on the commutant M' of M (we suppose M is a von Neumann algebra on the Hilbert space H'P); these technical results (2.2.5) will be useful in Chap. 5.
2.1 An Overview of Weight Theory 2.1.1 Definitions and Notations. (i) A weight on a W* -algebra M is an additive mapping cp : M+ --+ [0, +oo] such that cp( Ax) = Acp( x) for all A in Jll+ and x in M+. We shall use the notations and constructions of the Takesaki-Tomita theory associated to weights (cf. [158], [14], [114], [16], [56] or [150]) hereafter briefly summarized: Let us define:
rott = {x EM+, cp(x) < +oo} ~
= {x EM,
cp(x*x) < +oo}
46
2. Kac Algebras
Then ~c,o is a left ideal in M, and the involutive algebra ~~ ~ is the linear span of rott; it is denoted by rolc,o, and it is possible to extend cp to a positive linear form on rolc,o, which will be still denoted by cp. The weight cp is said to be faithful if cp( x) = 0 for x in M+ implies x = 0, semi-finite if rolc,o is ultra-weakly dense in M (finite if VJtcp = M), and normal if cp(supa xa) = supa cp(xa) for all increasing directed bounded nets {xa} in M+. If cp is normal, the set of all projections p in M such that cp(p) = 0 has a greatest element po, and q = 1- Po is called the support of cp. We may consider then cp as a faithful weight on the reduced algebra Mq. (ii) In what follows, cp will denote a faithful, semi-finite, normal w~ight on a W*-algebra M. Then, the left ideal~' equipped with the scalar product (x, y) -+ cp(y*x) (x, y E ~), is a pre-Hilbert space; let us denote Hc,o the associated Hilbert space (or H ifthere is no confusion), and Ac,o the canonical injection~-+ Hc,o. The set 21c,o = Ac,o(~ n ~~), equipped with the product T and the involution U defined by:
Ac,o(x)T Acp(y) = Acp(xy) Ac,o(x)U = Acp(x*)
(x, y E ~ n ~~) (x E ~c,o n ~~)
is an involutive algebra, dense in Hc,o, such that its involution is an antilinear preclosed mapping, and such that the left-multiplication representation of 21c,o is non-degenerate, bounded and involutive. The W* -algebra M is then isomorphic to the von Neumann on Hc,o generated by this representation ([14], th. 2.13). (iii) Generally speaking, we shall call left Hilbert algebra an involutive algebra 21, equipped with a scalar product, such that the involution is an antilinear preclosed mapping in the Hilbert space H associated, and such that the left-multiplication representation 1r of !2t is non-degenerate, bounded and involutive ([158], de£. 5.1). Let us then callS the closure of U, and F the adjoint S* of S, whose domains will be respectively denoted by vU and 1J. We shall say that in H is right bounded if the linear application from !21 to H defined by '7-+ 1r(71)e (71 E !21) is bounded; we shall then denote by 1r1(e) the element of .C(H) defined, for all7J in !21, by 7r'(e)'7 = 1r(71)e. Let us then define:
e
!21'
= {eE 'D~,
eis right bounded}
Then, 21', equipped with the product 6Te2 = 1r'(e2)6 (6,6 E !21'), and the involution e~ = Fe (e E !21') is a right Hilbert algebra (with the obvious definition). Repeating this construction, we obtain the definition of left bounded elements in H, and another left Hilbert algebra !2111 , containing !2t as an involutive subalgebra, the closed operator S being again the closure
2.1 An Overview of Weight Theory
47
of the involution of !!11 • A left Hilbert algebra is called achieved if !2t = !1!11 • Moreover, for any left Hilbert algebra !X, the left (resp. right) Hilbert algebra !!'' (resp. !X') is an achieved left (resp. right) Hilbert algebra, and the left Hilbert algebra !Xrp constructed in (ii) is achieved. Conversely, for any left Hilbert algebra !X, the left-multiplication representation 1r generates a von Neumann algebra M on the Hilbert space H, and the formula, for x in M+:
cp(x)
= { 11e11 2 +oo
if there exists e in !!11 such that elsewhere
X=
1r(e)
defines a faithful, semi-finite normal weight on M, with the left Hilbert algebra !2trp isomorphic to !!11 ([14], th. 2.11). For any in v# (resp. v"), it is possible ([14], def. 2.1) to define a left (resp. right) multiplication bye, which will be a closed operator on H, affiliated to M (resp. M'). (iv) Starting from !2trp, the polar decomposition S = Jrp.tJ}/ 2 gives rise to the antilinear isomorphism Jrp (or J if there is no confusion) from Hrp to Hrp, and the modular operator L1rp (or L1 if there is no confusion). If we consider the elements of Mas operators on Hrp, we have then JMJ = M' (and M is then in a standard position in H, in the sense of 1.1.1 (iii), thanks to the closed convex cone Prp generated by {xJrpArp(x); X E mrp}), an~ JxJ =:= x* for any x in the centre of M. Moreover, for all t in R, we have L1 at M ..1-at = M, this last formula leading to the definition of the modular automorphi8m group u'f by: ui(x) = L1itxL1-it (x E M,t E R)
e
Moreover, we have:
cp = cp o u'f and the modular automorphism group satisfies the Kubo-Martin-Schwinger condition ([14]), that is, for every x, y in ~ n m~. there exists a bounded function f on the strip {z E C, 0 ~ 1m z ~ 1}, holomorphic in its interior, · such that, for all tin R, we have:
f(t) = cp(u'f(y)x) f(t + i) = cp(xui(y)) These two properties characterize the modular automorphism group. An element x of M will be called analytic with respect to cp if the function t-+ u'f(x) has an extension to an analytic function z-+ uf(x) from C toM. We then define M'P = {x E M, u'f ( x) = x for all t in JR.}. Moreover, for any X in mrp and a in M'P' xa* belongs to ~. and we have ~( xa*) = J aJ Arp( X). For technical reasons, it is useful to know there exists a maximal subalgebra !2to of 2111 n 21', which is both a left and right Hilbert algebra, with ~ = !2t'
48
2. Kac Algebras
and !X~ = !X", and which is globaly invariant under the linear closed operators Llz for all z in C. The algebra 2lo is called the maximal modular subalgebra of !X". If 'P is a trace (i.e. ~.p(x*x) = ~.p(xx*) for all x in M), then '1tc,o = 'Jt~, Sc,o = Jc,o, Llc,o = 1, and the modular automorphism group reduces to the identity i; the algebra 'Jtcp is both a left and a right Hilbert algebra; we recover the Hilbert algebras of [24]. If trn is the canonical trace on £(H), then '1ttrH is the algebra of Hilbert-Schmidt operators on H; it is therefore complete, and we may consider HtrH as H®H, the identification AtrH being given, for all in H, by AtrH(Pe) = ® t, where Pe is the one dimensional orthogonal projection on ce. Conversely, every complete Hilbert algebra is a direct sum of algebras of that type ([24], I §8, prop. 7). For x in M'+, the formula ~.p1 (x) = ~.p(Jc,ox* Jc,o) defines a faithful semifinite normal weight on M', with 'Jlc,o~ = Jcp'Jlc,oJc,o; we identify Hc,o' with Hc,o by writing Ac,o'(Jc,oxJc,o) = Jc,oAc,o(x). If 'Pi ( i = 1, 2) is a faithful, semi-finite, normal weight on the W* -algebra Mi, then ([16]) there exists a unique faithful semi-finite normal weight 'Pl ®'P2 on M1 ® M2 such that:
e
e
('Pl ® 'P2)(xl ® x2)
= 'Pl(xl)'P2(x2) (x1 E M1, = ufl ® ur
x2 E M2)
ufl®Cf'2
If the weights are not faithful, we may define the tensor product of the reduced weights on the tensor product M1p1 ® M2p2 , which may be identified to (M1 ® M2)Pt®F.! (where Pi is the support of 'Pi), and, by composition with the reduction from M1 ®M2 to (M1 ®M2)Pt®F.!• we have a definition of the tensor product of 'Pl ® 'P2. (v) If 1.p, .,P are two faithful semi-finite normal weights on a W* -algebra M, it is possible ([16]) to define a cocycle Ut in M, such that, for all s, t in JR., and x in M, we have:
= Utuf( u uf(x) = utuf(x)ut Ut+s
8)
If 'P = 'Po uf, for all t in JR., this cocycle is a one parameter group of unitaries in MCf', which leads, thanks to Stone's theorem, to a positive self-adjoint operator h affiliated to MCf', such that Ut = hit. We have then, for all x in M+:
where he= h(1
+ eh)- 1 belongs to MCf'. Moreover: 1J(h 112 )
= {e E H;
sup(hee Ie)< +oo} E
2.1 An Overview of Weight Theory
and, for any
For any
X
49
ein "D(hl/2):
llh 112 ell 2 = sup(hee Ie)= lim(he:e Ie) e: e: in IJl~p n !Jl,p, we have: ,P(x*x)
= e-o lim lt'(h!1 2 x*xh!1 2 ) = lim IIJh!/ 2 JA~p(x)ll 2 e:-o
and so JA~p(x) belongs to "D(h 112 ), and: ,P(x*x)
= IIJhl/2JAtp(x)ll2
We shall write then ,P = lt'(h·), in the sense of ([114], th. 5.12). The operator h will be called the Radon-Nikodym derivative of ,P with respect to It'· If we have ,P ~ lj', then 0 ~ h ~ 1. If uf = uf for all tin R, then the operator his affiliated to the centre of M; as then Jh 112 J = h 112 , we get that, for any x in ~Jttpn!Jl,p, Atp(x) belongs to "D(h 1 12 ), and ,P(x*x) = llh 1 1 2 A~p{x)ll 2 . 2.1.2 Lemma. Let e; be a family of elements in M, belonging to IJl~p, analytic with respect to lj', such that, for allj, lle;ll ~ 1, and, for all zinC, Uz(e;) is weakly convergent to 1 (such a family exists, thanks to {[150], 2.16). Then, for all a in rol~p, we have:
lii;Illt'( aej) J
Proof Let x, y be in IJltp IJ'(y*xe;)
= It'(a) .
n IJl~. We have:
= (A~p(xe;) IAtp{y)) = (SA~p(ejx*) I A~p(y)) = (SejA~p(x*) I A~p(y))
= (SejSA~p(x) IAtp(y)) = (JL1 1 1 2 ejLl- 1 1 2 JA~p(x) I A~p(Y)) = (Ju~i/ 2 (ej)JA~p(x) I A~p(y))
--7 (A~p(x) I A~p(y)) = lt'(y*x) J
which, by linearity, completes the proof. 2.1.3 Lemma. Let {3 be an antiautomorphism of M (i.e. a linear mapping from M to M such that for all x, y in M, we have f3(xy) = {3(y)f3(x) and f3(x*) = f3(x)*). Let us put 9 = lt'of3; then(} is a semi-finite, faithful, normal weight on M and for all t in R, we have:
uf = tr 1 0 u"!..t 0
{3 .
50
2. Kac Algebras
'.no'
Proof. Let X and y in 'Jle n then (3( X) and f3(y) belong to IJtp n 'Jl~; therefore there exists ([14], prop. 4.4) a bounded continuous function f on the strip {z E C; 0 :::; 1m z :::; 1}, holomorphic in its interior, such that for all t in R, we have: f(t) = cp(ai(f3(y))f3(x)) f(t + i) = cp(f3(x)ai(f3(y))) It can also be written as follows:
= cp(f3(y)a~t(f3(x))) = 8((3- 1 a~tf3(x)y) f(t + i) = cp(a~t(f3(x))f3(y)) = 8(y(3- 1 a~tf3(x)) f(t)
As, on the other hand, we have, for all x in M+:
8((3- 1 a~tf3(x))
= cp(a~tf3(x)) = cp(f3(x)) = 8(x)
we see that (} satisfies the K.M.S. conditions with respect to the group of automorphisms t--+ (3- 1 a'!_tf3, then ((14] cor. 4.9) allows to conclude. 2.1.4 Lemma. Let ,P another faithful, semi-finite, normal weight on M, such that ,P is invariant under a'f for all t in R. Then A('Jtp n ~) is dense in H~.p.
Proof. Let h be the Radon-Nikodym derivative of ,P with respect to cp. Let us put h = J000 sdE8 ; en = Jt;n dEs; hn = hen. Let x be in IJtp; we have:
1/2
because en he
belongs to M
= IIJh:/ 2 JA~.p(x)ll 2
because heen is weakly convergent to hn when c goes to 0
and then xen belongs to~' and as en belongs to VJt'f', by ([114], th. 3.6), we get that xen belongs to '.Yt~.pn'.YttJ>· Moreover A~.p(xen) = JenJA~.p(x) converges
2.1 An Overview of Weight Theory
to Acp(x), because en -+ 1 when n goes to infinity. Thus, Acp(IJtcp dense in Acp(IJtcp) and therefore in Hcp.
51
n ~) is
2.1.5 Lemma. Let .,P be a semi-finite, normal weight on M, invariant by the modular automorphism group a'f. Let E be a subspace of!Jtcp, such that Acp(E) is dense in H, and that, for all x in E, we have:
cp(x*x)
= .,P(x*x)
Then, the weights
IJtcp n ~t/>, therefore J Acp( x) belongs to 'D( h 112 ), and we have:
then, using the same arguments as ([23], lemma 23), we get that Jh 112 J = 1, therefore h = 1 and
2.1.6 Definitions. (i) For every w in M*, we define: llwllcp
= sup{l(x*,w)l,
x EM, cp(x*x)::; 1}
Since the weight
= 0 implies w = 0.
Thus, for every w in Icp, there exists a unique vector acp(w) in Hcp such that, for all x in ~cp: Moreover, we have: An element of Icp will be called square-integrable. H there is no confusion, we shall write a(w) instead of acp(w).
2.1.7 Proposition. With the hypothesis and notations of 2.1.6, we have: (i) For any"' in 'D'rJ and in V(1r 1 ("1)*), we,, belongs to Icp, and we have:
e
(ii) The set Icp is a dense subspace in M*, and a(Icp) a dense subspace in
Hcp.
52
2. Kac Algebras
(iii) Let w be in lcp, x in M. Then, with the notations of 1.1.1 {ii), x · w belongs to lcp, and we have: a(x ·w) = xa(w) (iv) Let w be in lcp, x in 'Jtcp· We have, with the notations of 1.1.1 {ii): Qa(w),A'P(z)
= w . x*
·
Proof. Let x in 'Jtcp. We have: (x*e 177) = ce 1x77) = (e l1r'(11)Acp(x)) = (1r'(11)*e I Acp(x))
for 17 is in 1)~ fore is in 1J(7r1 (77)*)
From that (i) follows immediately. The result (i) implies that lcp contains a least all states of the form we, 11 where thee and 17 are elements in!!', which implies that a(Icp) contains !!' 2 ; so (ii) is proved. Let y be in '.Ytcp; we have:
(y*, x · w}
= (y*x,w} = (a(w) IAcp(x*y)) = (a(w) Ix* Acp(y)) = (xa(w) I Acp(y))
so (iii) is proved. For ally in M, we have:
(y, na(w),A'P(z)}
= (a(w) I y* Acp(x)) = (a(w) I Acp(y*x)) = (x*y,w} = (y,w · x*}
which is (iv). 2.1.8 Definitions and Notations. We shall use the constructions and notations of Haagerup's theory of operator-valued weights ([59) or [150)): (i) Let M be a von Neumann algebra; the extended positive part .M+ of M is the set of all lower semi-continuous additive functions m: Mt --t [0, +oo), which satisfies m( ..\w) = .Am(w) for all .A in JR.+ and w in Mt. Given m, n in .M+, .A in JR.+ and x in M, it is straightforward to define m + n, .Am and x*mx in ..M+, and it is clear that M+ is naturally imbedded in ..M+. Moreover, ([59), cor. 1.6), for every m in ..M+, there exists an increasing sequence of elements of M+ converging up tom. Let H be an Hilbert space on which M is standard (1.1.1 (iii)); then, ([59), lemma 1.4), for every m in ..M+, there exists a closed subspace H' of H, and a positive self-adjoint operator Ton H' such that 'D(T)- = H' and:
m(we) = IIT1/2ell2 m(we) = +oo
ce E 'D(Tl/2)) (e
rt 'D(Tl/2))
2.1 An Overview of Weight Theory
53
Now, let N be a von Neumann subalgebra of M; an operator valued weight on M with values in N is a mapping E : .M+ -+ :&+ such that:
E(m + n) = E(m) + E(n) (m,n E .M+) E(.\m) = .\E(m) (.\ E R+, mE _M+) E(a*ma) =a* E(m)a (a EN, mE .M+) Every weight on M may be considered as an operator valued weight on M with values in the von Neumann subalgebra ClM. Faithfulne88, 8emi-finitene88 and normality of operator valued weights are defined exactly the same way as for weights. Those operator valued weights such that E(lM) = lN are called conditional expectation8; as we have then E(M+) = N+, E can be extended to a positive linear map from M toN, which satisfies E 2 =E. (ii) Let M be a von Neumann algebra, N a von Neumann subalgebra of N,
PAcp(x) = Acp(Ex) Let A be a von Neumann algebra and w be a positive element in A., such that w(l) = 1. We may then define a conditional expectation (i ® w) from M ® A to M ® I(! ~ M, such that, for any faithful semi-finite normal weight on M, we have: <po(i®w)=
54
2. Kac Algebras
2.1.9 Lemma. For all w in
M.,
we have:
(i) (w ® i)(roti®<,c>) C rot'P (ii) (w ® i){IJti®<,c>) C IJt'P. Proof. Let us suppose w is positive; for any X in rot~'P' then (w ® i)(X) is positive and we have: tp({w ® i)(X))
= w(i ® tp)(X) <+co
from which it results that (w ® i)(X) belongs to rott, and (i) follows, by linearity. Let us suppose now that w is positive and w{l) = 1. Then w ® i is a normal conditionnal expectation from M ® M to M, and we have, for all
XinM®M:
((w ® i)(X))*(w ® i)(X):::; (w ® i)(X* X) whence, if X belongs to !Jli®<,c>:
tp({(w ® i)(X))*(w ® i)(X)) :::; tp(w ® i)(X* X) :::; +co
by {i)
from which {ii) follows, by linearity. 2.1.10 Lemma. Let x be in IJttp andy be in
rolwA
then (x* ® l)y(x ® 1)
belongs to roltp®tp, and we have:
Proof. For all a in M+, we have:
and this equality remains true if a is in the extended positive part of M; in particular, for ally positive in M ® M, if we replace a by {i ® tp)(y), we get:
which may be written, using {[59], 5.5):
Then, both sides are finite if y belongs to rottA
2.2 Definitions
55
2. 2 Definitions 2.2.1 Definitions. Let (M, r) be a Hopf-von Neumann algebra and cp be a faithful semi-finite normal weight on M. We shall say that cp is a left-invariant weight with respect to if it satisfies the following property:
r
(i®cp)F(x) =cp(x)l
(LIW)
Then cp satisfies the weaker property: (HWi) Let ill= (M, r, ~>.)be a co-involutive Hopf-von Neumann algebra, and cp be a faithful semi-finite normal weight on M. We shall say that cp is a Haar weight on llll if it complies with (HWi) and the two following axioms: (HWii) (i ® cp)((l ® y*)r(x)) = ~>.((i ® cp)(F(y*)(l ® x)))
(x,y E ~)
(left and right-hand sides of the equation in (HWii) make sense, thanks to (HWi))
(t E R)
(HWiii)
2.2.2 Proposition. Let G be a locally compact group and lllla(G) = (£C:le(G), Fa, K.a) the associated abelian co-involutive Hopf-von Neumann algebra {1.2.9). Let us consider the faithful semi-finite normal weight cp 0 on L 00 (G), which is the trace arising from the Haar measure on G by:
C(Ja(i)
=!a
f(s)ds
It is a left-invariant and a Haar weight on lllla(G). Moreover, the weight cp is finite if and only if the group G is compact. Proof. We identify Hrp,. with L 2 (G), ~.. with L 2 (G) n L 00 (G), rolrp,. with L 1 (G) n L 00 (G); let fin L 00 (G)+; since fa llfll 2 (st)dt =fa llfll 2 (s)ds, it is clear that (LIW) holds. The translation of (HWii) is: lag(t)f(st)dt = lag(s- 1t)f(t)dt
which results straightforwardly from the left-invariance property of ds. And (HWiii) is trivial, cp being a trace.
56
2. Kac Algebras
2.2.3 Proposition. Let E = (M, r, /\:) be a co-involutive Hopf-von Neumann algebra, and 1.{) be a faithful semi-finite normal weight on M. If 1.{) satisfies (LIW) (resp. (HWi), (HWii), (HWiii)), then ~.p1 satisfies the same property. Proof. Let x be in M'+, and
1.{)
a left-invariant weight; then:
(i ® ~.p')r'(x) = (i ® ~.p')((J ® J)r(JxJ)(J ® J))
= J(i ®~.p)T(JxJ)J = ~.p(JxJ)l = ~.p1 (x)1 So ~.p1 is a left-invariant weight. Proving axiom (HWi), (HWii) and (HWiii) is just a straightforward calculation of the same kind, using m
2.2.4 Proposition. Let E = (M, r, /\:) be a co-involutive Hopf-von Neumann algebra, and 1.{) be a faithful semi-finite normal weight on M. If 1.{) is a leftinvariant (resp. a Haar) weight onE, then 1.p o 1\: is a left-invariant (resp. a H aar) weight on E'. Proof. In 1.2.10{ii), we have seen that the application u(x) = Jl\:(x)* J from M to M' is an E-isomorphism from E' to E'. Furthermore, we have ~.p 1 o u( x) = 1.p o /\:( x) for all x in M+; so this Eisomorphism exchanges the weight 1.p o 1\: with the weight ~.p 1 • The result is then a corollary of 2.2.3. 2.2.5 Definitions and Notations. Let E = ( M, r, 1\:) be a co-involutive Hopfvon Neumann algebra, and 1.p a Haar weight on E. We shall then say that the quadruple K = ( M, r, 1\:, 1.p) is a K ac algebra. By 2.2.2, for any locally compact group G, (£<X>(G), Fa, 1\:a, l.{)a) is a Kac algebra, denoted Ka(G). For any Kac algebra K = (M, r, 1\:, ~.p ), we denote K' = (M', r', 1\:1, ~.p1 ) and K' = ( M, ~ r, 1\:, 1.p o 1\:) the Kac algebras associated respectively in 2.2.3 and 2.2.4. They will be called respectively the commutant Kac algebra of K, and the opposite Kac algebra of K Evidently, we have Ka(G)' = Ka(G) and Ka{G)' = Ka(G0 PP). Let K = ( M, r, 1\:, ~.p) be a Kac algebra, and a > 0. It is clear that ( M' r, 1\:, a~.p) is a Kac algebra, which will be denoted a:K. Let K1 = (Mt, T1, 1\:t, 1.{)1) and K2 = (M2, T2, /\:2, 1.{)2) be two Kac algebras. We shall say that K1 and K2 are isomorphic if there exists an E-isomorphism: u:
{Mt,T1,/\:1) - t (M2,r2,/\:2)
and a > 0 such that 1.{)2 o u = a~.p1. Clearly, then, u is implemented by a unitary u from H 'Pl onto H 'P2 defined for all X in m'Pl ' by:
UAp1 (x) = a- 112 Ap2 (u(x)) Therefore K and a:K are isomorphic.
2.2 Definitions
57
2.2.6 Proposition. Let lK = (M, r, 11:, It') be a K ac algebra and R a projection in the centre of M, such that: F(R) ~ R®R ~~:(R) = R We shall denote by IKR the quadruple (MR,rR,II:R,tt'R) where, for all x in
M: {i) MR is the usual reduced algebra; the canonical surjection M --+ MR will be denoted by r; {ii) {iii)
FR(r(x)) = (r ® r)(F(x)) II:R(r(x)) = r(~~:(x))
(iv) the weight It' R on M R is obtained by reduction from It' on M as in
{[16], def. 9.2 ..4). Then IKR is a Kac algebra, called a reduced Kac algebra of IK, and r is a surjective l8l-morphism. Proof. It is trivial to check that (MR, FR, t~:R) is a co-involutive Hopf-von Neumann algebra and that r is an l8l-morphism. We have rot'PR = r(rot'P) which implies:
Let x, y be in S)l'P. We have: (i ® tt'R)((1 ® r(y)*)rR(r(x))) = (i ® tt'R)(r ® r)((1 ® y*)r(x)) = (i ® tt')(r ® r )((1 ® y*)r(x )) = (i ®It')(( R ® Ry*)F(Rx)) = R(i ® tt')((1 ® Ry*)F(Rx)) = R~~:(i ® tt')(F(Ry*)(1 ® Rx)) = ~~:(i ® tt')(F(Ry*)(R ® Rx)) = ~~:(i ® tt')((R ® R)F(y*)(1 ® x)) = II:R(i ® tt'R)(r ® r)((F(y*)(1 ® x)) = 11:R(1 ® tt'R)(F(r(y)*)(1 ® r(x)))
As
ruf = ufRr, the axiom (HWiii) is trivially proved.
2.2.7 Proposition. Let lK = ( M, r, ~~:,It') a K ac algebra and M a sub von Neumann algebra of M such that:
58
2. Kac Algebras
{i} r(M) c (ii) ~~:(M) =
M®M M (iii} u'f(M) = M (t
E JR.)
(iv) the restriction cp I£"1+, which will be denoted cj;, is a semi-finite weight. We shall denote lK the quadruple (M, it, cj;) where and it are respectively the restrictions of r and 11: to M; the canonical one-to-one morphism M --+ M will be denoted by j. Then, i is a Kac algebra, called a sub-Kac algebra of :K, and j is a oneto-one Ill-morphism.
r,
r
Proof. Everything is proved by restriction. In particular, using ([16], 3.2.6),
we have
uf = uilv·
2. 3 Towards the Fourier Representation In what follows, (M,r) is a Hopf-von Neumann algebra and cp a faithful, semi-finite, normal weight on M, satisfying (HWi). 2.3.1 Lemma. For all w in M., we have: {i} (w ® i)r(rot"') c rot"' {ii} (w ® i)F('Jk,o) C 'Jkp. Proof. It is clear that (HWi) implies F(rot'l') C roti®
2.3.2 Definition. Thanks to 2.3.1, we can define an unbounded linear operator l(w) on H'P by:
V(l(w)) = A'P('Jkp) l(w)A'P(x) = A
l(w *w') = l(w1 )l(w). Proof. Let x be in 'Jk,o; we have:
l(w * w1 )Ap(x) = A'P((w * w1 ® i)F(x)) = A"'((w ® w' ® i)(r ® i)r(x))
by 2.3.2
2.3 Towards the Fourier Representation
59
= Arp((w ® w' ® i)(i ® T)T(x)) = Arp((w' ® i)(w ® i ® i)(i ® T)T(x))
= Arp((w' ® i)T((w ® i)T(x))) = l(w')Arp((w ® i)T(x)) = t(w')t(w)Arp(x)
by 2.3.1 and 2.3.2 by 2.3.2
2.3.4 Lemma. For all w in M., x, y in IJtrp, we have:
(t(w)Arp(x) I Arp(y)) = (w ® cp)((1 ® y*)T(x)) Proof. We have, if w is positive and w(1)
= 1:
(t(w)Arp(x) IArp(y)) = (Arp((w ® i)(T(x)) IArp(y)) = cp(y*(w ® i)T(x)) = (w ® c,o)((1 ® y*)r(x))
by 2.3.2
and so the result is proved, by linearity.
r,
2.3.5 Definition. Let lEI = (M, ~~:) be a co-involutive Hopf-von Neumann algebra, and cpa faithful, semi-finite, normal weight on M, satisfying (HWi). Then, we define, for all win M., an unbounded operator .X(w) by:
.X(w)
= t(w o ~~:)
We have then:
'D(.X(w)) = Arp(IJtrp) .X(w)Arp(x) = Arp((w o 11: ® i)T(x)) (x E !Jtrp) .X(w * w') = .X(w).X(w') (w,w' EM.) (.X(w)Arp(x) IArp(y)) = (w o 11: ® c,o)((1 ® y*)T(x)) (wE M.; x, y E Nrp) 2.3.6 Example. Let G be a locally compact group; let us consider the Kac algebra Ka(G) defined in 2.2.5; for all fin L 1 (G) the operator..\(!) defined in 2.3.5 is equal to the restriction of .Xa(f) to L 2 (G) n L 00 (G), where ..\a is the left regular representation of L 1 (G) (cf. 1.1.5). Proof. By 2.3.5, we have, for fin L 1 (G), gin L 2 (G)
(.X(f)g)(t)
=
Ia
f(s)g(s- 1 t)dt
n L 00 (G), tin G:
= (.Xa(f)g)(t)
.
by 1.1.5
60
2. Kac Algebras
2.3.7 Example. Let us compute the mapping >..' associated to the co-involutive Hopf von Neumann algebra l!ll' and the weight r.p1• Let w be in. M.; we note w' the element of M' defined, for all x in M', by:
(x,w')
= (Jx* J,w)
We easily get (w o K) 1 = w1 o K 1 , and w'(JxJ) Then, we have, for x in ~:
J>..'(w')JA,.,(x)
= w(x) for all x in M.
= J>..'(w')Acp~(JxJ) = J A,.,,((w' o K 1 ® i)T'(JxJ)) = JA,.,,(((w o K) 1® i)(J ® J)r'(x)(J ® J)) = A,.,((w o K ® i)T(x)) =
>..(W)Atp( X)
by 2.3.5
and then:
>..'(w') = J>..(w)J
2.4
The Fundamental Operator W
In what follows, (M, T) is a Hopf-von Neumann algebra and r.p a left-invariant weight with respect tor (so r.p satisfies (HWi)). 2.4.1 Lemma. For all x, y in
~'
we have:
(r.p ® r.p)((x* ® 1)T(y*y)(x ® 1)) = r.p(x*x)r.p(y*y). Proof. By definition (2.2.1), we have:
(i ® r.p)T(y*y) = r.p(y*y)1 whence, by ([59] 2.1 (3); cf. 2.1.8 (i)):
r.p(y*y)x*x = x*(i ®r.p)T(y*y)x = (i ® r.p)((x* ® 1)T(y*y)(x ® 1)) Then:
r.p(x*x)r.p(y*y) = (r.p ® r.p)((x* ® 1)T(y*y)(x ® 1)).
2.4 The Fundamental Operator W
61
2.4.2 Proposition. (i) There ezi8ts a unique i8ometrg W such that, for every
x, y in '.ntp, we have:
(ii) For all x in M, we have: F(x)W
= W(1 ® x)
(iii) W belongs toM® .C(HIP) Then, W is called the fundamental operator associated to (M, r, cp). Proof. By polarization and linearity it follows from 2.4.1 that, for all
Xt,
x2,
Yl , Y2 in '.ntp we have:
which can be also written as: (~®tp(F(yt)(xt ® 1))
I Atp®tp(F(y2)(x2 ® 1)))
= (~(xt) ® Atp(Yt) I AIP(x2) ® Atp(Y2)) As Atp('.ntp) ® Atp('.ntp) is dense in HIP® HIP, there exists an isometry W in .C(HIP ®HIP) such that, for all x, yin '.ntp, we have:
which is (i). Let x be in M, y, z in '.niP. We have:
F(x)W(Atp(Y) ® Atp(z))
= F(x)Atp®tp(F(z)(y ® 1))
by (i)
= AIP®IP(r(zz)(y ® 1))
= W(Atp(Y) ® ~(xz)) = W(1 ® x)(Atp(y) ® ~(z))
by (i)
By linearity, continuity and density, we obtain (ii). Let a, b be in '.ntp. We have:
(JaJ ® JbJ)W(Atp(x) ® Atp(y))
= F(y)(x ® 1)(J~(a) ® JAIP(b)) = F(y)(JaJAIP(x) ® J~(b)) = F(y)(JaJ ® 1)(Atp(x) ® JAIP(b))
This equality still holds, by continuity, when AlP( x) converges to any vector e in HIP and when a strongly converges to 1. Therefore, we have:
(1 ® JbJ)w(e ® ~(y)) = r(y)(e ® J~(b))
62
2. Kac AlgebrBB
which is valid for ally in IJtcp and e in Hcp. Let x' be in M' and let us replace by x' in the above formula:
e
e
(1 ® JbJ)W(x' ® l)(e ® Acp(y))
= r(y)(x' ® l)(e ® JAcp(b)) = (x' ® 1)r(y)(e ® JAcp(b) = (x' ® 1)(1 ® JbJ)W(e ® Acp(Y))
which still holds when b strongly converges to 1, then we get:
W(x' ® 1)(e ® Acp(y)) = (x' ® 1)W(e ® Acp(Y)) By linearity, density and continuity we can conclude that W and x' ® 1 commute, therefore W belongs toM® C(Hcp)· 2.4.3 Lemma. We have, for all x, y
w((i ® cp)(r(y*)(1 ® x)))
in~
and w in M.:
= (Acp(x) I(w ® i)(W)Acp(y)).
Proof. Let a be in 'Jtcp; we have: wA
by 2.1.10 by (LIW) by 2.4.2(i)
and so the result is proved by continuity and linearity arguments. 2.4.4 Corollary. Let (M, r) be a Hopf-von Neumann algebra, and cp a left-
invariant weight with respect to r. Then: (i} for all w in M., the linear mapping i(w) defined in 2.3.2 by: Acp(x)-+ Acp((w ® i)r(x)) is bounded; we shall still note l(w) its unique continuous extension to Hcp; it satisfies: l(w) = (w ® i)(W) and lll(w)ll ~ llwll (ii) the fundamental operator W satisfies: (r ® i)(W) = (1 ® W)(u ® 1)(1 ® W)(u ® 1)
2.4 The Fundamental Operator W
63
which can also be expressed as the so-called "pentagonal relation": (1 ® W)(u ® 1)(1 ® W)(u ® 1)(W ® 1) = (W ® 1)(1 ® W).
Proof. Let w be in
Mt, z, y in 'Jlrp; we have:
(Arp(x) ll(w)Arp(y)) = (Arp(z) I Arp((w ® i)F(y))) = c,o(((w ® i)F(y*))x) = w(( i ® c,o )F(y*)(1 ® z)) = (Arp(x) I(w ® i)(W)Arp(Y))
by 2.3.2
by 2.4.3
By linearity, we see that, for all win M., l(w) is the restriction of (w ® i)(W) to Arp('Jlrp); so (i) is proved. If we apply {1.5.1 (i)), to the linear mapping w--+ l(w) = (i ® w)(uWu) and to the Hopf-von Neumann algebra (M,(F), we get:
(i ® (F)(uWu) = (uWu ® 1)(1 ® u)(uWu ® 1)(1 ® u) from which we deduce:
(r ® i)(W)
= (1 ® W)(u ® 1)(1 ® W)(u ® 1)
The pentagonal relation comes then from 2.4.2 (ii). 2.4.5 Theorem. Let (M, F) be a Hopf-von Neumann algebra, and c,o a leftinvariant weight with respect to Then lrp (defined in 2.1.6) is a left ideal in M.; moreover, we have, for all w in M* and w0 in lrp:
r.
a(w * w0 ) = l(w)*a(w0 ) and then:
Proof. Let z in 'Jlrp; we have: (z* ,w * w 0 } = (z,w * w0 } -
by 1.2.1
= ((w ® i)F(x),wo}= (((w®i)F(x))*,wo} = (a(wo) I Arp((w ® i)F(z))) = (a(wo) ll(w)Arp(x)) = (i(w)*a(wo) I Arp(z))
by 2.3.2 by 2.4.4(i)
64
2. Kac Algebras
which gives, by density, the first result; the norm inequality is just then a staightforward application of 2.1.6 (ii) and 2.4.4 (i). 2.4.6 Proposition. Let l!ll = (M, r, ~~:) be a co-involutive Hopf-von Neumann algebra, and cp a left-invariant weight with respect to Then: (i) For all w in M., the linear mapping A(w) defined in 2.3.5 by:
r.
A(w )A'P(x) = Atp((w o 11: ® i)F(x ))
(x E !Jtp)
is bounded; we shall still note A(w) its unique continuous extension to H'P; it satisfies: A(w) = (w o 11: ®i)(W) and !IA(w)ll :5 llwll (ii) For all w in M. and all w0 in Itp, we have:
(iii) I'P n ~ is a dense involutive subalgebra of M •. (iv) The weight cp satisfies (HWii) if and only if A is involutive; we have then, for all w in M. and all w1 in Itp:
a(w * w') = A(w)a(w') (v) The weight cp satisfies (HWii) if and only i/, for all w in
M.,
we have:
(w ® i)(W*) = (w o 11: ® i)(W). Proof. The assertion {i) is just a rewriting of 2.4.4(i). Thanks to 1.2.5, (ii) comes from 2.4.5. By 2.1. 7 (ii), we see that Itp is a dense left ideal of M.; so ~is a dense right ideal of M., and the algebraic tensor product I; 0 I'P is a dense subspace of (M ® M) •. Let X be in M such that (x, wr * W2) = 0 for all WI. W2 in Itp. We get (r( X), wr ® w2) = 0 which therefore implies X = 0. Thanks to the Hahn-Banach theorem, it follows that the subspace generated by ~ * Itp is dense in M •. But, by 2.4.5, this subspace is included in I'P n and so (iii) is proved. Using 2.3.5, we get, for any w in M., x, y in S)l'P:
I;,
(A'P(x) I A(w 0 )Atp(y))
= (A(w )Atp(Y) IAtp(x))= (w o 11: ® cp)({l ® x)F(y))0
0
= (w ® cp)({l ® x*)r(y))= (w ® cp)(F(y*){l ® x))
Using again 2.3.5, we have:
(A'P(x) I A(w)* Atp(y)) = (w o 11: ® cp)((l ® y*)r(x))
by 1.2.5
2.4 The Fundamental Operator W
65
And, by density and continuity, we get (iv). If we apply 1.5.1 (ii) to the linear mapping w -+ ,\(woK)= (i ® w)(uWu) and to the co-involutive Hopf-von Neumann algebra 18[', we get that ,\is involutive if and only if, for all win M., we have: (i ®w o K)(uWu) = (i ®w)(uW*u) and so (v) comes from (iv). 2.4.7 Proposition. Let G be a locally compact group. (i) The functions-+ -\a(s ), considered as an element ofCb(G, C(L 2(G))), which can be identified to a subspace of C(L 2(G))®L 00 (G), is equal to uW(iu, where Wa is the fundamental operator of the Kac algebra Ka(G). {ii) The ideal ltp,. of the Kac algebra Ka(G) is L 1 (G)nL 2 (G); the mapping a is then the canonical injection from L 1 (G) n L 2(G) into L 2(G).
Proof. Let g1,g2 be in L 00 (G)
n L 2(G);
by definition, we get, for s,t in G:
(Wa(gl ® g2))(s, t) where we identified L 2( G) ® By continuity, for all fin
= g2(st)g1(s)
L 2( G) with L 2( G X G). L 2 (G x G), we get:
(Waf)(s,t)
= f(s,st)
and:
(uWauf)(s,t) = f(ts,t) We identify L 2 (G,L 2(G)) with L 2(G) ® L 2(G) (or L 2(G x G)): ~ in L 2(G,L 2 (G)) will be identified to the function f: (s,t)-+ ~(t)(s). We have then: (-\a(C 1 )~(t))(s)
= ~(t)(ts) = f(ts, t) = (uWauf)(s, t) = (uWau~(t))(s)
so we get (i); and (ii) is trivial. 2.4.8 Proposition. The fundamental operator W' associated to ( M', r', cp1) as defined in 2.2.3 is: W' = (J ® J)W(J ® J) .
Proof. It is an easy calculation, using the identification Arp~ ( J x J) = J Atp( x ), for all x in ')111'. 2.4.9 Lemma. Let (M, F) be a Hopf-von Neumann algebra, cp a left-invariant weight with respect to r, w the fundamental operator associated by 2.4.2; let A be a von Neumann algebra, .,P a faithful semi-finite normal weight on A,
66
2. Kac Algebras
X in IJ'l,p®
= (1 ® W)(1 ® a)(AI/J®
.
Proof. We have: (t/J ® cp ® cp )((1 ® y* ® 1)(i ®F)( X* X)(1 ® y ® 1)) = cp(y*(t/J ® i ® cp)(i ® F)(X* X)y) = cp(y*(t/J ® cp)(X* X)y) = cp(y*y)(t/J ® cp)(X* X)< +oo which gives the first part of the result. After polarization, we get, from the calculation above, the existence of an isometry U sending AI/J®
AI/J®
2.5 Haar Weights Are Left-Invariant In the sequel, llli = (M, r, K) is a co-involutive Hopf-von Neumann algebra, and cp a faithful, semi-finite, normal weight on M, satisfying (HWi) and (HWii); so, we can use the results 2.3.5 (but not 2.4 which is about leftinvariant weights, which may not satisfy (HWii)). 2.5.1 Lemma. For all w in M., and x in !Jtp, we have:
Proof. Let us assume that w is positive and w(1) = 1. We have, for all y in !Jtp: (.A(w)~(x) I ~(y))
= (w o"' ® cp)((1 ® y*)r(x))
by 2.3.5
2.5 Ha.ar Weights Are Left-Invariant
= (w ® cp)(F(y*)(l ® x)) = cp((w ® i)(F(y*)(l ® x)))
67
by (HWii)
= cp((w ® i)F(y*)x)
= (Ar,c(x) I Ar,c(w ® i)F(y)) = (Arp(x) I .\(w )Ar,c(Y)) 0
by 2.3.5 and the hypothesis So, we get .\(w)*
~
.\(w0 ) which is the lemma.
e
2.5.2 Lemma. Let be in Ar,c(SJlr,c)· The linear form ve on M*, defined by ve(w) = (.\(w)ele) is continuous, and llvell:::; llell 2 . Proof. By adding a unit e, let us consider the involutive Banach algebra
M* EB Ce. We can extend ve to ve, by putting, for all w in M*, a in C:
Then, we have:
ve((w 0
+ ae) * (w + ae)) = ve(w *W + aw +aw + aa) = ve(w * w) + ave(w) + ave(w + aallell 2 = (.\(w * w)e Ie)+ (.\(w )e Iae) + (.\(w )ae Ie)+ ilaell 2 = ll.\(w)ell 2 + (.\(w)e Iae) + (ae I .\(w)O + llaell 2 by 2.3.5 and 2.5.1 = ll.\(w)e + aell 2 2: 0 0
0
0
0
0
)
0
Therefore ve is positive on M* EB Ce, so is continuous, and we have:
which implies the lemma, by restriction toM*. 2.5.3 Theorem. With the hypothesis of 2.5, for all w in M* the operator .\( w) defined in 2.3.5 by:
.\(w)Ar,c(x) = Ar,c((w o K ® i)F(x))
(x E SJtrp)
is bounded; we shall still note .\(w) its unique continuous extension to Hr,c. The mapping A from M* to C(Hr,c) then defined is a representation of the involutive Banach algebra M*, which will be called the Fourier representation. We shall write M the von Neumann algebra on Hr,c generated by>..
68
2. Kac Algebras
Proof. Thanks to 2.5.2, for all
ein A'f'('.n""), we have:
which implies, using 2.3.5 and 2.5.1:
So .A(w) is bounded, and and 2.5.1.
II.A(w)ll
~
llwll·
The whole proof is given by 2.3.5
2.5.4 Example. Let G be a locally compact group; let us consider the Kac algebra lKa (G) defined in 2.2.5; using 2.3.6, we see that the Fourier representation of lKa (G) is the left regular representation of L 1 (G), and the algebra L 00 (Gr is then the von Neumann algebra C(G) defined in 1.1.7. 2.5.5 Lemma. For all w in M*, and t in R, we have: (i) .A(w)Ll~ = Ll~.A(w) (ii) J.A(w)* J = .A(w o 11":).
Proof. Let
X
be in 'Jl'f' n '.)1~. Let us assume that w is positive. Then, we have:
S'f'.A(w)A'f'(x) = S'f'A'f'((w o II":® i)F(x)) = A'f'((w o II":® i)F(x*)) = .A(w)A'f'(x*) = .A(w)S'f'A'f'(x)
by definition
Since A""(~ n 'Jl~) is a core for S'f', we get:
By substituting w0 tow (w 0 is still positive), we get:
or, using 2.5.3: and, by transposing: Then, we get:
from what (i) follows for w positive and by linearity in the general case.
2.5 Haar Weights Are Left-Invariant
We have also, for all w in M* and
Sc,o.A(w)* Sc,oAc,o(x)
X
69
in 'Jlcp n '.)1~:
= Sc,o>.(w )Ac,o(x*) = Sc,oAc,o((w o"' ® i)F(x*))
by 2.5.3
= Ac,o((w ® i)F(x)) = >.(w o t>,)Ac,o(x)
by 1.2.5
0
0
Since Ac,o('Jlc,o n 'Jl~) is a core for Sc,o, we get:
which, by (i), implies:
J>.(w)* J
= >.(w o "')
which completes the proof. 2.5.6 Corollary. For all t in R, we have:
ruf
= (i ® uf)r .
Proof. Let x be in 'Jlcp, w in M*;we have:
Ac,o((w o"' ® i)(Fuf(x)))
= .A(w)Ac,o(uf(x))
by 2.3.5
it
= >.(w)LlcpAc,o(w) •t = Ll~>.(w)Ac,o(w)
= Ll~Ac,o((w
by 2.5.5(i)
o"' ® i)(F(x)))
by 2.3.5
= Ac,o(uf((w o "'® i)(F(x))) = Ac,o((w o"' ® i)((i ® uf)(F(x))) which implies:
(w o "'® i)(Fuf(x))
= (w o "'® i)((i ® uf)r(x)))
therefore, for all x in 'Jlcp, we get:
ruf(x)
= (i ® uf)r(x)
which completes the proof, by density and continuity. 2.5.7 Corollary. Let x in M, analytic with respect to 'P· Then, for all w in M*, the element (w ® i)F(x) is analytic with respect to((', and, for all z in C, we have:
ur((w ® i)F(x))
= (w ® i)r(ur(x))
.
70
2. Kac Algebras
Proof The function z - t (w ® i)F(uf(x)) defined on Cis analytic, and, by 2.5.5, it extends to C the function defined on R by t - t ui((w ® i)F(x )).
r, ~t) be a co-involutive Hopf-von Neumann algebra, r.p a faithful, semi-finite, normal weight on M, satisfying {HWi) and {HWii). Then r.p is left-invariant.
2.5.8 Theorem. Let llll = ( M,
Proof Let w be in Mf, y in ~ and x; a family of elements satisfying the hypothesis of 2.1.2. We can write down:
r.p(((w ® i)F(y))(x;)) = r.p((w ® i)(F(y)(1 ® x;)) = (w ® r.p)(F(y)(1 ® x;)) = (w o ~t ® r.p)((1 ® y)F(x;)) = r.p(w o ~t ® i)((1 ® y)F(x;)) = r.p(y(w o ~t ® i)(r(x;)))
by (HWii)
It follows from 2.3.1 (i) that (w ® i)F(y) belongs to ~, and from both 2.3.1 (ii) and 2.5.7 that (woK.® i)F(x;) satisfies the hypothesis of 2.1.2. By passing to the limit, we get, for all win Mf andy in~:
r.p((w ® i)F(y))
= r.p(y)w(1)
with w being fixed, these two expressions will define two faithful semi-finite, normal weights on M which coincide on ~. The modular group of the second one is and we have:
ur
(w ® r.p)F(ut(y))
= (w ® r.p)(i ® uf)F(y) = (w ® r.p)F(y)
by 2.5.6
Therefore, the first one is invariant by that group. It follows from ([114], prop. 5.9), that they are equal. Then we have, for ally in M+ and win Mf:
w(( i ® r.p )F(y))
= w( r.p(y)1)
By linearity it is true for all win M., which completes the proof. 2.5.9 Example. The Fourier representation >..' of llll' satisfies:
(w 1 ) = ..\(w),
..\1
Vw EM*
where w'(x) = w(Jx* J), for all x in M'.
Proof. In 2.3.7, we have got ..\1 (w 1 ) = J..\(w)J; using 2.5.5(ii), 2.5.3 and 1.2.5, we get the result.
2.6 The Fundamental Operator W Is Unitary
71
2. 6 The Fundamental Operator W Is Unitary We keep on with the same hypothesis as in 2.5. The weight t.p being leftinvariant, thanks to 2.5.6, it is possible to use all the result of 2.4, in particular the construction of the fundamental operator W (2.4.2), the links between W, .X, and the ideal Icp (2.4.6). Let us recall that M is the von Neumann algebra generated by the Fourier representation .X (2.5.3). 2.6.1 Proposition. The fundamental operator W, the Fourier representation .X and the ideal Icp are linked by: {i) For all w in M., we have:
.X(w) (ii) For all
n in M.,
= (w o 11: ® i)(W) = (w ® i)(W*) we have:
.x.(n) = (i ® n)(W*)
= ~~:(i ® n)(W)
(iii) For a, {3, "'(, 6 in Hcp, we have the connection formula:
(W(a ® {3) 11 ® 6)
= (f31.X(w-y,a)6)
{iv) The fundamental operator W belongs to M ® (v) For all w in M. and w1 in Icp, we have:
.X(w)a(w')
M
= a(w * w') .
Proof. We have already proved (i) in 2.4.6(i) and (v); then (ii) and (iii) are straightforward corollaries of (i), (iv) is a consequence of (i) and 2.4.2(iii), and (v) is a rewriting of 2.4.6 (iv). 2.6.2 Corollary. The fundamental operator W is unitary. More precisely, for any antilinear isometric involution .7 of Hcp implementing 11: (i.e. such that ~~:(x) = .7x* .7 for all x in M), we have:
W* = (.7 ® J)W(.7 ® J). Proof. Let us recall that, for all a, {3 in Hcp, W:Ja,:T/3 = w~,/3 (1.2.8). So:
(w ® i)((.7 ® J)W(.7 ® J)) = J(w0 ® i)(W)J = J.X(w 0 o ~~:)J = J.X(w o ~~:)* J = .X(w) = (w ® i)(W*) and the result is proved, by linearity and density.
by 2.6.1 (i) by 2.5.3 by 2.5.5 (ii) by 2.6.1 (i)
72
2. Kac Algebras
2.6.3 Corollary. {i) The Fourier repre8entation A uW*u i8 it8 generator in the 8en8e of 1.5.2. {ii) For all x in M, we have:
&8
non-degenerate, and
F(x) = W(1 ® x)W* .
Proof. The assertion (i) is clear by 2.6.1 (i), 2.6.2 and 1.5.3 and (ii) is clear by 2.4.2 (ii), thanks to W being unitary. 2.6.4 Corollary. Let A be a von Neumann algebra. For all X in A® M, we have: (i ® F)(X) = (1 ® W)(1 ® u)(X ® 1)(1 ® u)(1 ® W*) .
Proof. Let a be in A, x in M. We have: ( i ® F)( a ® X)
= a ® T( X) = a® W(1 ® x )W* by 2.6.3 (ii) = (1 ® W)(a ® 1 ® x)(1 ® W*) = (1 ® W)(1 ® u)(a ® x ® 1)(1 ® u)(1 ® W*)
which completes the proof, by linearity, continuity and density. 2.6.5 Proposition ([79]). Let A be a von Neumann algebra, V a unitary of A ® M 8uch that: (i ® F)(V) = (V ® 1)(1 ® u)(V ® 1)(1 ® u)
Then, for every w in M, we have:
(i ® w o ~~:)(V) = (i ®w)(V*) The mapping r : M* -+ A defined by r(w) = (i ® w )(V) i8 a non-degenerate repre8entation of M, and V i8 the generator of r. Proof. By 2.6.4, we have:
(1 ® W)(1 ® u)(V ® 1)(1 ® u)(1 ® W*) = (V ® 1)(1 ® u)(V ® 1)(1 ® u) which can also be written: (V ® 1)(1 ®u)(1 ® W*)(1 ®u)(V* ® 1) = (1 ®u)(1 ® W*)(V ® 1)(1 ®u) (*)
2.6 The Fundamental Operator W Is Unitary
73
Let us consider the representation of M* in A® .C(H) defined by:
= V(1 ® .X(w))V*
J-L(w)
Therefore, for all w in M't 1 , we have, using 2.6.1 (i):
J-L(w)
= (i ® i ® w)((V ® 1)(1 ® uW*u)(V* ® 1)) =
For all
(i ® i ® w)((1 ® u)(1 ® W*)(V ® 1)(1 ® u))
by ( *)
fh in At 1, il2 in .C(H)t1 , and w in M., we have:
(J-L(w), ilt ® .G2) = (.Gt ® il2 ® w)((1 ® u)(V* ® 1)(1 ® W)(1 ® u)) = (.Gt ®w ® .G2)((V* ® 1)(1 ® W)) = (.Gt ®w)((i ® i ® .G2)((V* ® 1)(1 ® W))) = (ilt ®w)(V*(1 ® (i ® .G2)(W))) = w((.Gt ® i)(V*(1 ® (i ® il2)(W))) = w((ilt ® i)(V*)(i ® .G2)(W)) Therefore, by 1.5.1 (ii), we have: ~t((.Gt ® i)(V*)(i ® il2)(W)) = ((.Gt ® i)(V*)(i ® .G2)(W))*
which can be written as follows:
since
nl and .a2 are positive.
H we apply 1.5.1 (ii) again, we get:
or:
(i ® il2)((~t(il1 ® i)(V) ® 1)W)
= (i ® il2)(((il1 ® i)(V*) ® 1)W)
which, by linearity, will still holds for all (~t(.Gt ®
n2 in .C(H)., and therefore implies:
i)(V) ® 1)W = ((ilt ® i)(V*) ® 1)W
As W is unitary, we have: ~t(.Gt ®
i)(V) = (.Gt ® i)(V*)
74
2. Kac Algebras
and by 1.5.1 (i) and (ii), r is a representation and we have, for all win M.:
(i ®w o ~~:)(V) = (i ®w)(V*) As V is unitary, r is non-degenerate (1.5.3).
2.6.6 Corollaries. (i) Let u be in the intrinsic group of (M, r, ~~:). Then, we have: ~~:(u) = u* and the application w -+ w( u) is a one-dimensional representation of M., which has u as generator. (ii) If u, v are two elements of the intrinsic group of (M, r, ~~:) the Kronecker product of the two one-dimensional representations w -+ w( u) and w -+ w( v) is w -+ w( uv). So, the Kronecker product, restricted to the intrinsic group, is the usual product. (iii) If J1. is a representation of M., we have, for all w in M., with the notations of 1.1.1 (ii): (JJ. x u)(w) = JJ.(u · w) (u X JJ.)(w) = JJ.(w · u).
Proof. By 1.2.3, u is a unitary and satisfies F( u) = u ® u. We can then apply 2.6.5, with A = C, and (i) is proved; then (ii) is just a corollary of 1.5.5; (iii) is an application of 1.4.3, with the representations J1. and u.
2.6. 7 Proposition. Let R be in the centre of M, such that:
F(R)
~
~~:(R)
Then, we have: (i) W(R®R) = (R®R)W (ii) F(R)(R ® 1) = F(R)(1 ® R)
R®R = R
= R ® R.
Proof. By hypothesis, we have, using 2.6.3 (ii):
R®R
= (R ® R)F(R) = (R ® R)W(1 ® R)W*
Using 2.6.2, we get:
R®R
= (R ® R)(.:T ® J)W*(.:T ® J)(1 ® R)(.:T ® J)W(.:T ® J)
2.6 The Fundamental Operator W Is Unitary
75
and, using the fact that J RJ = R (because R belongs to the centre of M) and that :JR:J = R (because ~~:( R) = R), we have: R®R= (R®R)W*(1 ®R)W Taking adjoints, we get: R®R= W*(1 ®R)W(R®R) and, as W is unitary: W(R®R) = (1 ®R)W(R®R) = (R®R)W(1 ®R) = (R®R)W
by 2.6.1 (iv) by(*)
So, (i) is proved. We have then: F(R)(R ® 1)
= W(1 ® R)W*(R® 1) = W(R®R)W* =R®R
by 2.6.3 (ii) by 2.6.1 (iv) by (i)
Applying this result to (M, r;F, ~~:,
=R ® R
which ends the proof. 2.6.8 Corollary. Let P, Q two projections in the centre of M, such that: F(P)?::. P®P r(Q)?::.Q®Q P+Q?::.1
Then, either P or Q is equal to 1. Proof. From 2.6. 7 (ii), we have:
F(P)((1- P) ® P) = 0 r(Q)(Q ® (1- Q)) = o which implies, as 1- Q :5 P and 1- P :5 Q, by hypothesis: F(P)((1- P) ® (1- Q)) = 0 r(Q)((1- P) ® (1- Q)) = o
76
2. Kac Algebras
Taking the sum of these equalities, as F(P)
+ r( Q) ~ 1, we get:
(1 - P) ® (1 - Q)
=0
which ends the proof.
2. 7 Unicity of the Haar Weight
r,
Let (M, ~~:) be a co-involutive Hopf-von Neumann algebra and cp, t/J two faithful semi-finite normal weights on M+, satisfying both (HWi) and (HWii). 2.7.1 Lemma. Let z be in M such that F(z) = z ® 1; then z is scalar. Proof. We have, for all t in R:
r(uf(z))
= (i ® uf)r(z)
by 2.5.6
= (i ® uf)(z ® 1) =z®1 = r(z)
As
r
is one-to-one, uf(z) = z and z belongs toM"'. Let x, y be in 'Jlr.p; we have:
W(1 ® Jz* J)(Ar.p(x) ® Ar.p(y)) = W(Ar.p(x) ® Ar.p(yz)) = Ar.p®r.p(F(yz)(x ® 1)) = Ar.p®r.p(F(y)F(z)(x ® 1)) = Ar.p®r.p(F(y)(zx ® 1)) = W(Ar.p(zx) ® Ar.p(y)) = W(z ® 1)(Ar.p(x) ® Ar.p(y)) by linearity and density, we get:
W(1 ® Jz* J)
= W(z ® 1)
and then: 1 ®Jz*J = z®l
which implies z being a scalar.
by 2.4.2 (i)
by 2.4.2 (i)
2.7 Unicity of the Haar Weight
2.7.2 Proposition. The relative position of M and (i) MnM' = c (ii) M' n M' =c.
Proof. Let
X
be in M
M is
77
such that:
n M'; we have, then: F(x) = W(1 ~ x)W*
by 2.6.3 (ii) by 2.6.1 (iv)
=1~x
and then: r(~~:(x)) = c;(~~: ~ ~~:)r(x)
= c;(~~: ~ ~~:)(1 ~ x) =
~~:(x) ~
1
Using lemma 2.7.1, we see that ~~:(x) is a scalar, and so is x too; therefore (i) is proved. If we apply result (i) to nn', and to the weight c.p1 which satisfy the same hypothesis by 2.2.3, we get (ii), using 2.5.9. 2.7.3 Corollary. (i) Let A be a von Neumann algebra. Let P be a projection of A~ M such that (i ~ c;F)(P) ~ P ~ 1 {resp. (i ~ c;F)(P)?: P ~ 1). Then there exists a projection Q in A such that P = Q ~ 1. (ii) Let P be a projection of M such that F(P) ~ P ~ 1; then, we have either P = 0, or P = 1.
Proof. From 2.6.4, we have (i ~ c;F)(P) = (1 ~ uWu)(P ~ 1)(1 So, the hypothesis (i ~ c;F)(P) ~ P ~ 1 may be written:
(1 ~ uWu)(P ~ 1)(1 ~ uW*u)(P ~ 1)
~ uW*u).
= (1 ~ uWu)(P ~ 1)(1 ~ uW*u)
or: (P ~ 1)(1 ~ uW*u)(P ~ 1) For all w in
(i ~ i
= (P ~ 1)(1 ~ uW*u)
Mt1 , we get:
~ w)((P ~
1)(1 ~ uW*u)(P ~ 1)) = (i ~ i ~ w)((P ~ 1)(1 ~ uW*u))
Or, by 2.6.1 (i):
P(1
~
,\(w))P = P(1
~
,\(w))
which, by linearity, is true for all win M*. Taking the adjoints, we get, thanks to ,\ being a representation:
P(1
~
,\(w))
= (1 ~ ,\(w))P
which, by continuity, implies that P belongs to A~ M'; the result (i) comes then from 2.7.2(i). With the hypothesis (i ~ c;F)(P) 2: P ~ 1, the proof if analogous; taking A= C, one gets (ii).
78
2. Kac Algebras
2.7.4 Proposition. (i) Let x in C(H). Then, x belong& toM' if and only if:
(1®x)W=W(1®x) (ii) Let x be in
M',
&uch that:
W*(1®x)W=1®x then x i& &calar. Proof. We have the following sequence of equivalences: x E MA' # x *.X(w")',a)
= .X(w")',a)x* Va, '"'{ E H (.8lx*.X(w")',a)8) = (.8I.X(w"Y,a)x*8)
Va,{3,'"'{ E H # (xf31 .X(w")',a)8) = ({31 .X(w")',a)x*8) Va,{3,'"'{ E H # (W(a ® xf3) I'"'{® 8) = (W(a ® {3) I'"'{® x*8) by 2.6.1 (iii) # W(1 ®x) = (1 ®x)W #
So, (i) is proved; (ii) is then clear from (i) and 2. 7.2 (ii).
2.7.5 Proposition. For all t in R, we have:
= KO'!.tK rarOIC = (afOIC ® i)r C{) 0 O'tOIC = C{)
(i)
afOIC
{ii} {iii}
(iv) The &pace~(~ n ~o~~:) i& den&e in HIP. (v) The &pace~(~ n ']liPOIC n 'Jl~ n ~OIC) i& den&e in HIP. Proof. The assertion (i) is just an application of 2.1.3. We have:
raf0 1t
= FKa!.t"' = ~c"' ® "')ra!.t"'
by 1.2.5
= ~(K ® K)(i ® a!.t)FK
by 2.5.6
=~ell,®
by 1.2.5
=
K)(i ® a!.t)(K ® K)~r
~(i ® ailt)~r = (a fOil: ® i)r
So (ii) is proved.
by (i)
by (i)
2.7 Unicity of the Haar Weight
79
Let x be a positive element in M. We have: cp(af0 "(x))1 = (i ® c,o)raf0 "(x) = (i ® cp)(af0 " ® i)F(x)
by 2.5.8 by (ii) applied to '1/J
= af 0 "((i ® cp)F(x)) = af 0 "(cp(x)1) =cp(x)1
by 2.5.8
So (iii) is proved; then (iv) is then an application of 2.1.4. Now, let X be in mcp n ~OIC• As ~ n ~" = ~+cpo~e and as, because of (iii) and ([114], prop. 5.6. and 5.10), the weight
= (M, r, "'•
2. 7.6 Theorem. Let lK
(x ® 1)W = W(x ® 1) (vi) W*(L1~ ® 1)W = .:1~ ® .:1~ (vii) W*(L1cp ® 1)W = L1cp ® L1cp. Proof. The assertion (i) is a straightf9rward consequence of (HWiii) and 2.7.5(i). Then (ii) comes from 2.5.6, 2.7.5(ii), and (i). As we have ai®cp = ai ® ai, (ii) implies ai®cp r = raft and so we get (iii). Let us put N = (F(M) U (M ®C))". It results from (iii) that N is a aF'P-invariant subalgebra of M ® M. For x andy in~. by 2.4.1 we get that r(y )(X ® 1) belongs to mcp®cp n N. From what it is easy to deduce that the restriction of
80
2. Kac Algebras
As W is unitary (2.6.2), this subspace is Hcp ® Hcp. Therefore we have successively P = 1, E = i and N = M ® M. Let us apply this result to K5 (2.2.5), and we get (iv). Let now x in M'; we have W(x ® 1) = (x ® 1)W by 2.4.2(ii). Conversely, let us suppose W( x ® 1) = (x ® 1)W. For any y in M, we have:
(x ® 1)F(y)
= (x ® 1)W(1 ® y)W* = W(1 ® y)(x ® 1)W*
= W(1 ® y)W*(x ® 1) = r(y)(x ® 1)
by 2.6.3 (ii) by hypothesis by hypothesis by 2.6.3 (ii)
So (x ® 1) commutes with F(M); as it commutes with C ® M, by (iv), it commutes with M ® M and we get (v). Let z, y in 'Jlcp. We have:
(..1~ ® 1)W(Acp(z) ® Acp(y)) = (..1~ ® 1)Acp®cp(F(y)(z ® 1))
by 2.4.2(i)
= Acp®cp((uf ® i)((F(y)(z ® 1))) = Acp®cp(F(uf(y))(uf(z) ® 1))
= W(Acp(uf(z)) ® Acp(uf(y)))
by (ii) by 2.4.2(i)
= W(..1it ® ..1it)(Acp(z) ® Acp(Y))
from what follows (vi), by density and because W is isometric. It leads directly to (vii) because then the infinitesimal generators of those two continuous groups of unitary operators are equal. 2. 7. 7 Theorem. Let lK = (M, r, "'• r.p) be a K ac algebra, and let .,P be a faithful, 8emi-finite, normal weight on M+, 8ati8fying (HWi) and (HWii). Then, .,P and r.p are proportional. Proof. We have, for all t in :R.: .,P
= .,P o uj" =.,Po uf
by 2.7.5(iii) applied to the weights .,P and r.p by 2.7.6(i)
then, using ([114], prop. 5.6 and 5.10), we see that the weight (J = r.p + .,P is semi-finite. As 'Jte = 'Jtcp n 'Jtt/1, it is clear that (J satisfies (HWi) and (HWii); we have, then, for all t in :R.: (J
= (J o u i " = (Jo uf
by 2. 7.5 (iii) applied to the weights (J and r.p by 2.7.6(i)
2.7 Unicity of the Haar Weight
81
As <.p ~ 0, there is an injective positive operator h in M 0 , 0 ::::; h ::::; 1, such that <.p = 0( h· ). It implies, for x in IJlo:
thus, for x, yin
IJlo, since <.p 0 <.p = (0 0 O)((h 0 h)·), we shall have:
(<.p 0 <.p)((x* 01)F(y*y)(x 01))
= II(Joh1/ 2 Jo 0 = II(Joh 112 Jo 0
Joh 112 Jo)Ao®o(F(y)(x 0 1))11 2 Joh 112 Jo)WoAo(x) 0 Ao(Y)II 2
by 2.4.2 (i) applied to (M, r, 0), where Wo stands for the fundamental operator associated to (M, r, 9). By applying 2.4.2 (i) to (M, r, <.p ), it is also worth:
<.p(x*x)<.p(y*y) = II(Joh1 / 2 JoAo(x)II 2 II(Joh 112 JoAo(Y)II 2
= II(Joh1/ 2 Jo 0
Joh 1/ 2 Jo)(Ao(x) 0 Ao(y))ll 2
Therefore, we have:
IIW8(Joh 112 Jo 0 Joh 112 Jo)Wo(Ao(x) 0 Ao(Y))II 2
= II(Joh 1/ 2 Jo 0
Joh 112 Jo)(Ao(x) 0 Ao(y))ll 2
and, by the unicity of the polar decomposition, we get:
As Wo belongs to 7f'o(M) 0 .C(Ho) by 2.4.2(ii) applied to (M,F,9), and Joh 112 Jo belongs to 7f'O(M)', we have:
and, since h is one-to-one:
W8(10 JohJo)Wo = 10 JohJo By applying 2.7.4 (ii) to the quadruple (M, r, "'' 9) and the operator JohJo, we see that this operator is a scalar, so is h too; it means that <.p and 9 are proportional, and so are <.p and t/J too. 2.7.8 Corollary. Let IK1 = (Mt, F1, "-b 'Pl) and IK2 = (M2, F2, "-2, 'P2) be two Kac algebras, u a surjective llll-morphism from M1 to M2, Pu the greatest
82
2. Kac Algebras
projection of the ideal Keru, Ru that: '{)2
= 1- Pu.
o u( z) = ar.p1 (Rux)
Then, there ezi8u a
>0
8uch
Vx E Mt.
Proof. From 1.2.7 we get Ft(Ru) ~ Ru®Ru and ~tt(Ru) = Ru. So, from 2.2.6, the quadruple ][{lR,. = (MtR,. 'rlR,.' ll:lRu ''-PlRu) is a Kac algebra. Let r be the canonical surjection Mt -+ MtR,.. We can define a bijective l!l!-morphism v from MtR,. to M2 by v(r(x)) = u(x). Clearly,
0 such that, for all x in MtR,.:
'{)2 0 v( X) So, for all x in
= Cl!'{)lR,.( X)
M:{:
'{)2 o u(x) = '{)2 o v(r(x)) = ar.p1R,. (r(x)) = ar.pt(Rux). 2.7.9 Corollary.Let Kt = (Mt,Ft,~tt. 0 8uch that '{)2 o u = ar.p1, and 80 the Kac algebra8 OCt and K2 are i8omorphic in the 8en8e of 2.2.5.
Chapter 3 Representations of a Kac Algebra; Dual Kac Algebra
In this chapter, we shall use the notations hereafter: lK = (M, r, "• cp) will be a Kac algebra, .X its Fourier representation, Wits fundamental operator and M the von Neumann algebra generated by .X. This chapter deals with the representations of the Banach algebra M., following Kirchberg ([79]) and de Canniere and the authors ([21]), and the construction of the dual Kac algebra, as found independently by the authors ([34]) and Va.lnermann and Kac ([180]). This chapter begins with a Kirchberg's important result on Kac algebras: every non-degenerate representation of the involutive Banach algebra M* has a unitary generator (9.1.4). For the Kac algebra lKa(G) constructed with L 00 ( G), one recovers the well-known result that every non-degenerate representation of L 1 (G) is given by a unitary representation of G. As a corollary, we get that, for any non-degenerate representation p., the Kronecker product .X X p. is quasi-equivalent to .X (9.!.£); in the group case, that means that, for every unitary representation Jl.G of G, the tensor product .Xa ® P.G is quasi-equivalent to .Xa, which is Fell's theorem ([48]). When we choose p. =.X, we then get a coproduct i' on M (9.B.B). A co-involution K on M is then defined, for all win M., by the formula:
K(.X(w))
= .X(w o ~t)
The triple (M,F,K) obtained is a co-involutive Hopf-von Neumann alg'lbra. For locally compacts groups, that means (9.9.6) that the von Neumann algebra £(G) generated by the left regular representation .Xa has a co-involutive Hopf-von Neumann structure given by a coproduct T 8 and a co-involution "• such that, for all sinG:
r.(.Xa(s)) = .Xa(s) ® .Xa(s) "•(.Xa(s)) = .Xa(s-1 ) By predualizing the ~canonical surjection from the von Neumann algebra generated by M* onto M, one obtains an isometric, multiplicative and involutive morphism from the Banach involutive algebra M* in the Fourier-Stieltjes
84
3. Representations of a Kac Algebra; Dual Kac Algebra
algebra defined in Chap. 1, the image of which will be called the Fourier algebra of the Kac algebra, and is a self-adjoint ideal of the Fourier-Stieltjes algebra. For locally compact groups, we recover the situation of the Fourier algebra A( G) and the Fourier-Stieltjes algebra B( G) defined and studied by Eymard in [46]. Using the canonical Tomita-Takesaki construction, we define, starting from the left ideal lcp of M., a left Hilbert algebra dense into the Hilbert space Hcp, which generates the von Neumann algebra Manda faithful semi-finite normal weight rjJ on M (9.5.2), satisfying a Plancherel-type relation:
rfi(-\(w)*-\(w))
= lla(w)ll 2
for all win lcp
Moreover, the modular operator L1,p is affiliated to the centre of M, and is, in the sense of [114], the Radon-Nikodym derivative of the weight cp with respect to the weight cpot>, (9.6. 7). We prove that this weight is a Haar weight (9. 7.4 ), and we have so defined a dual Kac algebra :i = (M, f, K, rji), the fundamental operator W of which is 0'W* u, and the Fourier representation 5. of which is given by (where ,\* : M. -+ M is obtained by predualizing -\). As ,\ is non-degenerate, 5. is faithful. On ..C( G), the weight so constructed is equal to the Plancherel weight cp8 studied by Haagerup in [58]. So, 'K8 (G) = (..C(G),rs, "-s,cps) is another example of a Kac algebra (9.7.5), which is symmetric and will be studied in Chap. 4. Another essential result about Kac algebras is the following: characters on M. (that is, elements x of M such that x 'f= 0 and r(x) = x ® x), are unitaries, verify "-(X) = x*, and, with the weak topology of M, form a locally compact group, called the intrinsic group of the Kac algebra (9.6.10). See also (1.2.2), (1.2.9) and (2.6.6).
,,x.
3.1 The Generator of a Representation 3.1.1 Lemma. With the definition of 1.9.6, we have:
Proof. Let X be in p R(JK) n 1)1~. It follows from 1.3.6 that there exists a Hilbert space 1-£, a non-degenerate representation p. of M. on 1i and a vector in 1-£, such that, for all w in M.:
e
(x,w} = (p.(w)e I e)
3.1 The Generator of a Representation
85
As the space I,.,ni~ is norm dense in M* by 2.4.6{iii), the algebra Jl(I,.,n~) is dense in Jl( M* ). Thanks to Kaplansky's theorem, it exists a sequence {wn}neN of elements of Irp such that IIJ.l{wn)ll :::; 1 and that Jl(wn) strongly converges to 1. Let us consider the linear forms on M* defined by:
4in(w) = (x,w~*W*Wn)
(wE M*)
We have, by hypothesis on x:
4in(w 0 * w) = (x, (w * wnt * (w * Wn)) 2:: 0 Therefore 4in is positive definite. Moreover, since x belongs to by 2.4.5, w~ * w * Wn belongs to I,.,, we get:
4in(w)
Let us put J.?n and therefore:
=
= (a(w~ * w * Wn) IA,.,(x*)) = (.A(w~ * w)a(wn) I A,.,(x*)) = (.A(w)a(wn) I.A(wn)A,.,(x*)) = (.A(w), {.?a(wn),A(wn)A
f.?a(wn),A(wn)A
'.)1~
and since,
by 2.1.6 (ii) by 2.6.1 (v) by 2.5.3
4in(w) = (.A*(J.?n),w),
4>n = .A*(J.?n) E .A*(M*) From 1.6.10, we get the existence of :z:f.? in B(JK)+ such that x = 11'*(:z:J.?). Let us also consider the decomposition.,\*= 11'*(sAh {1.6.1 {ii)). We have: l(:z:J.?- (sA)*(J.?n), 1l'{w))l = 1(11'*{:z:J.?)- .A*{J.?n),w)l = l(x- 4in,w)l = l(x,w)- (4in,w)l
= l(x,w)- (x,w~ *W*Wn)l = I(Jl(W )e Ie) - (Jl( Wn)* Jl(W )Jl(wn)e Ie)l = I(Jl(w), ne- np(wn)e)l :::; liJl(w)li lie- Jl(wn)eli lie+ Jl(wn)ell by {[57] Proof 2.11) :::; 2IIJ.l(w)li 11e11 11e- Jl(wn)ell :::; 21i1l'{w)li 11e11 11e- Jl(wn)ell Therefore, thanks to Kaplansky's theorem, we get:
As, by 1.6.2 {ii ), {sA)* is an isometry, its image is closed in B(JK), therefore :z:J.? belongs to it. Which is to say that there is an element n in .M* such that :z:f.? ={sA)*( f.?) which implies that x = 11'*(:z:il) = .A*(il).
86
3. Representations of a Kac Algebra; Dual Kac Algebra
3.1.2 Lemma. (i) Let X be in 'Jtpn'Jlrpolt and win lr.p; let us write "1 = a(w). Then A.(ne, 11 ) belongs to 'Jl~ and we have:
{ii) The set {n EM., A.(n) E ~} is dense in Proof. For all w1 in
M.,
e= ~(x),
M•.
we have:
{A.(ne, 11 ),w') = (A(w')e 1"1)
= (Ar.p(x) I A(w')*a(w)) = (Ar.p(x) I A(w )a(w) = (Ar.p(x) Ia(w * w)) = {x*,w *W)= {~t(x),w 0 *w') 10
10
10
by 2.5.3 by 2.6.1 (v) by 2.1.6 (ii)
= {(w0 ® i)F~t(x),w')
Therefore: A.(ne, 11 )*
= ((w ® i)F~t(x))* = (w o ~t ® i)F(~t(x*)) 0
by 1.2.5
As ~t(x*) belongs to 'Jtp, A.(ne, 11 )* belongs to 'Jtp by 2.3.1 (ii), and 2.3.5 gives the completion of (i). By using 2.1.7(ii) and 2.7.5(iv), (ii) is an immediate corollary of (i). 3.1.3 Proposition. Let p. be a representation of M.; let us suppose that the set {6 E (A,..).; p..(6) E '.Yl~} is dense in (A,..).; then, p. is quasi-equivalent to a subrepresentation of A.
Proof. Let 6 be in (A,..);t such that p..(6)* belongs to 'Jt,c.. Therefore, by lemma 3.1.1, we get the existence of an element n in !VI. such that:
A.( n)
= ,.,..( 6)
Then, we have, for all win M.: (p.(w),6)
= (p..(6),w) = (A.(n),w) = (A(w),n)
which can be written as well:
(s,..(1r(w )), 6}
= (s.x(1r(w)), n}
3.1 The Generator of a Representation
87
By density, for all x in W*(JK), we shall have:
(s 11 (x), B)= (sA(x), il) Let us ass tune sA ( x) = 0; then we shall have (s 11 (x), B) = 0 for all B satisfying the above hypothesis; by linearity and density, it implies that s11 (x) = 0; therefore Ker sA C Ker s 11 ; so, there exists a morphism iP from M to A 11 such that iP( .A( w)) = J.L(w) and the lemma is proved. 3.1.4 Theorem. Any representation J.L of M* has a generator. Proof. Let
wbe in M., n in (A/J).; we have:
As ~ is a right ideal, we see, using 3.1.2 (ii), that the representation .A x J.L satisfies the hypothesis of 3.1.3. So, .A x J.L is quasi-equivalent to a subrepresentation of .A, and, by 1.5.4 (i) and (ii), we get the existence of a partial isometry U in AAXIJ ® M C M ® A 11 ® M such that, for all w in M.:
(.A X J.L)(w) = (i ® w)(U) UU* = U*U = PAXIJ ® 1 where PAXIJ is the projection on the essential space of .A X J.L· Now, let wbe in Mt, w(1) = 1, and n in (AIJ)t; we have:
(w ® n ®w)(U) =((.Ax J.L)(w),w ® n) = (.A.(w)J.L.(il),w)
by 1.4.3
Therefore:
(w ® n ® i)(U) = .A.(w)J.L.(n) = (w ® i)(uW*u)J.L.(il) = (w ® i)(uW*o-(1 ® J.L•(n))) and, by linearity and density:
(i ® n ® i)(U) So: 1 ® J.L•(il)
= uW*o-(1 ® J.L•(il))
= uWu(i ® n
® i)(U) = (i ® n ® i)((u ® 1)(1 ® uW u)(u ® 1)U)
by 2.6.1 (i)
88
3. Representations of a. Ka.c Algebra.; Dual Ka.c Algebra.
Therefore, we have: (1 ® JJ(w),w ® n) = w(1)(JJ.(n),w} = (w ® n ®w)((u ® 1)(1 ® uWu)(u ® 1)U) and, eventually: 1 ® JJ(w) = (i ® i ®w)((u ® 1)(1 ® uWu)(u ® 1)U) Let x be in C(H
(u ® 1)(1 ® uWu)(u ® 1)U = 1 ® V and then, for all w in M.:
JJ(w) = (i ®w)(V) Then, we easily get: 1 ® V*V
= U*U = P>.x1-1 ® 1
and so we deduce that the projector P>.x1-1 may be written 1 ® Q, where Q is in Aw Moreover, we get: 1 ® VV* = (u ® 1)(1 ® uWu)(u ® 1)UU*(u ® 1)(1 ® uW*u)(u ® 1) = (u ® 1)(1 ® uWu)(u ® 1)(P>.xp ® 1)(u ® 1)(1 ® uW*u)(u ® 1) = (u ® 1)(1 ® uWu)(u ® 1)(1 ® Q ® 1)(u ® 1)(1 ® uW*u)(u ® 1) =1®Q®1 And so we get that V*V = VV* = Q ® 1 and the theorem is proved. 3.1.5 Corollary. (i) Let JJl and JJ2 two non•degenerate representations of M.; then JJl X JJ2 is non-degenerate. (ii) The triple (W*(K), c;s11'X11'• s;r ), with the definitions of 1.6.5 and 1.6.6, is a co-involutive H opf-von Neumann algebra; it is symmetric if K. is abelian, and abelian if ][{ is symmetric. Proof. The assertion (i) is a direct corollary of 1.5.5 and 3.1.4, as degenerate and (ii) is a trivial application of 1.6.7 and 3.1.4.
11"
is non-
3.2 The Essential Property of the Representation A
89
3.2 The Essential Property of the Representation A 3.2.1 Lemma. Let J.L be a non-degenerate representation of M* with a generator V. For all w in M*, we have:
,(Ax J.L)(w)
= V(10 A(w))V*
.
Proof. It results from 1.5.5 that the generator U of A X J.L is equal to:
(, 0 i)(i 0 aW*a)(10 V) =(a 01)(10 a)(10 W*)(10 a)(a 01)(10 V) =(a 01)(10 a)(1 0 W*)(V 01)(10 a)(a 01) Therefore, we have, for all win M*: '(A
X
J.L)(w)V = (, 0 i)(i 0 i 0w)((a 01)(10 a)(10 W*)(V 01)(10 a)( a 01))V = (i 0 i 0 w)((10 a)(1 0 W*)(V 01)(10 a))V = (i 0 i 0 w)((10 a)(1 0 W*)(V 01)(10 a)(V 01)) = (i 0 i 0w)((10 a)(10 W*)(i 0 F)(V)(10 a)) by 1.5.1(i) = (i 0 i 0 w)((10 a)(10 W*)(1 0 W)(10 a)(V 01) (1 0 a)(10 W*)(10 a)) by 2.6.4 = (i 0 i 0 w)((V 01)(1 0 aW*a)) = V(10 (i 0 w)(aW*a)) = V(10 A(w)) by 2.6.1 (i)
which completes the proof. 3.2.2 Theorem and Definitions. Let J.L be a non-degenerate representation of M* with a generator V. Then: (i) The mapping which sends any element x of M to V(10x)V* is a oneto-one normal morphism from M to AJS 0 M which shall be denoted by ,:YJS; (ii) The representations A and A x J.L are equivalent; moreover, we have, for all w in M*: (iii) We have: "fJS 8 A
= 'SAXJS
(iv) The mapping ,:YA is a coproduct on M; it shall be denoted by i' and by transposition it induces a product* on M*. For all x in M, we have:
F(x)
= aW*a(1 ® x)aWa
90
3. Representations of a Kac Algebra; Dual Kac Algebra
(v) For all u in the intrin,ic group of K., and w in M., we have:
.Yu(.\(w)) = .\(u ·w). Proof. As Vis unitary, it is enough to check in which space belongs V(1®x )V* for any x in M. It results from 3.2.1 that for all w in M., we have: .Y,(.X(w))
= ~(.\ x JL(w)) E A,® M
which by density and continuity gives (i) and (ii) on our way; the assertion (iii) is straightforward, (iv) results immediately from the associativity of the Kronecker product and (v) is the application of (ii) and 2.6.6(iii). 3.2.3 Lemma. Let JL!. JL2 be two non-degenerate repre.5entatiom of M., with, re8pectively, generator, V1, V2; then: (i) For all x in M, and t in Hom(JLl, JL2), we have:
{ii) For any morphi8m ~ : A, 1
-+
A,2 , .5uch that ~ o JLl = JL2, we have:
Proof Let w be in M*. We have: 1)~(.\
by 3.2.2 (ii) by 1.4.5(i) by 3.2.2 (ii)
= (~®i)(VI(1 ®x)Vt) = V2(1 ® x)V2*
by 3.2.2(i) by 1.5.6 (ii) by 3.2.2(i)
(t ® 1)1, 1 (.\(w)) = (t ®
X JLl)(w) = ~(.\ x JL2)(w)(t ® 1) = '1,2 (.\(w))(t ® 1)
by continuity and density, we get (i). Let x be in M; we have: (~®i)7, 1 (x)
=.Y,2(x) which completes the proof. 3.2.4 Lemma. Let JL a non-degenerate repre8entation of M., in (A,) •. We have:
w in M.
and
n
3.2 The Essential Property of the Representation
~
91
Proof. Let w be in M •. We have: {-X.((fl ® w) o -)'p),w}
= {1-p(.\(w)), n ® w} = {.\ x JL(w),w ® fl} = {-X.(w)J.L•(fl),w}
by 3.2.2 (ii) by 1.4.3
which completes the proof.
3.2.5 Lemma. Let Ill and 1-'2 be two non-degenerate representations of M •. We have:
Proof. Let w be in M., fl1 in (Ap 1 )., fl2 in (Ap 2 )* and win
M•. We have:
{-)'pl Xp2(.\(w)), fl1 @ fl2@ w}
={.\X Ill X JL2(w),w ® fl1 ® fl2}
by 3.2.2 (ii)
= {.X.(w )(J.Ll)•( fll)(J.L2)•(fl2), w}
by 1.4.3
= {.X.((fll ® w) o -)'p1 )(J.L2).(fl2),w}
by 3.2.4
by 1.4.3 ={(.\X J.L2)(w), (fll ® w) O-)'p1 ® fl2} = (,(.\X J.L2)(w), (fl2 ® fl1 ® w) O (i ® -)'p1 )} = {-)'p 2 (-X(w)), (fl2 ® n1 ® w) o (i ® -)'p 1 )} by 3.2.2 (ii) = ((, ® i)(i ® -)'p1 )-)'p 2 (.\(w)), n1 ® n2 ®w}
which completes the proof by linearity, density and continuity.
3.2.6 Proposition. Let 1-' be a non-degenerate representation of M •. We have, with the notations of 3.2.2:
(i'p ® i)i' = (i ® r).y~'
.
Proof. We have:
(.y~' ® i)r =
( .y~' ® i)i'~
by 3.2.2 (iv)
= i'.y,.o~
by 3.2.3 (ii)
= .Y,~xp = (' ® i)i'~xp
by 3.2.2 (ii)
= (i ® 1'~)-)'p
= (i ®T)i'p
by 3.2.3 (ii) by 3.2.5 by 3.2.2 (iv)
92
3. Representations of a Kac Algebra; Dual Kac Algebra
3.2.7 Fell's Theorem ([48]). Let G be a locally compact group, >.a the left regular representation of G, fJa a unitary representation of G. The representation fJa ®>.a is then quasi- equivalent to >.a. More precisely we get, for any sinG: V(1 ® >.a(s))V* = fJa(s) ® >.a(s) where Vis the unitary in C('Hp)®L 00 (G) defined by the continuous bounded functions-+ fJa(s). Proof. Let us apply theorem 3.2.2 to the Kac algebra Ka( G) defined in 2.2.5. We have seen that V is the generator of the non-degenerate representation fJ associated to fJa (1.5.10), that M is then the von Neumann algebra£( G) generated by the left regular representation>. of L 1 (G) (2.5.4). So the morphism ,:Yp satisfies: ,:Yp(x)
= V(1 ® x)V*
,:Yp(>.(J)) =~().X fJ){f)
(x E C(G)) (! E L 1(G))
From this last relation, we deduce, using 1.4.7, for the unitary representations of G, associated to :Yo>. and~>. X fJ respectively: ,:Yp(>.a(s))
= fJa(s) ® >.a(s)
3.2.8 Corollary. Let G be a locally compact group, C( G) be the von Neumann algebra generated by the left regular representation >.a. There exists a unique normal injective morphism F8 from C( G) to C( G) ® C( G) such that, for all s in G, we have: Fa(>.a(s)) = >.a(s) ® >.a(s) and F8 is a coproduct {in the sense of 1.2.1} on C(G).
3. 3 The Dual Co-Involutive Hopf- Von Neumann Algebra 3.3.1 Proposition and Definitions. The mapping from C(Hr.p) onto itself defined by x -+ Jx* J for all x in C(Hr.p) is an involutive anti-automorphism of C(Hr.p)· The restriction of this mapping to M is an involutive antiautomorphism of M in the sense of 1.2.5. It shall be denoted it. Moreover, the involutions "' and it are linked by the following relations: it(>.(w)) =>.(woK.) ,(>..(w)) = >..(w o it)
(wE M.)
(wE M.).
3.3 The Dual Co-Involutive Hopf-Von Neumann Algebra
Proof. For all x in M, let us put i\:(x)
= Jx* J.
93
Let w be in M •. We have:
11:(-\(w)) = J,\(w)* J = -\(w o ~~:)
by 2.5.5{ii)
which altogether provides the first equality and ensures, by continuity, that for all x in M, i\:(x) belongs toM; the involutive character of i\: is trivial. For all win M., we have: (~~:(-\.(w)),w}
= (-\.(w),w o ~~:}
= (-\(w o ~~:),w} = (il:(-\(w)),w}
by the first equality
= (-\(w),w oil:}
= (-\.(w o il:),w} which completes the proof. 3.3.2 Theorem. The triple ( M, i', i\:) is a co-involutive H opf-von Neumann algebra. It will be called the dual co-involutive Hopf-von Neumann algebra of
:K. Proof. Let w be in M*, WI. w2 in M*. We have:
(Fil:-\{w),w1 ® w2} = (F-\(w o ~~:),w1 ® w2} = ((,\ X -\)(w o ~~:),w2 ® w1} = (-\.(w2)-\.(wi),w o ~~:)
by 3.3.1 by 3.2.2 (iv) by 1.4.3
= (~~:-\.(wl)~~:-\*(w2),w} = (-\.(wl o il:)-\.(w2 o il:),w}
by 3.3.1 = ((-\ x -\)(w), (w1 ® w2) o {il: ® il:)} by 1.4.3 = (~(il: ® il:)r-\(w), (w1 ® w2)} by 3.2.2 (iv) Therefore we have:
Fi\:,\(w) = ~{il: ® ii:)F-\(w) and we can complete the proof by continuity. 3.3.3 Proposition. The mappings.\ is an E-morphism from (W*(.K), ~s1rx1r, s;r) to ( M' i\:). If .K is abelian, ( M' i'' i\:) is symmetric, and if K. is symmetric, (M, i', il:) is abelian. ·
r'
Proof. Let w be in M*. We have: i's.\1r(w)
= :Y.\s.\1r(w) = ~s.\x.\7r(w) ~(,\ x ,\)(w)
= = ~(s.\ ® s.\)(1r x 1r)(w)
= (s.\ ® s.\)~s1rx1r1r(w)
by 3.2.2 (iv) by 3.2.2 (iii) by 1.6.1 {ii) by 1.6.4 (iii) by 1.6.1 {ii)
94
3. Representations of a Kac Algebra; Dual Kac Algebra
and, we get: And: KS~1r(w)
= KA(w) = .\(w o ~~:) = s~1r(w o ~~:) = s~1i"(w)
by 1.6.1 (ii) by 3.3.1 by 1.6.1 (ii) by definition of 1i" (1.6.6)
= S~S,r1r(w) and we get: KS~
= s~s,r
At last, since.\ is non-degenerate, we have s~(1) = 1, by 1.6.1 (iii); ass~ is one-to-one, it completes the proof, together with 3.1.5 (ii). 3.3.4 Definition and Notations. Let us denote Cl (IK) the C* -algebra generated by the Fourier representation .\ of lK. By 1.6.1 (i), we may identify its dual (Cl(IK))* with a closed subspace B~(IK) of B(IK). Mo~e precisely, to e~ element (}of (Cl(JK))*, one associates the element (} o .\of B(K) (where.\ denotes the restriction of s~ to C*(IK); cf. 1.6.1 (i) and (ii)). By 1.6.2 (ii), the mapping (s~)· is an isometry from M. into B(IK), the image of which is contained in B~(IK). By transposing 3.3.3, (s~). is then an isometric Banach algebra morphism, the image of which will be denoted by A(IK) and called the Fourier algebra associated to lK. Every element of A(IK) vanishes over Kers~; conversely, let(} be in B(K), such that (x, 0) = 0 for all x in Ker s~. We can define a linear mapping won M by writing: (s~(z),w)
= {z,O)
(z E W*(JK))
In fact, w appears as the composition of the restriction of(} to the reduced algebra W* (K)supp 8 ~, with the canonical isomorphism between W* (K)supp 8 ~ and M. Therefore w is ultraweakly continuous and belongs to M., and we have(}= (s~).(w). So, an element of B(K) belongs to A(K) if and only if it vanishes over Ker s ~. 3.3.5 Proposition. The mapping 11:.\* is a non-degenerate faithful representation of M. in M. Iu generator is W and we have:
{i) {ii)
(i ® i')(W) = (W ® 1)(1 ® u)(W ® 1)(1 ® u) (i ®w o K)(W) = (i ®w)(W*) (wE M.).
Proof. By 1.6.1 (ii) we have ~~:.\* = 11:1r*(s~)., it then results from 1.6.9 and 3.3.4 that it is a faithful representation. By 2.6.1 (i), for all w in M., and all
3.3 The Dual Co-Involutive Hopf-Von Neumann Algebra
win
M*,
95
we have:
therefore: KA*(w)
= (i 0
w)(W)
and we get (i) and (ii) through a straightforward application of 1.5.1 (i) and (ii). 3.3.6 Theorem. Let G be a locally compact group, .C(G) be the von Neumann algebra generated by the left regular representation Aa. There exists a unique normal morphism Fa from .C(G) to .C(G) 0 .C(G), and a unique normal antiautomorphism Ka in .C( G) such that, for all s in G: Fa(Aa(s)) = Aa(s) 0 Aa(s) Ka(Aa(s))
= Aa(s- 1 )
Then, (.C( G), Fa, Ka) i.<J a .<Jymmetric co-involutive Hopf-von Neumann algebra; we shall denote it JH[a( G). It is the dual co-involutive Hopf-von Neumann algebra associated to the Kac algebra Ka(G). Moreover, the morphism s_x from the envelopping W* -algebra W* (G) to .C( G) .<Juch that we have, for all sinG: s_x('7ra(s)) = Aa(s) i.<J an JH[-morphi.<Jm from the Erne.<Jt algebra of G (cf. 1.6.8} to JH[a(G). Proof. The existence of K8 is the only non-trivial result; applying 3.3.1, we see there exists a co-involution P;, on (.C(G),F8 ), defined in 3.2.8, such that, for all fin L 1 (G):
K(A(j)) = A(j o Ka) From 1.1.3, we have, for any sinG, fin L 1 (G):
and so, we have:
We then get, for any sinG:
96
3. Representations of a Kac Algebra; Dual Kac Algebra
3.3.7 Theorem. Let][{= (M, r, K,cp) be a Kac algebra, :K' the commutant Kac
algebra. Then the dual co-involutive Hop/ von Neumann algebra (M'~, r'~, ,'~) is equal to (M, c;F, K.). Proof. Usin~ 2.5.9, we see that the Fourier representation >..' generates M; so M'~ = M. By 2.4.8, the fundamental operator W' associated to K:' is W' = ( J ® J) W( J ® J). The coproduct r'~ is, then, using 3.2.2 (iv ), such that: r'~(x) =
(J ® J)uW*u(J ® J)(1 ® x)(J ® J)uWu(J ® J) = (K. ® K.)FK(x) by 3.3.1 and 3.2.2 (iv) = c;F(x) by 1.2.5 applied to (M,r,K.)
As, by the identification of HIP with HIP'' the associated antilinear isomorphism JIP and JIP, are equal, we see, by 3.3.1, that ,_,~ = K, and the theorem is proved.
= (M' r, K, cp) be a K ac algebra, and i. = (M' i'' 'k, cp) be a Kac subalgebra in the sense of 2.2.7. Let us denote j the canonical imbedding from M into M, which is an 18!-morphism. There is then a canonical surjective 18!-morphism r from (M,F,K.) to (M~,t~,'K~) such that r(>..~w)) = X(w o j), for all w in M., where X is the Fourier representation of:K.
3.3.8 Proposition. Let lK
Proof. Let us call I the isometry from Hrp to HIP defined, for all element x in 1Jtrp = ~ n M by: (i) IArp(x) = Atp(j(x)) As j is an 18!-morphism, the application w -+ w o j is multiplicative and involutive from M* to (M) •. This application is surjective because, for any a, 'Y in HIP, we have w-y,a o j = WJ•-y,I•a· Let x in !Jtrp, w in M •. We have:
IX(w o j)Arp(x)
= IArp((w o j o 'k ® i)i'(x))
by 2.3.5
= IArp((w o K o j ® i)i'(x)) = AIP((w o K o i))F(j(x)))
by (i)
= >..(w)AIP(j(x)) = >..(w)IAq,(x) therefore:
IX(w oj) Let us put, for X in M, r(x) because I is an isometry.
= >..(w)I
(ii)
= rxi. We have r(l) = 1 and r(>..(w)) = X(woj),
3.4 Eymard Algebra.
Let now
97
wbe in (M~) •. We have: {j(X.(w)),w) = {X.(w)),woj)
= {X(w o i), w) = {r(A(w)),w) = {A.r.(w),w) then:
A.r.(w)
= jX.(w)
As A* is injective, it can be then easily deduced that r* is involutive and multiplicative, and therefore that r is an Ill-morphism.
3.4 Eymard Algebra 3.4.1 Lemma. The .set B_x(K) i.s a .self-adjoint part of B(K). Proof. For all x in M and w in M*, we have:
by 1.2.5 and, by 3.3.1:
IIA(w 0 o ~~:)II= IIK(A(w))*ll = I!A(w)ll By using 1.6.2 (i) we then see that thanks to 1.3.4, we get the result.
'~~"•(B.x(K))
is stable by involution, and,
3.4.2 Proposition. (i) Any norm-one po.sitive element of B.x(K) i.s the limit, for the u(B(K), C*(K)) topology, of norm-one po.sitive element8 of A(:K). {ii} The .space B_x(K) i8 the u(B(K), C*(K))-do.sure of A(K). {iii) The .space B_x(K) {re.sp. 1r*(B_x(:K))) i.s compo.sed of the element8 of the form (sp).(il) {re.sp. 1-'•(il)}, where 1-' i.s a repre.sentation of M. weakly contained in A, in the .sen.se of [47), and n an element of (Ap)•· Proof. Any element of B_x(K) vanishes on Ker X (cf. 1.6.1 (ii)). By ([25], 3.4.2(i)) any norm-one positive element of B_x(K) is thus the limit, for the u( B(K), C* (K)) topology, of elements of the form ( s .x)• ( il), where n is normone positive in M.; at last 3.3.4 allows to deduce (i). It results from (i), by linearity that B_x(K) is contained in the closure, for the cr(B(:K), C*(K)) topology, of A(K); on the other hand, since B_x(K) is the annihilator of Ker A, it is u(B(K), C*(K)) closed, which completes the proof of (ii).
98
3. Representations of a Kac Algebra; Dual Kac Algebra
Let I" be a representation of M* weakly contained in>. and n in (Ap);t". By ([25], 3.4.4), (sph(il) is the u(B(lK),C*(JK)) limit of elements of the form (sA)*(w) where w belongs to M;t", i.e. of positive elements of A(JK). By using (ii), we can conclude that (sp)*(il) belongs to BA(JK); by linearity this conclusion still holds for any n. To prove the converse, let us note that those elements of the form (sp)*(il) with I" weakly contained in >. and n is in (Ap)* compose a vector space. Indeed if I" (resp. IL') is weakly contained in >. and n (resp. il') belongs to (Aph (resp. (Ap' )*), it is easy to check that:
(sp)*(il)
+ (sp' )*(il') =
(spEilp' )*(il
EB il')
and that I" EB I"' is weakly contained in .>.. By linearity, it is therefore enough to consider x_ in BA(JK)+. Let y be the positive linear form on C~(JK) such that x =yo>. (cf. 3.3.4). The Gelfand-Naimark-Segal construction allows to associate toy a triple ('H, I")· Then I" o >.is a representation of M* and we have: ((I" o >.)*(n~),w) = (l"(>.(w))e I0
e,
= (y,>.(w)) =(yo .\,1r(w)) = (x, 1r(w)) = {1r*(x),w)
Therefore, we have: and:
x
= (spoA)*(il~)
since it is clear that I" o >. is weakly contained in
.>., it completes the proof.
3.4.3 Lemma. The product of B(JK) is u(B(JK), C*(JK)) separately continuous on the bounded part8. Proof. Let 8i be a bounded family of elements of B(JK) converging to an element 8 in B(JK). For all 81 in B(JK) and w in M*, we have:
{1r(w), 8i * 8')
= {1r*(8i * 8'),w) = (7r*(81 )7r*(8i),w) = {1r*(8i),w · 1r*(81 ))
= {1r(w · 7r*(81)),8i) which converges to:
thanks to the same computation.
by 1.6.9
3.4 Eymard Algebra
99
Since the ll9i * 9'11 are bounded by 119'11 sup ll9ill which is finite, by the density of 1r(M.) in C*(JK), we get that 9i * 91 converges to 9 * 91 for O"(B(JK), C*(JK)). The left multiplication is dealt with in the same way. 3.4.4 Theorem. The sets A(JK) and B,x(lK) and are norm-closed self-adjoint ideals of B(JK). Specifically, B,x(lK) is an involutive Banach algebra which we shall call the Eymard algebra associated to K. Proof. Let
wbe in M., 9 in B(JK) and win M •. We have: by 1.6.9 and 1.6.1 (ii) (1r.(6 * (s.x).(w)),w) = (-X.(w)7r.(9),w) by 1.4.3 = ((.\ x 1r).(w ® 9),w) = ((.\ x 1r)(w),w ® 9) by 3.2.2 (ii) = (,Y,...\(w),9®w) = (.\(w),(9®w)o.y,..) = (-X.((9®w)o,Y,..),w)
From what we obtain that:
By 3.3.4, it follows that A(JK) is a left ideal of B(JK). As A(JK) is self-adjoint and norm-closed, the first part of the theorem is secured. Now, let 9 be in B,x(JK)+l. By 3.4.2 (i), 9 can be O"(B(JK), C*(JK)) approximated by norm-one positive elements of A(JK). Applying the first part of this proof, as well as 3.4.3 and 3.4.2 (ii), we find that, for all 91 in B(JK), 91 * 9 belongs to B,x(lK). By linearity, we can conclude that B,x(lK) is a B(JK)-left ideal; since, by 3.4.1 and 1.6.1 (i), B,x(JK) is norm-closed and self-adjoint, the proof is completed. 3.4.5 Proposition. The restriction to B,x(JK) of the Fourier-Stieltjes representation of B(JK) is the transposed of the mapping iU from M. to Cl (JK) (once B,x(lK) is identified with the dual ofCl(JK)). Proof. Let 9 be in (Cl(JK))* and w in M •. Then by 3.3.4, 9 o Xbelongs to B,x(K) and we have: (~~:7r.(9o.X),w)
= (7r(wo~~:),9o.X) = (.\(w o ~~:),9) = (iU(w))),9)
which completes the proof.
by 3.3.1
100
3. Representations of a Kac Algebra; Dual Kac Algebra
3.4.6 Eymard's Theorem ([46]). Let G be a locally compact group, and f an element of B( G). The following assertions are equivalent: (i) There exists a Hilbert space H, vectors e, 11 in H, and a unitary representation fiG of G on H, weakly contained in the left regular representation, such that, for all t in G:
f(t)
= (Pa(t)e l11)
(ii) We have: sup{la lf(t)h(t)dtl, hE L 1 (G), IIA(h)ll :::; 1} < +oo
The space of such functions is noted B>. (G); it is a closed ideal of the FourierStieltjes algebra of G, and will be called the Eymard algebra of G. Moreover, iff is in B>.(G), its B(G)-norm is equal to: sup{
!a
lf(t)h(t)dtl, hE L 1 (G), IIA(h)ll ::=; 1}
The space B>.(G) can be identified with the dual of C~(G) (the C*-algebra generated by the left regular representation of L 1 ( G)), the duality being given, if f(t) = (pa(t)e 111), and h in CHG), by:
u. h) = (p(h)e l11) where fL denotes again the associated representation of C~ (G) (recall that fL is weakly contained in A). Moreover, for every w in £(G)*, the set: A(G)
= {s-+ (A(s- 1 ),w)
(s E G)}
is a norm-closed ideal of B>.(G). It will be called the Fourier algebra of G, and its dual is £(G). Proof. Using 3.4.2(iii), we see that property (i) characterizes 11'*(B>.(Ka(G))); by 1.6.3 (iii), it is the same for property (ii). So, we get B>.(G) = 11'*(B>.(0Ca(G))) = l\':a11'*(B>.(0Ca(G))) and all other properties of B>.(G) come then from 1.6.3(iii) and 3.4.4. Let us now consider l\':a11'*(A(OCa(G))). Using definition 3.3.4, it is the set of all elements which may be written, for all [lin £(G)*:
In 1.3.10, we have seen that A*(il) is the functions -+ (Aa(s), il); by then
1\':aA*(il) is the functions-+ {Aa(s- 1 ), il).
3.5 Construction of the Dual Weight
IOI
So, we get A( G)= Ka'll'*(A(lKa(G))), and all properties of A( G) come from 3.4.4.
3.5 Construction of the Dual Weight In that paragraph, we consider the set 2:l = a(Irp n I~). 3.5.1 Proposition. Let w, w' be in Irp
n ~· The formulas:
a(w )T a(w') = a(w * w1 )
{i)
{ii}
a(w)U = a(w 0 )
allow us to equip 2:l with a structure of left Hilbert algebra, dense in Hrp. Let us denote by 7r the left multiplication of 2:l. We have, for all w in Irp n I~:
?r(a(w)) = .X(w)
{iii)
and the von Neumann algebra generated by 7r(2:l) is equal toM. Proof. (a) We have seen in 2.4.6 (iii) that Irp n I~ is an involutive subalgebra of M*. As a is a bijection from Irp n I~ to 2:l, we see that 2:l, equipped with
T and # is an involutive algebra. (b) Let e be in Hrp, orthogonal to 2:l; by 2.4.5, we have, for all WI,W2 in Irp: 0 = (a(w!
* W2) Ie)= (.A(wi)*a(w2) I e)
by 2.4.6 (ii)
= (a(w2) I.A(wi)e) because of the density of a(Irp) in Hrp (2.1.7 (ii)), it implies .A(wi)e = 0 for all in Irp; because of the density of Irp in M* (2.1.7 (ii)), it implies, for all win M*, .X(w)e = 0, which, in turn, because of A being non-degenerate (2.6.3(i)) implies e = 0. Therefore 2:l is dense in Hrp. (c) For all WI fixed in Irp n ~the mapping a(w)-+ a(wi)Ta(w) is continuous from 2:l to 2:l. In fact, we have:
WI
a(wi)Ta(w) = a(wi *W) = .X(wi)a(w) (d) For all WI,W2,W3 in Irp
by definition by 2.6.1 (v)
n I~, we have:
(a(wi)Ta(w2) I a(w3)) = (.A(wi)a(w2) Ia(w3)) = (a(w2) I.A(wl)a(w3)) = (a(w2) I a(wl)Ta(w3)) = (a(w2) I a(wi)UTa(w3))
by (c) by 2.5.3 by (c) by definition
102
3. Representations of a Kac Algebra; Dual Kac Algebra
(e) Let e in HI(J orthogonal to ~t~. We have, for all WI,W2 in II(J n I;:
o=
(a(wl)Ta(w2) Ie)= (.X(wl)a(w2) Ie) = (a(w2) Ia(wl)e)
by (c) by 2.5.3
By (b) it implies .\(w1)e = 0, since a(I'fJ n I;) is dense in H by (b); by continuity, it implies .\(w)e = 0, for all win M.; therefore because of.\ being non-degenerate, it implies e = 0; so, ~T~ is dense in H'fJ. (f) Let w be in Itp n I; and X in 'Jltp n 'Jltpolt• We have:
(Atp(x)la(w)j) = (Atp(x)la(w 0 )) = (x*,w 0 ) = (~~:(x),w) = (a(w) I ~(~~:(x*)))
by definition by 2.1.6 (ii) by 1.2.5 by 2.1.6 (ii)
because SJltpo~e = ~~:('Jt~). Therefore, the mapping j has an adjoint, the restriction of which to ~(SJltp n 'Jltpo~e) is the mapping A'fJ(x) -+ A'fJ(~~:(x*)). By 2.7.5(iv), this adjoint mapping is densely defined, therefore j is closable. Following ([158), def. 5.1; cf. 2.1.1 (iii)), if we remark that (iii) has been proved in (c), and that, thanks to 2.4.6(iii), 11'(~) generates M, we have completed the proof. 3.5.2 Definitions. We shall denote~' the right Hilbert algebra associated to~' ~ 11 the achieved left Hilbert algebra, and ~ 0 the maximal modular 8ubalgebra of ~11 (cf. 2.1.1 (iii)). We shall still note T and j (resp. b) the product and the involution on ~ 11 (resp. ~'). We shall note S and F, the closures of j and b, with respective domains denoted to vi and vb. In particular' we have, for all X in SJltp n SJltpo~~::
He in Htp is left bounded with respect multiplication" by ([14), def. 2.1).
e
to~'
we shall still note 7r(e) the "left
3.5.3 Definitions. We shall note~ the faithful, semi-finite normal weight on canonically associated to ~ ([14), th. 2.11), and call ~ the dual weight a88ociated to IK. For all w in I'fJ n by 3.5.1 (c), .\(w) belongs to 11'(~) and therefore to ~ n sn~. Moreover, for WI and W2 in Itp n I;, we have:
M
:r;,
3.5 Construction of the Dual Weight
I03
To the weight cj; we associate the Hilbert space Hq, and the canonical oneto-one mapping Ac:p : 'Jtc:p -+ Hq,. We shall note !i the left Hilbert algebra associated to cj;, i.e. Ac:p('Jtcp n 'Jt~), which is isomorphic to ~11 (2.1.1 (iii)). More precisely, the mapping which, to every in ~", associates the vector:
e
can be uniquely prolonged into a unitary operator from Hcp to Hc:p still denoted by :F. It will be called the Fourier-Plancherel mapping and will allow us to identify Hc:p and Hcp, and, through this identification of Hcp, we have ~" = !i. Using the definition 2.1.6 (ii), we shall note a instead of aq,. 3.5.4 Proposition. For all win Icp, a(w) is left-bounded with respect to~' and we have: (i) 11-(a(w)) = A(w) {ii) a(w) = Ac:p(A(w)) {iii) for all a,"( in 2l1, 7To.b is left-bounded with respect to~ and we have:
Proof. Let
WI
be in Icp, W2 in Icp n I;, e in ~'. We have:
A(w2)1r'(e)a(wi) = 1r'(e)A(w2)a(wi) by 2.6.1 (v) = 1r'(e)a(w2 * wi) = 11-( a(W2 * WI ))e because W2 * WI belongs to Icp n /; = A(w2 * wi)e by 3.5.1 (c) = A(w2)A(wi)e As 1 is in the closure of A(lcp n ~)
= 11-(~), we have:
which yields (i), and (ii) immediately, then (iii) follows by applying 2.1.7(i). 3.5.5 Remark. Thanks to 2.1.1 (iv), we know that M is in a standard position in H; so, by 1.1.1 (iii), every element of M. can be written nat,/31 M for some vectors a, fJ in Hcp. This element shall be written wat,/3· 3.5.8 Corollary. {i) The algebra C(G) is in a standard position in L 2(G). {ii) The predual C(G). is equal to the set {n1,9 IC(G), j,g E £2(G)}.
104
3. Representations of a Ka.c Algebra; Dual Ka.c Algebra
(iii) For all fin L 2 (G), let us put j(s) = f(s- 1 )- for all sin G. The set of all {! * g, f, g E £ 2 ( G)} is the Fourier algebra A( G) defined in 3.4.6. Proof. The aBsertions (i) and (ii) are just applications of 3.5.5 to IKa(G). In 3.4.6, A(G) haB been defined aB the set offunctions s-+ (>.a(s- 1 ),w}, for all w in £(G) •. But we have: (>.a(s- 1 ),wf,g} = (>.a(s- 1 )flg) = laf(st)g(t)dt = laf(t)g(s- 1t)dt
fa
= f(t)g(C 1s)dt = (! * g)(s) And so, (ii) implies (iii).
3. 6 Connection Relations and Consequences 3.6.1 Proposition (Connection relations). The operations T and by the following relations: (i) For any a, 'Y in !211 and fJ, 5 in H'P, we have:
(ii) For any a, 'Y in !211 and
fJ, 6 in !B1,
f
are linked
we have:
(iii) The set !B'T!B' is included in A'P(~) and, for any a, 'Y in H'P and we have:
fJ, 5 in !B1,
Proof. Combining 2.6.1 (iii) and 3.5.4 (iii), we get (i). Therefore, with the hypothesis of (ii), we have:
(W(a ® fJ) I'Y ® 6) = (fJ I11-'(6)('YT ab)) = ( 11-'(6)* fJ I 'YTab)
= (1r1(6b)fJ I'YTab) = (fJT5G 1'YTab) which is (ii).
3.6 Connection Relations and Consequences
105
It can also be written as follows:
It follows that:
Thus, {3T 8~ is left-bounded with respect to 21 (cf. 2.1.1 (iii)), and we can write: (W(a ® /3) I7 ® 8) = (1r(/3T8~)a I7) which, by continuity, still holds for any a, 7 in H cp· This completes the proof. 3.6.2 Lemma. {i} The set 23 1T23 1 is included in Acp('Jlcp n 'Jlcpo~t)· More precisely, for {3, 8 in 23 1 , we have:
and:
~t( 1r(f3T 8~ )*) = 1r(8T 13~)
(ii) The space Acp('Jlcp n 'Jlcpo~t) is a core for
F.
Proof. Let {3, 8 be in 231 , a, 7 in Hcp. We have:
(..\.(wcS,,B)*a h)= (..\.(wcS,,B),w-y,a)= (..\(wooy,a), WcS,,B)= (f31..\(wooy,a)8) = (W( a ® {3) I"{ ® 8)
by 2.6.1 (iii)
= (1r(/3T 8~)a I7)
by 3.6.1 (iii)
from what follows the first equality. For win M., we have, then:
(~t( 1r(/3T8b)), w) = (K(..\.(wcS,,B)*), w)
= (..\.(wcS,,B),w = (..\(w WcS,,B)-
0 )-
0 ),
= (..\(w)*,w6,,8)= (..\(w),w,B,cS) = (..\.(w,B,cS),w)
by 1.2.5
106
3. Representations of a Kac Algebra; Dual Kac Algebra
Therefore, we have:
by the first equality; as !Jtpo~~: = ~~:(~), the proof of (i) is completed. As, by ([158], p. 17), ~~t~' is a core for F, (ii) is immediate. 3.6.3 Lemma. Let x bt;. in M. • (i} For any a in TJ, xa belong& to 'Db, and we have:
Fxa = ~~:(x)* Fa (ii} For any /3 in
vi,
x/3 belong.s to
vi,
and we have:
Sxf3=~~:(x)*Sf3.
Proof. Let a be in Acp(IJ'tp n IJ'tpo~~:), w in lcp
(xa Ia(w)i)
n I~, x in M. We have:
= (xa Ia(w = (A;p 1 (xa)*,w 0 ))
0 )-
by 3.5.1 by 2.1.6 (ii)
= (~~:A;p 1 (xa),w)
= (~~:(xA;p 1 (a)),w)
= ((~~:(x*)~~:(A;p 1 (a)*))*,w) = ((~~:(x*)A;p 1 (Fa))*,w) = (A;p 1 (~~:(x*)ra)*,w)
by 3.5.2(i)
= (a(w) l~~:(x*)F~a)
by 2.1.6 (ii)
Therefore xa belongs to 1Jb and:
Fxa = ~~:(x*)Fa As Acp(mcp n mcpo~~:) is a core for Let a be in
vb' /3 in vi
and
X
F by 3.6.2 (ii), we have proved (i). in M. We have:
(x/31 ra) = (/31 x* F~a) = (/31 r~~:(x)a)
= (~~:(x)a I S~/3) =(a l~~:(x*)S~/3) which completes the proof.
by (i)
3.6 Connection Relations and Consequences
3.6.4 Proposition. The modular operator Proof. Let a be in
~0
xLia =xi'Sa
as ~ 0 is a core for
Li =
107
L1rf> is affiliated to M 1 •
and x in M. We have:
= P,.(x*)Sa = FSxa = Lixa
by 3.6.3 (i), because Sa belongs to vG by 3.6.3 (ii)
Li, we have xLi C Llx, which completes the proof.
3.6.5 Corollary. For all t in R, we have:
rat= (i ® af)i'. Proof. For all x in
M,
and t in R, we have:
i'af(x) = aW*(af(x) ® l)Wa = aW*(LiitxLi-it ® l)Wa = aW*(Liit ® l)(x ®!)(,&-it® l)Wa
by 3.2.2(iv)
Now, by 2.6.1 (iv), W belongs toM® M and by 3.6.4, jit belongs toM', therefore, we have:
i'af(x)
= a(Liit ® l)W*(x ® l)W(..&-it ® l)a = (1 ® jit)aW*(x ® l)Wa(l ®,&-it) = (1 ®Liit)r(x)(I 0Li-it)
by 3.2.2(iv)
= (i ® af)F(x) which completes the proof. 3.6.6 Corollary. For any x in M, we have:
(i) (ii) Proof. Let a be in
"'(x*) = JxJ (l®J)W(J®J)=W*. ~o
and x in M. We have:
xla = xLi112 sa = ..& 112 xSa = jl/2§,.(x*)a
= J"'(x*)a by density, we get (i). By (i) and 2.6.2, we get immediately (ii).
by 3.6.4 by 3.6.3 (ii)
108
3. Representations of a Kac Algebra; Dual Kac Algebra
3.6. 7 Theorem. The modular operator ..& i, affiliated to the centre of M; moreover, it i.'J the Radon-Nikodym derivative of the weight cp with re8pect to the weight cp o K-, in the .'Jen.'Je of [114] (cf. 2.1.1 (v)).
Proof. By 2. 7.6 (i), we have arK. = u'f for all t in lll. The theorem 5.4 of [114] gives then the existence of a unique positive self-adjoint operator h, affiliated to the centre of M, such that cp o K. = cp(h·) (cf. 2.1.1 (v)). Let x be in '.ltp n '.ltpoK. j then Acp( X) belongs to V( h1/ 2 ), and we have:
llh 112Acp(x)ll 2 = cp o K-(x*x) = 11Acp(K-(x*))ll 2 = IIFAcp(x)ll 2
by 3.5.2(i)
= IIi,&-1/2 Acp(x)ll2 = 11..&-1/2 Acp(x)il2 Now, by 3.6.2 (ii), we see that Acp('Jlcp n 'JlcpoK.) is a core for F, thus also for ..&- 112; on the other hand, 3.6.4 implies that h112 and ..&- 1/ 2 commute. Using the same arguments as in ([23] lemma 23), we can conclude that h1/2 = ..& - 1/2, and so h = ..& - 1. The operator ..& is therefore affiliated to the centre of M and cp o K-(Ll·) = cp, which completes the proof. 3.6.8 Corollary. For all t in lll, we have: ~
(p
K.O't =
(p
~
0' -tK.
.
Proof. By 3.6.7, for all tin lll, ,&it belongs to the centre of M. We then have for all x in M:
~uf(x) = Juf(x)J = J..&itx,&-it J = ,&-itJxJ..&it
by ([14], 4.10) by 3.3.1
= ,&-it~(x)..&it
= u~t~(x) which completes the proof. 3.6.9 Lemma. Let for all w in M*:
(i) {ii) {iii}
X
in M,
X
=/:-
o,
8Uch that r(x)
.X(w)x = x.X(K-(x) · w) x.X(w) = .X(x · w)x .X(K.(x) · w) = K-(x).X(w)x .
= X® X.
Then, we have,
3.6 Connection Relations and Consequences
109
Proof. Let y be in l)tp. We have:
.X(w)xA'f'(y)
= .X(w)~(xy) = A'f'((w o "'® i)r(xy)) = A'f'((w o"' ® i)((x ® x)r(y))) = xA'f'((w o "') · x ® i)r(y)) = xA'f'(((K(x) · w) o "'® i)r(y))
= x.X(,.,(x) · w)A'f'(y)
by 2.3.5
by 1.2.5 by 2.3.5
which yields (i ), by continuity. Taking the adjoints in (i), one gets, using 2.5.3:
x* .X(w 0 ) = .X((K(x) · w) 0 )x* = .X(x* · w0 )x*
by 1.2.5
and, changing w to w0 , x to x* (which satisfies the same hypothesis), we get (ii). Let us now assume that w is in I'f', and let w1 be another element of I'f'. We have:
A(K(x) · w)ia(w1)
= 1r'(ia(w1 ))a(K(x) · w) = i.X(w')iK(x)a(w) = i.X(w')x* ia(w)
by 3.5.4(ii) and 3.5.1 (c) by 3.5.4(i) and 2.1.7(iii) by 3.6.6(i) by (i) applied to w 1 and x = ix* .X(K(x*) · w')ia(w) by 3.6.6(i) = K(x)i-X(K(x*) · w')ia(w) 1 and 3.5.1 (c) by 3.5.4(i) = K(x)1r'(ia(K(x*) · w ))a(w) by 3.5.4(i) = K(x).X(w)ia(K(x*) · w1) by 2.1.7(iii) = K(x).X(w)iK(x*)a(w1) 1 by 3.6.6(i) = K(x).X(w)xia(w )
By continuity, we get:
.X(K(x) · w) = K(x).X(w)x for all win I'f', and by continuity again, for all w in M., which is (iii). 3.6.10 Theorem. Let K. be a Kac algebra. The intrinsic group of K. is equal to the set of characters on M., that is the set of all x in M, such that x =/: 0
andr(x)=x®x. Proof. Let x be a character on M., that is, x belongs toM, and is such that X =F 0 and r(x) =X® x. As K(x) satisfies the same hypothesis, we have, for
110
3. Representations of a Kac Algebra; Dual Kac Algebra
all win M.: x~~:(x ).>.(w)
= x.>.(~~:(x) · w)~~:(x) = .>.(w)x~~:(x)
by 3.6.9 (ii) applied to ~~:(x) by 3.6.9(i)
So, by continuity, x~~:(x) belongs toM', and, then, by 2.7.2(i), it is equal to a scalar a. But then, for all w in M., we have:
.>.(w)x =
x.>.(~~:(x)
· w)
by 3.6.9(i) by 3.6.9 (iii)
= x~~:(x).>.(w)x = a.>.(w)x By continuity, we get x =ax, and, as x #- 0, we have a= 1. So x~~:(x) = 1, xis invertible, and the theorem is proved.
3.6.11 Proposition. Let G be a locally compact group, IKa(G) the abelian Kac algebra associated to G in 2.2.5; the dual co-involutive Hopf-von Neumann algebra associated to 'Ka(G) is (.C(G),T8 ,~~: 8 ) (cf. 3.3.6}, and the dual weight (C,Oar on .C( G) is the Plancherel weight C,Os studied in [58], associated to the left Hilbert algebra JC( G) of continuous functions on G with compact support.
Proof. By definition 3.5.2, the weight (c.oar is associated to the left Hilbert algebra a(I'P n I~), that is, by 2.4.7(ii) and 1.1.2(ii), the set:
{! E L 1 (G) n L 2 (G); the functions-+ r(s)
= /(s- 1 )L1a(s- 1 )
belongs to L 1 (G) n L 2 (G)} equipped (by 3.5.1 and 1.1.2 (ii)) with the usual convolution product and the involution °. We have JC(G) C a(I'P n ~),and the operations on JC(G) being the restrictions of those on a(I'P n ~).Both generate the same von Neumann algebra£( G). Thanks to 3.6. 7 and 2.2.2, it appears that the modular operator associated to ( C,Oar is the Radon-Nikodym derivative of the left Haar measure with respect to the right Haar measure, that is the modular function L1a. It is also the modular operator associated to c,o 8 • Therefore, using ([114], prop. 5.9), we have:
3.6.12 Theorem. Let G be a locally compact group; the set of continuous characters of G (i.e. continuous multiplicative functions from G to C, except the function 0}, is a locally compact abelian group, which is the intrinsic group of the Kac algebra 'Ka(G).
Proof. The intrinsic group of 'Ka(G) is, by 3.6.10, the set of all such that I =F 0 and Fa(/)= I® I (i.e. l(st) = l(s)l(t), a.e.)
I in L 00 (G)
3.7 The Dual Kac Algebra
Let now g in /C( G) such that (g, f)
(g,J)f(s)
=
111
-:f. 0; we have:
(fa g(t)f(t)dt) f(s) =fa f(st)g(t)dt =fa f(t)g(s- t)dt 1
We then see that f is almost everywhere equal to a continuous function, and we get the result.
3. 7 The Dual Kac Algebra 3. 7.1 Lemma. Let x, y in ~; then F(y )( x ® 1) belongs to ~rp®rp and we have:
Arp®,p(F(y)(x ® 1)) = uW*u(A,p(x) ® A,p(y)) . Proof. Let /31, P2 and 6 in 23'; let w be in IV' and x in ~· We have:
((1 ® 1i-1(6))uW*u(A,p(x) ® a(w)) I/31 ® /32) = (uW* u(A,p( x) ® a(w)) I /31 ® /32 T6b)
= (W*(a(w) ® A,p(x)) I /32 T6b ® P1) = (.X(Da(w),,B2 TcS&)A,p(x) IP1) = (.X(w · 1r(/32 T6b)*)Arp(x) I/31) = (.X(w · .x.(wo,p2 ))A,p(x) IP1) = (.X(w . .x.(wo,p2)),wAq,(z),.Bl) = (.X.(wAq,(z),,Bl),w . .x.(wo,.B2)) = (.X.(wo,,B2).X.(wAq,(z),pJ, w) = (.X.(wAq,(z),p *w0,p2 ),w) = (.X(w),wAq,(z),.B1 *wo,p2)
by 2.6.1 (iii) by 2.1.7(iv) by 3.6.2
1
by 3.3.5
= (i'(.X(w))(A,p(x) ® 6) IP1 ® P2)
So, by linearity and density, we have:
(1 ® 1i-1(6))uW*u(A,p(x) ® a(w))
= F(.X(w))(A,p(x) ® 6)
Let 61 be in 231 • We have:
(71-'(61) ® 1i-1(6))uW*u(A,p(x) ® a(w))
= (71-1(61) ® 1)(F(.X(w))(Arp(x) ® 6) = F(.X(w))(1i- (61)Arp(x) ® 6) = F(.X(w))(x ® 1)(61 ® 6) 1
112
3. Representations of a Kac Algebra; Dual Kac Algebra
Then, we can deduce that 17W* u( A
~®
M defined for
.,P(y)
y in
(i)
u+ by:
= (cp ® cp)((x* ® 1)T(y)(x ® 1))
We have, for all w in M*:
.,P(..X(w)* ..X(w)) = (cp ® cp)((x* ® 1)i'(..X(w))* i'(..X(w))(x ® 1)) = IIA
u+,
.,P(uf'(y)) = (cp ® cp)((x* ® 1)i'(uf'(y))(x ® 1)) = (cp ® cp)((x* ® 1)(i ® uf')i'(y)(x ® 1))
by 3.6.5
= (cp ® cp)((x* ® 1)T(y)(x ® 1)) = .,P(y)
Therefore '1/J is uf'-invariant, we can apply the proposition 2.1.5 to the von Neumann algebra M, the two weights cp(x*x)cp and .,P, with E = .\(I
u+:
.,P(y) = cp(x*x)cp(y) that is, for all
X
in l)l
(cp ® cp)((x* ® 1)T(y)(x ® 1)) By polarization, for
Xt. x2, Yt. Y2 in~.
= cp(x*x)cp(y)
we shall have:
Therefore, there is some isometry U in .C(H
3.7 The Dual Kac Algebra
113
= U(Acp(x) ® Acp(y)) By (i) and by density, we see that U = uW*u, which completes the proof. Acp®cp(.i'(y)(x ® 1))
3. 7.2 Proposition. The weight
rott. By 2.1.8(i), it exists a sequence
{an}neN of
positive elements in M which are monotonely converging up to (i®cp)(i'(y)). We have:
((i ® cp)(i'(y))(w Aq,(z))
= li~r{an,wAq,(z)) = limr(anAcp(x) IAcp(x)) n = lim tcfJ(x* anx) n = cp(x*((i ® cp).i'(y))x) = cp((i ® cp)((x* ® 1)i'(y)(x ® 1))) = (cp ® cp)((x* ® 1)i'(y)(x ® 1))
by ([59], 2.1(3))
=cp(x*x)cp(y) = (cp(y)Acp(x) I Acp(x))
by ([59], 5.5) by 3.7.1 (i)
Now, we know by 2.1.8 (i) that is exists a closed subspace H' C Hcp and a positive selfadjoint operator Ton H' such that:
'D(T)-
= H'
and:
= IIT112ell 2 (i ® cjJ)(r(y))(we) = +oo
(i ® r:p)(r(y))(we)
<e E V(T 112 )) (e rt 'D(Tlf2))
It follows from (i) that 'D(T 112 ) contains Acp('Jlcp), which is dense in Hcp; therefore H 1 = Hcp. Moreover, for all in Acp ('Jlcp), we have:
e
IIT112ell 2 =
Tl/2
= cjJ(y )112 1
or:
T = cjJ(y)1
by (i)
114
3. Representations of a Kac Algebra; Dual Kac Algebra
Therefore (i 0 proof.
cp)(i'(y)) is bounded and is worth cp(y)1, which completes the
3. 7.3 Theorem. The quadruple ( M, i', K., cp) is a K ac algebra. Its Fourier representation is ). = K.A*j its fundamental operator is H
W = O"W*O"
{once
Proof. We got in 3.3.2 that (M, i', K.) is a co-involutive Hopf-von Neumann algebra and in 3. 7.2 that cp is a left-invariant weight on it. We can therefore apply 2.4 to (M, i', cp); the fundamental operator associated to (M, i', cp) appears to be O"W*O", by 3.7.2. Applying 2.4.6. to (M, i', K., cp) we can then define a bounded linear mapping >.(w), for all win M., by: >.(w) =(woK. 0 i)(O'W*O')
= (i 0
w o K.)(W*)
Using 2.6.1 (ii) and 3.3.1 we get:
Using 3.3.5, we see that ). is a faithful non-degenerate representation; so, by 2.4.6 (iv), the weight cp satisfies (HWii); it satisfies (HWiii) by 3.5.3, and then (M, i', K., cp) is a Kac algebra. 3. 7.4 Definition. The Kac algebra Kac algebra of K
lK
= ( M, i', K., cp) will be called the dual
3.7.5 Theorem. Let G be a locally compact group. Then the quadruple (.C(G), F8 , K- 8 , cp 8 ), where .C( G) is the von Neumann algebra generated by the left regular representation >..a of G, F8 , K- 8 and cp 8 have been defined respectively in 3.2.8, 3.3.6 and 3.6.11, is a symmetric Kac algebra, denoted lK8 ( G), and we have:
Ks( G) = !Ka(
or
Its fundamental operator is equal to the functions--+ >..a(s) (s E G), considered as an element of .C(G) 0 L 0 '>(G). Its Fourier representation >.(w) is defined as being, for all win .C(G)., the functions--+ (>..a(s- 1 ),w} (s E G); the Fourier representation). oflK8 (G) is then surjective on the Fourier algebra defined in 3.4.6 and 3.5.6 (iii). Proof. By 3.7.3 and 3.6.11, we see that (.C(G),F8 ,K- 8 ,cp8 ) is the dual Kac algebra of Ka(G). Therefore, we get, by 2.4.7(i) that O"W*O" is the function s --+ >..a(s), when considered as an element of .C(G) 0 L 00 (G). By 3.7.3, it is the fundamental operator of lK8 ( G). By 2.6.1 (i), applied to OC8 ( G), we get then the Fourier representation of K 8 (G).
3. 7 The Dual Kac Algebra
115
3.7.6 Proposition. Let K1 = (Mt.Tt,~Ct,cpl) and OC2 = (M2,T2,"2,cp2) be two Kac algebras, U a unitary HIP 1 -+ HIP 2 which implement& an llli8omorphi&m from OC1 to OC2 (cf. 2.7.9 and 2.2.5). Then U implement& an lll-i8omorphi&m from lK1 to lK2, too. Proof. Let u an lEI-isomorphism from M1 to M2, such that 'P2 o u (cf. 2.7.9) and let U be defined by (cf. 2.2.5):
= acp1
U ~~ (x) = a- 1 / 2 ~ 2 (u(x)) Then, for any w in I'P 2 , w o u belongs to IIP 1 , and we have:
So, we get, using 2.6.1 (v), for all w 1 in M2*:
U>.t(w 1 o u)at(w o u)
= Ua1(w 1 o u * w o u) = Uat((w 1 * w) o u) = a 1l 2a2(w1 * w) = a 1l 2>.2(w1 )a2(w) = >.2(w')Ua1(w o u)
from which we get that the application
u(x)
u defined, for
x in M2, by:
= U*xU
is a von Neumann isomorphism from M2 to
Mt
such that, for all w' in Mt.:
From that, by predualizing, we get u., a Banach space isomorphism from Mt. to M2*' such that, for all w1 in Mt.:
and, as >.2* is injective (3.3.5), we see that u. is multiplicative and involutive, and, so, u is an JBl-isomorphism. Moreover, we have then, for all win l'P2 :
.2(w)* >.2(w))
= tPt(>.t(w o u)* >.t(w o u)) = llat(w o u)ll 2
by 3.5.2 (ii)
= al!a2(w)1! 2 = acp2(>.2(w)* >.2(w))
by 3.5.2 (ii)
from that, we deduce, by 2.7.9, that
tPl o u = acp2.
116
3. Representations of a Kac Algebra; Dual Kac Algebra
3.7.7 Proposition. Let][{ be a Kac algebra, W its fundamental operator, the modular operator associated to the dual weight tP· Then: (i) We have: W(1 ® Ll)W* = Li ® Li
Li
{ii) For all t in R, .Jit belongs to the intrinsic group of K {iii) For all t in R, w in M*, we have:
(iv) For all x in M+, we have: (cp ® i)(F(x)) = cp(x).J-I {v) Let x be in 'Jl,., n '.ncpo~~: and "' in '.ncp n '.ncpo~~: and we have:
~o. Then ( i
® w 11 )F( x) belongs to
Proof. The fundamental operator of the dual Kac algebra i is uW*u (3.7.3); so (i) comes from 2.7.6(vii) applied to OC. From (i), we get W(1 ~~ .Jity~r:. =~ .Jit ~~lit; so (ii) comes from 3.2.2 (iv).. By 3.6.6(i) we have ~~:(Llzt) = JLldJ = Ll-zt; so, 3.6.9(iii) applied to Ll-zt yields (iii). Let w be in M't. We put w1 = w(Ll- 1 ·) in the sense of ([114], prop. 4.2). By 3.6.7 and ([16], 1.1.2(b)), we have: cp ® w = (cp o 11: ® w 1 )((Ll ® Ll)·)
Therefore, for all x in M+, by ([114], prop. 4.2), we have: ( cp ® w )(F(x ))
= e-+0 lim( cp o 11: ® w
1
)((Ll ® Ll)eF(x ))
where: We have: W(1 ® (1
and so:
+ c:Ll))W* = 1 ® 1 + c:W(1 ® Ll)W* = 1 ® 1 + c:(Li ® Ll)
by (i)
3.7 The Dual Kac Algebra
and:
(Li ® Li)e
117
= (Li ® Ll)W(1 ® (1 + e..&)- 1)W* = W(1 ® LnW*W(1 ® (1 + eLl)- 1)W* = W(1 ® L1A(1
+ e..&)- 1)W*
= r(Lie)
where: therefore, we get:
(c,o ® w )(F(x ))
= e-->0 lim( c,o o K ® w')r(Liex) = lim c,o o K(Liex)w 1(1)
by 2.2.4
= c,o(x)w(..&- 1 )
by 3.6.7
e-->0
which is (iv). We assume 111711 = 1. Then (i ® w'l) is a conditional expectation and we have:
(i ®w'l)(F(x*))(i ®w'l)(F(x)):::; (i ®w'l)(F(x*x)) which implies: c,o(( i ® w'l)(F(x*))( i ® w'l)(F(x))) :::; (c,o ® w'l)(F(x*x))
= IIL1A-1/217112c,o(x*x) therefore ( i ® W'l )( F( X)) belongs to
K((i ®w'l)F(x))*
~·
by (iv)
Furthermore, we have:
= (i ®w'l o K)(K ® K)F(x*) = (w'l o K ® i)F(K(x*))
which, by hypothesis, and by 2.3.1 (ii), belongs to 'Jlcp; therefore (i®w'l)(F(x )) belongs to have:
'.ncpo~~:, and by 3.5.2(i), ~((i ®w'l)F(x)) belongs to 1)~, and we F~((i ® w'l)(F(x)))
= Acp((w'l o K® i)F(K(x*)) = ,\(w'l)Acp(K(x*)) = -\(w'l)FAcp(x)
Therefore, we have: ~((i ® w'l)r(x)) = F-\(w'l)FAcp(x)
= j J.-1/2 ,\(w'7)J.1/2 j Acp(x)
by 2.3.5 by 3.5.2(i)
118
3. Representations of a Kac Algebra; Dual Kac Algebra
ein !Bo, we have:
On the other hand, for any
(.\(w.21-t/4,)e Ie)= (w.21-1/4'1 ® we)(W*)
by 2.6.1 (i)
= (W*(..::i- 1/ 4 71 ®e) 1.&- 1/ 4 71 ®e) = (w*(..::i- 11271 ®e) 111 ®e)
by 3.6.7
= (w*(..::i-1/271 ® ..:1-1/2 j1/2e) 171 ®e) = {{1 ® ..:1-ll2)w*(71 ® Li1t2e) 111 ®e) by (i) = (w*(71 ® Li 1t 2e) 111 ® .&- 1t 2e) = (.\(w,)..::i1/2e I j-1/2e) by 2.6.1 {i) Therefore we have the proof of (v).
.1- 1/ 2.\(w,)..::i1/ 2 C
.\(w.21-t/4'1) which allows to complete
3. 7.8 Theorem. Let K = (M, r, tt, r.p) be a K ac algebra, IK' the commutant Kac algebra. The dual Kac algebra K'A is equal to K'. Proof. We have seen in 3.3. 7 that the dual co-involutive Hopf-von Neumann algebra (M'A,r'A,tt'A) is equal to {M,d',K.). Let us now compute the dual weight cp'A; let us recall {2.3.7) that, if w belongs to M., if we define w' in M. by w1(x) = w(Jx* J), we have .\1(w1 ) = .\(w). Let us now suppose that w' belongs to lr.pt. We have, then, for any X in
m:,:
w(JxJ)
= w(Jx* J)= w'(x)= (a (w I Ar.pt(x*))1
1)
= (a 1(w 1 ) I JAr.p(Jx* J))-
= (Ja'(w') IAr.p(Jx* J)) sow belongs to lr.p, and a(w) = Ja1(w1 ). For w1, w1 in lr.p', we have: c,01(.\1(w')* .\1(w 1 ))
= (a 1(w 1) Ia1(w 1)) = (Ja(w2) I Ja(w1))
by 3.5.2(ii) applied to c,O'
= (a(wt) I a(w2))
= c,O(.\(w2)* .\(w1))
by 3.5.2 {ii) applied to c,O = c,O(.\(w2 o tt).\(w1 ott)*) by 1.2.5 and 2.5.3 = c,O(K.(.\(w2))K.(.\(w1)*)) by 3.3.1 = c,O o K.(.\(w1)* .\(w2))
= c,O o K.(.\1(w')* .\1(w1)) Using 2.7.7, we get c,O' = c,0 o K., which ends the proof.
3.7 The Dual Kac Algebra
119
3.7.9 Proposition. Let][{= (M, r, ~t, cp) be a Kac algebra and i: = CM, i', it, cp) be a K ac subalgebra in the sense of 2.2. 7. Then: (i) there is a unique faithful normal conditional expectation E from M to M such that cp o E = cp. Moreover, E satisfies:
i' o E
= (E ® E)r
itE
= E~t
and the projection P defined by P Acp( x)
= Acp( Ex)
belongs to the centre of
M· '(ii)
-
the canonical surjective E-morphism r from M toM~, defined in 3.3.8, has P as support, and identifies i:~ to Kp (cf. 2.2.6).
Proof. As M is o-f-invariant and cp!M semi-finite, by [160], there is a unique normal conditional expectation E such that cp o E = cp (cf. 2.1.8(ii)). As in 3.3.8, let j be the canonical imbedding M -+ M, and I the isometry defined, for all X in ~~ = ~ n M by: IA~(x)
= Acp(j(x))
Let P =II*. We know, from [160] that, for ally in
~cp:
A~(Ey) =I* Acp(y)
Acp( Ey)
= P Acp(Y)
and, moreover, that, for all x in M, Ex is the unique element of M such that (Ex)P = PxP. Let w be in Icp; it is easy to see that w o j belongs to I~, and that: a(w 0 j)
= I*a(w)
We have then, using 3.3.8: cp~(r(A(w)* A(w))) = cp~(.\(w o j)*.\(w o j)) = l!a(w o j)ll 2
by 3.5.2 (ii) applied to cp
= I!I*a(w)ll2 = I!Pa(w)l! 2
Let R be the support of r. Using 2.7.8, we know there ifl a> 0 such that: cp~(r(A(w)* A(w)))
= acp(RA(w)* A(w)) = ai!Ra(w)l! 2
by 3.5.2 (ii)
120
3. Representations of a Kac Algebra; Dual Kac Algebra
So, we have 11Pa(w)ll 2 = a11Ra(w)ll 2 for all win IV'; so we get P = R (and so P belongs to the centre of M), a= 1, cpA= c(;p, which ends the proof of (ii). Now, E ®Eisa faithful normal conditional expectation from M ® M to M ® M such that, for all X in ~IP®IP:
Atp®IP((E ®E)( X))= (P ® P)Atp®tp(X) So, for x,y
in~.
we shall have:
AIP®IP((E ® E)(F(y)(x ® 1))) = (P ® P)A~P®IP(F(y)(x ® 1)) = (P ® P)W(AIP(x) ® AIP(y)) = (P ® 1)W(1 ® P)(AIP(x) ® AIP(y)) because P belongs to Z ( M) = (P ® 1)W(A1P(x) ® Atp(Ey)) = (P ® 1)A1P®1P(F(Ey)(x ® 1)) As E ® i is also a faithful normal conditional expectation from M ® M to
M ® M such that, for all X in ~IP®IP: AIP®IP((E ® i)(X))
= (P ® 1)A~P®1P(X)
We have then:
AIP®IP((E ® E)(F(y)(x ® 1)) = Atp®IP((E ® i)(F(Ey)(x ® 1))) and, therefore:
(E ® E)(F(y)(x ® 1)) = (E ® i)(F(Ey)(x ® 1)) By continuity, we get, for all y in
(E ® E)(F(y))
~V':
= (E ® i)(F(Ey)) = F(Ey)
and, by continuity again, we have:
(E®E)F=FE Let now x be in M; we have: ~t(Ex)P = }(Ex)JP
= }(Ex)PJ
by 3.6.6(i) because P belongs to Z(M)
= JPxPJ
=PJxJP = P~t(X)P = E~t(X)P
because P belongs to Z(M) by 3.6.6(i)
3.7 The Dual Kac Algebra.
121
We have then K(Ex) = EK(x), which ends the proof of (i) and of the proposition.
3. 7.10 Proposition. Let ][{ = ( M' r, K, 'P) be a K ac algebra, R be a projection of the centre of M such that F(R) ~ R Q9 R, K(R) = R and JKR be the reduced Kac algebra in the sense of 2.2.6. There is a canonical one-to-one llll-morphism j from (JKRt to lK which identifies (KRt with a Kac subalgebra ofOC (cf. 2.2.7). More precisely, ifr denotes the reduction x - t XR of M on MR, we shall have, for all w in (MR)*: j(>..R(w))
= .\(w or)
where AR denotes the Fourier representation of JKR. Proof. Let us call I the projection R, considered as an element of .C(H'f', H'f'R). We have, then:
I* I= R II*= ln'PR IA~.p(x)
= A'f'R(r(x))
r(x) = Ixl*
(x E l)'tp) (x EM)
Moreover, if w is in I'f'R• it is easy to check that w or is in I'f' and that:
a(w or)= I*a(w) The reduction r is an llll-morphism, thus the mapping w - t wor from ( M R)* to M* is multiplicative and involutive; therefore, the set { .\( w or), w E (M R)*} is an involutive subalgebra of M; let us call N its weak closure. Let f3 in H 'f' such that {/31 xc5) = 0 for all x in N and c5 in H 'f'· We have, for all a, 1 in H'f'R• all c5 in H'f': 0=
{/31 A(WJ*-y,I•a)c5)
= (W(ra ® /3)
r1 ® c5) =((I ®l)W(ra Q9 /3) 11 Q9 c5) 1
which implies, for all a in H'f'R:
(I ®l)W(ra Q9 /3)
=o
(R ®l)W(I*a Q9 {3)
=0
or:
by 2.6.1 (iii)
122
3. Representations of a Ka.c Algebra; Dual Ka.c Algebra
as R is in the centre of M, W belongs toM® M and Rr = r, it implies:
W(ra®/3)
=o
As w is unitary, it gives r a ® /3 = 0, for all a in H cp R' which implies /3 = 0. Then, N is a non-degenerate algebra on Hcp; it is a von Neumann subalgebra of M. Let y be in '.>lcp, win (MR)•· We have:
D.(w o r)Acp(Y) = IAcp((w oro K ® i)T(y)) = AcpR((w o K ® i)(r ® r)T(y)) = AcpR((w o K ® r)T(ry)) = AR(w)AcpR(r(y)) = AR(w)IAcp(Y)
by 2.3.5
therefore:
D.(w or)= AR(w)I and:
RA(w or)=
r AR(w)I
By passing to the adjoints this equality yields that R belongs to N'. So R belongs to the centre of M, and, for all in!!~, we have:
e,"'
Now, let z be in
M such that zR = 0. For all e.., in!!~, we shall have: zA(wRe,R'I) = z1r(ReT R"'b)
= zR1r(eT.,b)
by 3.5.4 (iii) by the above remarks
which implies, by continuity, for all e,"' in Hcp:
zA(wRe,R'I)
=0
or, also, for all -y,a in HcpR:
that is, for all w in (MR)•:
zA(wor)=O and zN = 0, which ensures z = 0 by the above results on N. So, the reduction N -+ N R is an isomorphism. Let us call 9 the inverse isomorphism, and for x
3. 7 The Dual Ka.c Algebra
123
in ( M RY, let us put j ( x) = ~(I* xi). It is clearly a one-to-one homomorphism from (MRY toM such that j(l) = 1. Moreover, we have, for all win (MR)*:
j(>.R(w))
=~(I* >.R(w)I)
= ~(R>.(w or))= >.(w or)
The range of j is therefore equal to N. Now, let wbe in M*. We have:
and then r >.* = >. R*i*. From what it is straightforward to prove that j* is involutive and multiplicative and therefore that j is an lBI-morphism. For all t in JR, we have:
afp.(w or)) = >.(w oro L ~it) by 3.7.7 (iii) = >.(w o Lr(~it) or) which belongs toN therefore N is af-invariant. Let win I'PR" We have:
cp(j(>.R(w)* >.R(w)) = cp(>.(w o r)*>.(w or)) = lla(w o r)ll2 = III*a(w)ll 2 = lla(w)ll 2 = .R(w)* ).R(w)) Therefore j(>.R(I'PR) C SJl.p, which implies that cp IN is a semi-finite weight. Finally, we see that N is a Kac subalgebra of M, j is an lBI-isomorphism from (MRr toN and, by 2.7.9 and the above calculation cp o j =
Chapter 4 Duality Theorems for Kac Algebras and Locally Compact Groups
In that chapter, we obtain a duality theorem for Kac algebras, namely that the bidual Kac algebra is isomorphic to the original Kac algebra (4.1.1 ). From that, we can successively deduce that the Fourier representation A is faithful, and that M* is semi-simple (4.1.3). We also see that the dual Kac algebra of the Kac algebra lK8 (G) constructed in Chap. 3 is the Kac algebra !Ka (G) constructed in Chap. 2 (4.1.2). These results were found, independently, by the authors in [36], and Va:lnermann and Kac in [180]. Moreover, we obtain that the relative position of the von Neumann algebras M and M is such that (4.1.5):
M
n M = M n M' = M' n M = M' n M'
This result, from [136], leads, in the case of !Ka( G), to Heisenberg's theorem (4.1.6). The crucial link with duality of locally compact groups is given by Takesaki's theorem ([157]), which states that every symmetric Kac algebra lK is isomorphic to the symmetric Kac algebra constructed from the intrinsic group of lK (4.2.5). By duality, we get that every abelian Kac algebra lK is isomorphic to the abelian Kac algebra constructed from the intrinsic group of K. Applied to a standard Borel group with a left-invariant measure (4.2.6}, we get A. Weil's theorem [197]. Applied to lK8 (G), Takesaki's theorem leads immediately to Eymard's duality theorem ([46]), which states that G is the spectrum of the Fourier algebra A( G) (4.3.3), and, which, in turn, contains, in the commutative case, Pontrjagin's duality theorem (4.3.8). Eymard's duality theorem allows us to give a precise description of all the objects constructed from the symmetric Kac algebra .lK8 (G); in particular, the Fourier algebra of lK8 (G) is L 1 (G), and its Fourier-Stieltjes algebra is M 1 (G) (4.4.1), which leads, in the commutative case, to Bochner's theorem (4-4-3). We then, after [38], characterize all the morphisms which realize the quasiequivalence of A with A x p., for all non-degenerate representations p., as
4.1 Duality of Kac Algebras
125
proved in Chap. 3 (-4.5.6). When I' is of dimension 1, we then get another characterization of the intrinsic group of a Kac algebra (.4.5.8), due to De Canniere ([18]), which leads to Wendel's duality theorem ([199]) for locally compacts groups (.4.5.9). To each couple of non-degenerate representations of M* and M* respectively, we functorially associate a unitary operator belonging to the tensor product of the von Neumann algebras generated, as it was done in [40]; it is called the Heisenberg's pairing operator (.4.6.2). For one-dimensional representations, i.e. for the intrinsic groups, we get a bicharacter in a situation similar to Heisenberg's commutation relation (-4.6. 7). This Heisenberg's pairing operator allows us to construct the extension of any non-degenerate representation of M* to a representation of the Fourier-Stieltjes algebra B(K), just the same way non-degenerate representations of L1 (G) are extended to M 1 (G) (-4.6.8). This will be essential to define the arrows of the category of Kac algebras in Chap. 5. Chapter 4 ends with a Tatsuuma type theorem about Kac algebras (-4. 7.2), which gives, as corollaries, Ernest's duality theorem ([44]) and Tatsuuma's duality theorem ([168]) on locally compact groups.
4.1 Duality of Kac Algebras
r,
4.1.1 Theorem. Let ][{ = (M, K, r.p) be a K ac algebra. The bidual K ac algebra ][{M is isomorphic to ][{ (equal if we identify Hcp and Hq,), and the Fourier representations A and ,\ are linked by:
,\ = K 0 A*
A= k o ).* .
Proof. The von Neumann algebra MM is, by definition, generated by the Fourier representation>. which is equal to KA* (cf. 3.7.3), up to the isomorphism between Hcp and Hq,; we thus have: MMCM
The fundamental operator W associated to lK is equal to uW*u (cf. 3.7.3); so 2.7.4(i) applied to i: gives that, for any x in .C(H), x belongs to MM' if and only if (x ® 1)W = W(x ® 1). So, by 2.7.6(v), we get MM' = M' and then M =MM. Similarly, for any x in MM = M, we have:
rA(x)
= uW*o-(1 ® x)uWu = W(1 ®x)W* = r(x)
and:
by 3.2.2 (iv) applied to
lK
by 3.7.3 by 2.6.3 (ii)
126
4. Duality Theorems for Kac Algebras and Locally Compact Groups ~~:AA(x)
= Jx* J =
~~:(x)
by 3.3.1 applied to lK by 3.6.6(i)
By 2. 7. 7 the two Haar weights
cp(.X(w,a,cS)* .X(w,a,cS))
= cp(~~:.X.(w,a,cS)*~~:.x.(w,a,cS)) = cp(7r({3tob)*7r({3tob)) = llf3Tobll2 = IICi(w,a,cS)II 2 = cpAA(X(w,a,6)* .X(w,a,6))
by 3.7.3 by 3.6.2(i) by 2.1.7(i) applied to<(; by 3.5.2(ii) applied to
lK
Therefore
IK. 4.1.2 Corollary. Let G be a locally compact group, and JK..( G) the symmetric Kac algebra associated by 3.7.5. The dual Kac algebra IK8 (Gr is equal to IKa(G) (when L 2 (G) and Hr.p. are identified).
Proof. It is a combination of 4.1.1 and 3.7.5.
r,
4.1.3 Corollary. Let 1K = (M, ~~:, cp) be a Kac algebra, K = (M, t, K., <(;), its dual K ac algebra. Then: {i) The modular operator ..:1 is affiliated to the centre of M, and is the Radon-Nikodym derivative of<(; with respect to <(; o K.. {ii} The Fourier representation .X is injective and, therefore, M* is semisimple. (iii} For all t in :R., ..:1it belongs to the intrinsic group of K. (iv) Let X in IJty, n !Jty,o~ and.,., in 2lo. Then the element (i ® w,)i'(x) belongs to 1Jty, n !Jty,o~ and we have:
(v) Let A be a von Neumann algebra, 1/J 'a faithful semi-finite normal weight on A+, and let X in ~]t,p0y, n IJt,p®<,Oo~ and Tf in !2lo. Then the element (i ® i ® w11 )(i ® i')(X) belongs to !Jt,p0 y, n 1Jt,p 0 y,0 ~, and we have:
4.1 Duality of Kac Algebras
Proof. As, by 4.1.1,
][{M
127
= OC, the first assertion results of 3.6.7 applied to
OC. By 3.3.5, ~~:.\* is faithful, therefore so is .X, thanks to 3.7.3. Applying this result to OC and using 4.1.1, we get the second assertion. The third assertion is 3.7.7(ii) applied to][{= oc··. By applying 3.7.7(v) to OC, we get (iv). Let us assume 117711 = 1. As i ® i ® w.,., is a conditional expectation, we get:
(1/J e
e i e w.,.,)((i e f)(X))))
~ (1/; ® cj; o it)( i ® i ® w.,., )( i ® f)( X* X)
= (1/; ® cj; o it)(X* X) because cj; o It is left-invariant with respect to ~f (2.2.4). So (i ® i ® w.,., )( i ® f)( X) belongs to '.)lt/>®
c1/J e cf;)(((i e i e w.,.,)(i e f)(X*)((i e i e w.,.,)(( i e f)(X)))) ~ (1/; ® cj;)(i ® i ® w.,.,)(i ®f)( X* X)
= IILl-l/2 7711 2 (1/; ® cj;)(X* X)< +oo by 3.7.7(iv) applied to OC Then ( i ® i ® w.,., )( i ® f)( X) belongs to '.)lt/>®cp and the operator which sends AtJ>®®
which, by continuity and density, completes the proof of (v). 4.1.4 Corollary. (i) Let][{= (M,F,~~:,cp) be a Kac algebra, and let OC" be (M,~F,~~:,cp o ~~:) the opposite Kac algebra as associated in 2.2.5. Then the dual K ac algebra ][{"' is equal to OC1• (ii) The K ac algebra ][{ is abelian if and only if the dual K ac algebra OC is symmetric.
Proof. Let us apply 3.7.8 to the Kac algebra K; we get (K)'' = by 4.1.1; the second assertion is clear from 3.3.3 and 4.1.1. 4.1.5 Corollary. Let][{= (M, dual K ac algebra. Then:
M n
M=
M n
r, 11:, cp) M' =
be a Kac algebra,
M' n
M=
M' n
(KY"
= OC"
K = (M, f, it, cj;)
M' = c .
its
128
4. Duality Theorems for Kac Algebras and Locally Compact Groups
Proof. By 2. 7 .2, we have:
M n M'
= M' n M' = c
Applying 2.7.2 to the Kac algebra ][(5, we have, using 4.1.4:
and the result is proved. 4.1.6 Corollary (Heisenberg's Theorem). Let G be a locally compact group. If we con8ider the element8 of L 00 (G) a8 operator" on L 2 (G), we have:
Proof. We get the result by applying 4.1.5 to the Kac algebra IKa(G).
4.1.7 Proposition. The i8ometries J and j commute. Proof. Let X in ~ n IJl~ n ~Oit n IJl~Oit j by 3.5.2, i.e. to 2'(Ll112) n 2'(.&- 112). We have then:
~(X) belongs to 1)U n 1)b'
iJ~(x) = iLl 112 Acp(x*) = Ll112 iAcp(x*)
by 4.1.3(i)
= Lll/2j-1/2(Acp(x*))b
= Lll/2j-1/2Acp(~~:(x))
by 3.5.2(i)
= .,&- 1/ 2Ll 1/ 2Acp(~~:(x))
by 3.6.7 because ~(~~:(x)) belongs to 2'(Ll 112)
= j-1/2JJLll/2~(~~:(x)) = j-1/2JAcp(~~:(x*)) = J j-1/2 ~(~~:(x*))
by 3.6. 7
= J i(Acp(~~:(x*)))b
= Ji~(x)
by 3.5.2(i)
by density (2.7.5(v)), it completes the proof. 4.1.8 Lemma. The set Y = { xJ Acp( x ); x E IJlcp selfdual cone Pep defined in 2.1.1 (iv).
n IJlcpolt}
i,
dense in the
4.1 Duality
of
Ka.c
129
Algebras
Proof. Let us put ..1- 1 = f000 sdE8 ; en = J{/n dE11 ; from 3.6.7, en belongs to the centre of M. Let x in 'Jtp. We have:
and enX belongs to 'Jtp n IJtepo~t· Moreover:
because en belongs to the center of M. Thus, this vector, which belongs to Y, converges to x J Aep( x) when n goes to infinity. Therefore Y is dense in {xJAcp(x); x E IJtep}, the closure of which is Pep. 4.1.9 Theorem. The isometry sense o/[57] (cf. 1.1.1 (iii)).
J is
the canonical implementation of K. in the
Proof. Thanks to ([57] th. 2.18), 3.6.6 (i) and 4.1. 7, it remains to prove that Pep is invariant under J. Using the above lemma, it is enough to prove that Jy C Pep. As Aep('Jtp n IJtc,oo") c 'Db = TJ(Li- 112) and as ..1 is affiliated to the centre of M, we have J ..1- 1/ 2 C ..1- 1/ 2J, and therefore, for all x in 'Jtp n 'Jtpo". J Aep( x) belongs to 'Db; then, we have:
JxJAep(x) =
Li112 PxJAep(x)
= ..1112K-(x*)F J Aep(x) = _.11/2 K.(x*)J1/2 j J Aep(x) = _.11/2 K.(x*)J1/2 J j Aep(x) = _.11/2K.(x*)J..11/2rAcp(x)
= ..1
112 K-(x*)JAep(K-(x*))
by 3.6.3 (i) by 4.1.7 by 3.6.7 by 3.5.2(i)
Let us remark that K-(x*)JAep(K-(x*)) belongs toY C Pep. Let us put:
dn = {n sdE8 }1/n It results from 3.6.7 that dn belongs to the centre of M, and then that dn = d-:/2 Jd-:/ 2 J. By ((57]1.9.(3)), we get that dnPep is included in Pep and therefore dnK.(x*)J Aep(K-(x*)) belongs to Pep. As Pep is closed, it implies, when n goes to infinity, that Li 112 K-(x*)J Aep(K-(x*)) belongs to Pep, which completes the proof.
130
4. Duality Theorems for Kac Algebras and Locally Compact Groups
4.1.10 Corollary. The i.sometry J i.s the canonical implementation of K (in the sen&e of 1.1.1 (iii)). Proof. By 4.1.9 and 4.1.1.
4.2 Takesaki's Theorem on Symmetric Kac Algebras In all this paragraph ][{ = (M' r, K,
= {s,w) = gw(s- 1 )
by 2.6.6(i)
by definition of g and 1.2.3 which gives (i). Moreover: g(s ·w)(t)
= {t*,s ·w) = (t*s,w) = {(s- 1t)*,w) = gw(s- 1t)
by definition of g
which gives (ii). We have: g(w0 )(s) = {s*,w 0 )
= {n(s),w)= (s*,w)= (gw)-(s)
which gives (iii).
by 1.2.5 by 2.6.6(i) by definition of g
4.2 Takesaki's Theorem on Symmetric Kac Algebras
131
By (iii) we see that all characters on M* are hermitian; using ([12], I, §6), we get (iv). By 4.1.3 (ii) we see that the Fourier representation A is one-to-one; therefore the same holds for the universal representation, that is 9, by (iii), and it completes the proof. 4.2.2 Lemma. There in lr.p:
exi.~ts
on G a H aar measure ds such that, for all w1, w2
Proof. Let X be the canonical extension of A to C0 (G) (cf. 1.6.1 (i)). The space X(C0 (G)) is included in M which is an abelian von Neumann algebra. Therefore, the weight cj; is a trace, and cj; o X is a lower semi-continuous trace on Co( G). Let w in Ir.p. We have:
cj; o X(l9wl 2) = cj;(A(w)* A(w))
= lla(w)ll 2
(*) by 3.5.2 (ii)
Therefore 9w belongs to IJl.poX. The ideals IJl.po_x and rot.poX are dense in Co( G); by ([114], 5.6.3) they include the algebra K( G) of continuous functions on G with compact support. Therefore, cj; o X is a positive linear form on K( G) and, by [11], a Radon measure on G which will be noted p. As cj; oX is lower semi-continuous, we shall have, for any f in Co( G)+:
and, by ( *) we get, for all w in Icp:
therefore, 9w belongs to L 2 (G, p). Polarizing, we shall get, for
Ia
(9w2)-gwldJ-L
= (a(wl) Ia(w2)) = cp(A(w2)* A(wl))
WI,W2
in Icp:
132
4. Duality Theorems for Kac Algebras and Locally Compact Groups
Finally, Qlcp is dense in C0 { G), it is then also dense in L 2 {G, J.l) and (Qicp)-gicp will be dense in L 1(G,J.t). And, we have:
fa (Qw2)-(s- 1t)Qwt(s- 1t)dJ.t(t)
= fa(Q(s·w2))-(t)(Q(s·wt))(t) = (a(s · w1) Ia(s · w2)) = (sa(wt) Isa(w2)) = (a(wt) Ia(w2)) = fa (Qw2)-(Qwt)dJ.l
by 4.2.1 {ii) by(**) by 2.1.7{iii) by 1.2.3
By density, it follows that J.l is left-invariant, which completes the proof. 4.2.3 Proposition. There exist a Haar measure ds on G and an isomorphism U from Hcp to L 2 ( G, ds) defined by, for all w1 in Icp:
Ua(w')
= Qw'
such that we have, for all s in G, and w in M.:
{i) (ii)
UsU* = >.a(s) U>.(w)U* = Qw.
Proof. Thanks to 4.2.2 it is clear that U as defined in the proposition is unitary. Moreover, we have, for all s, tin G and win Icp:
(>.a(s)Qw')(t)
= Qw1(s- 1t) = (Q(s · w 1 ))(t)
by 4.2.1 {ii)
Therefore:
>.a(s)Qw'
= Q(s · w') = Ua(s · w1) = Usa(w 1)
by definition of U by 2.1.7(iii)
which gives {i). Let w in M., w1 in Icp. We have:
U >.(w )U*Qw1 = U >.(w )a(w1 ) = Ua(w * w1 ) = Q(w*w') = QwQw1
by 2.6.1 (v)
4.2 Takesaki's Theorem on Symmetric Kac Algebras
133
Since Qlcp is dense in L 2 (G), we have:
U>.(w)U* = Qw which completes the proof. 4.2.4 Takesaki's Theorem. Let lK = (M, r, ,.,,
Proof. Using 4.2.3 (ii) and considering the generated von Neumann algebras, we see that x --+ UxU* is an isomorphism from M to L 00 (G(IK)). More accurately, we have, for all w in M.: U>.(w)U*
= Qw
and we can deduce, thanks to 1.4.5 (ii), that we have:
(U ® U)F>.(w)(U* ® U*) = (U ® U)c;(>. x >.)(w)(U* ® U*) = c;Q x Q(w) Let J,h,h in L 1 (G(IK)). We have:
(Q.J,w) = (Qw,J) = faf(s)Qw(s)ds = fa(s*,w)f(s)ds which implies:
and:
(c;Q
X
Q(wl),h ®h)= (Q.hQ.h,w) = {
laxG
=
f
lax a
h(t)h(s)(t*s*,w)dsdt Qw(st)fl(s)h(t)dsdt
= (Fa9w, h
®h)
Thus we have, for all w in M.:
(U ® U)F(>.(w))(U* ® U*) and, by density, for all x in
= Fa(9w)
M:
(U ® U)F(x)(U* ® U*)
= Ta(UxU*)
134
4. Duality Theorems for Kac Algebras and Locally Compact Groups
Moreover, we have: Uk(.A(w))U* = U.A(w o K)U*
by 3.3.1
=Q(woK.)
= Ka(Qw) by density, for all x in
by 4.2.1(i)
M, we get: Uk(x)U* = Ka(UxU*)
Finally, by 4.2.2, we have, for all w in lr.p:
Since x --+ U xU* is an llll-isomorphism between two Kac algebras,
4.2.6 Corollary (Well's Theorem (197], (94]). Let G be a standard Borel group, and let there be au-finite left-invariant measure m on G. Then G is a locally compact group and m is a left H aar measure on G. Proof. The hypothesis allows us, using FUbini's theorem to see that the product measure m ® m is invariant under (x,y) --+ (y- 1 x,y), and then, successively, by (x,y)--+ (x,x- 1 y) and (x,y)--+ (y-l,xy). So, if the function f belongs to the abelian von Neumann algebra L 00 (G, m), the function (s, t)--+ f(st) on GxG belongs to L 00 (Gx G, m®m) (identified to L 00 (G, m) ®L 00 (G, m)), and the functions--+ f(s- 1 ) on G belongs to L 00 (G, m). We have then defined a coproduct rand a co-involution "' on L 00 ( G, m ). Moreover, the measure m defines a normal semi-finite faithful trace on L 00 (G,m)+, which is left-invariant with respect to the coprod-
4.2 Takesaki's Theorem on Symmetric Ka.c Algebras
135
uct r. The fundamental operator associated by 2.4.2 is defined, for all j in L 2 (G x G, m®m), x,y in G, by: (Wf)(x,y)
= f(x,xy)
Let us now consider, for all sinG, the unitary operator ~t(s) on L 2 {G,m) defined, for all fin L 2 {G,m), tin G, by: (~t(s)f)(t) = f(s- 1t)
By ([94], lemma 7.4), I' is injective. A straightforward calculation gives that the functions - ~t(s)* on G, if it is considered as an element of the tensor product L 00 (G,m) ®.C(L 2 (G,m)), is equal toW, and therefore, we get that W is unitary, and that (here~ is an automorphism because L 00 (G,m) is abelian): {~ ® i)(W) = W* which, by 2.4.6 (v), proves that (L 00 (G, m),r, ~. m) is an abelian Kac algebra. Moreover, for all s in G, we have: W*(~t(s) ®
l)W = ~t(s) ® ~t(s)
and, by 3.6.10 and 3.2.2, ~t(s) belongs to the intrinsic group G' of L 00 (G, mr. In fact, by 4.2.3 {i) and 4.2.5 {iii), there exists an isomorphism U from L 2 (G,m) to L 2 (G 1) such that, for all v in G': UvU* = >..a,(v)
and moreover, such that x - U xU* is an isomorphism from the Kac algebra (L 00 (G,m),r,~,m) to :Ka{G') and, therefore, from the dual Kac algebra (L 00 (G,m),r,~,mr to :Ka{G'). So, (U ® U)W(U* ® U*) is the fundamental operator of :Ka(G'), and, by 2.4.7{i), (U ® U)W*(U* ® U*) is the function v - >..a,(v) on G', and (U ® I)W*(U* ®I) is the identity function on G1, considered as an element of L 00 (G') ® L 00 (G,mt. So, for all win {L00 {G,mn., we clearly see that U(i ® w)(W)U* is the function v - (v,w) on G' and, as {i ® w)(W) is the functions- (~t(s),w) on G, we infer that the isomorphism f - U* JU from £ 00 ( G') to £ 00 ( G, m) is just the composition by I'· Using f = XG'-p(G)• we see then that G1 -~t(G) is of Haar measure 0. So, if v belongs to G1 -~t(G), v~t(G) ~ G1 -~t(G) is of Haar measure 0, and so is I'( G), which is impossible; so I' is surjective, and the theorem is proved.
136
4. Duality Theorems for Kac Algebras and Locally Compact Groups
4.3 Eymard's Duality Theorem for Locally Compact Groups In that paragraph G will denote a locally compact group. We shall apply the preceeding paragraph to OC8 ( G). 4.3.1 Lemma ([44)). The left regular representation >..a is a one-to-one homeomorphism from G to the intrinsic group of OC8 ( G). Proof. Lets be in G. Then, by 3.3.6, >..a(s) belongs to G(OCs(G)); the mapping >..a is also clearly one-to-one. Now let {>..a(s 11 )} 11 denote a net converging to the identity 1 = >..a(e), where e is the unit of the group G. We wish to show that s 11 converges to e.
Proceeding by way of contradiction, we suppose it does not. Then there exists a subnet, say Sn of s 11 and a compact neighbourhood V of e such that sn rf. V, for all n. Choose a compact neighbourhood U of e such that uu-l c v. Then, for each Sn, we have (xu denotes the characteristic function of U and 11 the Haar left measure on G):
(>..a(sn)xu Ixu)= faxsnuxud/1 = 1-L(snU nU) = 0 Thus:
l((>..a(sn) -1)xu I xu)l =(xu I xu) = 11(U) > 0
Thus >..a(sn) does not converge weakly to 1, which contradicts our first assumption and completes the proof. 4.3.2 Theorem. The left regular representation >..a is a bicontinuoul! bijection from G onto the intrinllic group of OC 8 ( G). Proof. Let denote G0 the intrinsic group G(OCs(G)). By 3.7.5, the Fourier representation >.. of OC8 (G) is the mapping, defined, for all w in .C( Gh, by:
s-+ (>..a(s)*,w) which belongs to L 00 ( G). By 4.2.4 applied to ][{8 ( G), there exists a unitary U from L 2 (G) on L 2 (Go) such that U>..(w)U* be equal to Qw, which is the mapping defined, for all t in G 0 , by:
t-+(t*,w) which belongs to L=(G0 ). Thus, we have:
U*QwU
= Qw o >.a
4.3 Eymard's Duality Theorem for Locally Compact Groups
137
and, by continuity, for all fin L=(G0 ): U*fU
= fo>.a
By selecting f = Xao-.Xa(a)• we see that f = 0, and Go- >.a( G) is a zeromeasure set. Let so EGo and So rf. >.a( G); we have so>.a(G) C Go- >.a( G); therefore s 0 >.a( G) is a zero-measure set as well as >.a( G) by left-invariance, but this is impossible by(*). Therefore >.a(G) = G 0 , which completes the proof. 4.3.3 Corollary (Eymard's Theorem [46]). Let G be a locally compact group. Let us recall ( cf. 9.4. 6 and 9. 5. 6{ that the Fo'll:..rier al~ebra A( G) has been defined as the set {f*g, f,g E L (G)} (where f(s) = f(s- 1 ) for all sinG, fin L 2(G)), equipped with the norm:
llall
=sup
{l.la
a(s)f(s)dsl, f E L 2(G),
l!>.a(f)ll
:51}
Then, the spectrum of A( G) is G; so every character on A( G) is involutive. Proof. By 4.3.2, the spectrum of£( G). is equal to >.a( G). By 3.4.6 and 3.5.6, there is an isomorphism between C(G)* and A(G), which, to each Wj,g in C(G)., associates the function f * g, linked by:
So, we see that the spectrum of A(G) is the set {s- 1 ,s E G}, that is G. 4.3.4 Corollary. Let G1, G2 two locally compact groups, u an JH[.morphism from lHl8 (G1) to lHl 8 (G2); then there exists a continuous group homomorphism a from G1 to G2 such that u( >.a1 ( s)) = >.a2 (a( s)), for all s in G1. The image a(Gl) is a closed subgroup of G2. If u is injective, then a is injective; if u is surjective, then a is surjective too. Proof. It is clear that u sends the intrinsic group of JH[8 (G1) into the intrinsic group of JH[8 ( G2 ), and that the restriction of u is a continuous homomorphism of groups. So the existence of a comes directly from 4.3.2. Then, the subset {x E £(G1); T 8 (x) = x®x} is a closed subset of the unit-ball of £(G1), and therefore, is compact for weak topology. Its image par u, that is:
by 3.6.10 and 4.3.2, is then also compact for the weak topology of £(G 2 ). So {>.a 2 (a(s)), s E GI} is locally compact, and by 4.3.1, a(GI) is a locally compact subgroup of G2, and so is a closed subgroup of G 2.
138
4. Duality Theorems for Ka.c Algebras and Locally Compact Groups
For every win .C(G2)., we have: gw(a(s)) = (.\a2 (a(s)*),w)
by 4.2.1 by de:fini tion of a
= (u.\a1 (s)* ,w)
= (.\a1 (s)*,w o u) by 4.2.1
= g(w o u)(s)
And: gw o a= g(w o u)
Therefore, gw o a = 0 implies w o u = 0, by 4.2.1 (v), and, so, if u is surjective, it implies w = 0. Using 4.3.3, we get then that a( G1) is dense in G2, and so that a(G1) = G2. If u is injective, a is trivially injective. 4.3.5 Corollary. Let G1 and G2 be two locally compact group8j then, the following a88ertion8 are equivalent: (i) There exi8t8 a bicontinuoU8 i8omorphi8m u : G1 -+ G2 (ii) There exi8t8 an JIJ.-i8omorphi8m from l!ll0 (G2) onto l!lla(Gl)· (iii) There exi8t8 an l!ll-i8omorphi8m from l!ll8 ( G1) onto l!ll8 ( G2). Proof. We have (i) => (ii) because the application f -+ f o u is an l!llisomorphism from l!ll0 (G2) onto l!ll0 (G1), (ii) => (iii) by 3.7.6, and (iii) => (i) by 4.3.2.
4.3.6 Corollary ([163]). Let G be a locally compact group. (i) Let (M, K) be a 8ub co-involutive Hopf-von Neumann algebra of l!ll8 ( G), 8uch that there exi8t8 a Haar weight
r,
r,
Proof. Let K = (M, r, K, 'P ); as K is a symmetric Kac algebra, by 4.2.5 (i), it is isomorphic to K 8 (G(K)). But it is clear, by 4.3.2, that: G(K) = M
n {.\a(s), s
E
G}
Let us put: G'
= {s E G;
Aa(s) EM}
4.3 Eymard's Duality Theorem for Locally Compact Groups
139
Then, G' is a closed subgroup of G, and M is generated by {Aa(s), s E G'}. Moreover, by 4.3.2, >..a IG' is a bicontinuous isomorphism from G' onto G(JK), and by 4.3.5, (M,r,K.) is 18I-isomorphic to 18I8 (G'), and we have (i). With the hypothesis of (ii), let E be the conditional expectation from .C(G) toM obtained by 3.7.9(i). For any sinG, we have:
r(E>..a(s))
= (E®E)rs(>..s(s)) = (E®E)(>..a(s)®>..a(s)) = E>..a(s)®E>..a(s)
So, by 3.6.10, either E>..a(s) = 0, or E>..a(s) is in the intrinsic group of lK, and is therefore unitary, which implies E>..a(s) = >..a(s). So, the subset {s E G; >..a( s) ~ M} is therefore equal to the subset {s E G; E>..a(s) = 0} and is closed. Then G' = {s E G; >..a(s) E M} is an open subgroup of G. So (ii) is proved and (iii) is proved using (ii) and 3.7.9(ii). 4.3. 7 Proposition. Let G be an abelian locally compact group, G the abelian locally compact group of all continuous characters of G (cf. 3.6.12), which will be called the dual group of G. Then: {i) The group G is the spectrum of L 1 (G), and the Gelfand repre8entation of L 1 (G) is given by the Fourier transform:
{ii) There exists a Haar mea8ure dx on
G,
and an isomorphism U from
L 2 (G) to L 2 (G) defined, for all fin L 1 (G) n L 2 (G), by:
called the Fourier-Plancherel transform, such that the mapping x -+ U xU* is an isomorphism from the dual Kac algebra lK8 (G) to Ka(G), such that, for all f in L 1 (G):
U>..a(f)U* =
j.
Proof. By 3.6.12, the set of continuous characters of G is the intrinsic group of Ka(G); as G is abelian, Ka(G) is symmetric, and, so, G' is the spectrum of the abelian Banach algebra L 1 ( G), and the Gelfand transform, taking a coherent definition with 4.2.1, will be given by:
for all fin L 1 (G),
x in G, which is (i).
140
4. Duality Theorems for Kac Algebras and Locally Compact Groups
Using Takesaki's theorem (4.2.3 and 4.2.4), we see that there are a Haar measure dx on G, and an isomorphism U from L 2 (G) to L 2 (G), defined by Uf = j for all fin L 1 (G) n L 2 (G), such that the mapping x-+ UxU* is an isomorphism of the dual Kac algebra lK8 (G) to Ka(G). By 4.2.3(ii), we get U>..a(f)U* = j, for all fin L 1 (G), which ends the proof. 4.3.8 Pontrjagin's Theorem ([121]). Let G an abelian locally compact group, G its dual group, as defined in 4.3.7. Then the group GM is isomorphic to G. Proof. The isomorphism defined in 4.3. 7 (ii) sends the intrinsic group of K 8 ( G) onto the intrinsic group of Ka(G), that is onto GAA. For all fin L 1(G), we have, using 4.3. 7 (ii ): kf(s)(x,s- 1 )ds
= ](x) = kf(s)U>..a(s)U*ds
from which we can deduce that U>..a(s)U* is the function on
G:
x-+ (x, s- 1 )
Using Eymard's theorem (4.3.2), we get that these functions are all the characters on G, and so that the group GAA is isomorphic to the group G.
4.4 The Kac Algebra lKs(G) It is now possible to describe the various objects associated to the Kac algebra lK8 (G) by the general theory.
4.4.1 Proposition. {i) The enveloping C* -algebra C*(Ks( G)) is the algebra Co(G) of continuous functions on G, vanishing at infinity; the canonical representation of C(G). into C*(JK8 (G)) is then the Gelfand transform Qw(s) = (>..a(s)*,w) for all win C(G)., s in G. {ii) The Fourier-Stieltjes algebra B(lK8 ( G)) is the algebra M 1 ( G) of bounded measures on G, and its Fourier-Stieltjes representation is the left regular representation of M 1 (G); an element x of C( G) is positive definite representable (in the sense of 1.3.6} if and only if there exists a {unique) positive bounded measure m on G s·uch that: x
=fa >..a(s)dm(s)
(iii) The enveloping W*-algebra W*(JK 8 (G)) is the dual M 1 (G)* of M 1 (G). This Banach space, which is a W*-algebra, being equal to the bidual ofC0 (G),
4.4 The Kac Algebra K,(a)
141
has then a structure of co-involutive Hopf-von Neumann algebra, given by:
(F(8), m1 0 m2)
= (8, m1 * m2)
(~~:(8),m) = (O,m 0 ) -
(8 E M 1 (G)*, mt. m2 E M 1(G)) (8 E M 1(G)*, mE M 1(G))
where* is the multiplication of M 1 (G), and 0 its involution, and where 0 is defined by (O,m) = (8,m)-, with m(f) = (J jdm)- for all fin C0 (G). (iv) The canonical imbedding (s.x)• from L 1 (G) to B(lK8 (G)) = M 1 (G) is the usual imbedding from L 1 (G) to M 1 (G). Proof. By 4.3.3, the enveloping C* -algebra of C( G). is the algebra of continuous functions on G, vanishing at infinity, and the canonical representation of C(G). is its Gelfand transform. So (i) results from 4.3.3. As B(lKs( G)) is the dual of C*(lK8 ( G)), we deduce, from (i), that B(lK8 (G)) is equal, as a Banach space, to M 1 (G). Let us compute its Fourier-Stieltjes representation K 8 7r*j if m is in M 1 ( G), j,g in L 2 (G), we shall have: (~~: 8 7r*(m),wf,g)
= (1r(wf,g o ~~:),m) =
Ia
=
la(>•a(s),wf,g}dm(s)
And so, we have: Ks7r•(m) =
Ia
(i(wf,g o
~~:)dm by 3.7.5 and (i)
.Aa(s)dm(s)
We can deduce from it that the multiplication and the involution of B(lK8 ( G)) are the usual ones on M 1 (G), which gives (ii), with the help of 1.6.10. Then (iii) is a straightforward application of 3.1.5 (ii)). Let f be in L 1 (G), and m = (s_x).(f). As ~(f) is fa .Aa(s)f(s)ds, and, by 1.6.1 (ii) and 3.7.3, equal to Ks7r•(s_x).(f) which is, by (ii), equal to fa .Aa(s)dm(s), we see that m is the measure f(s)ds, which gives (iv). 4.4.2 Proposition. {i) Every non-degenerate representation p. of C(G). is given by a spectral measure Pf.J on G, with values in 1if.J, as defined, for example in {[105], IV, §17.4), such that, for all w in C(G).: p.(w) =
fa(>. a(s)*,w)dPf.J(s)
{ii} Let p. be a non-degenerate representation of£( G)., Pf.J its associated spectral measure on gf.J, il an element of (Af.J)*; then ~~: 8 p..(il) is the image by the left regular representation of the bounded measure d(Pf.J ( s ), il) on G.
142
4. Duality Theorems for Kac Algebras and Locally Compact Groups
(iii} Let f-1.1, f-1.2 be two non-degenerate representations of C( Gh, P1411 P142 their associated spectral measures. The spectral measure associated to the Kronecker product f-1.1 X f-1.2 is the convolution product of P141 and P142 defined by:
Proof. By 4.4.1 {i) and 1.6.1 {i), there is a representation jl. of C0 (G) on 1£14 such that jl. o g = J.l.i by {[105], IV, §17.4), the representation p. is given by a spectral measure on G with values in 1£14 , which gives {i). We have then:
{Kaf..l.•{il),w}
= {J.I.(w o ~~: 8 ), il} = fa{>·a(s)*,wo~~:)d{P14 (s),n} =
fa {-Xa(s),w}d{P14 (s), il}
and therefore:
Kaf..l.•(n) =fa -Xa(s)d{P14 (s), il} which gives {ii). Let now il1 be in {A141 )., il2 in {A142 h; we have:
{(f-1.1
X
J.1.2)(w), il1 ® il2} = {J.I.t.(il1)J.I.2•(n2),w}
by 1.4.3
= {~~:af..1.2•(il2)~~:af..1.h(il1),w o ~~:a}
= fa{-Xa(s),wo~~:a}d({P142 ,il}*{P1411 il})(s) =
by{ii)
fa{.Xa(s)*,w}d({P142 ,il} * {P141 ,il}){s)
therefore, the measure d{P#Jl X#J2' n1 ® .02} is the convolution product of the measures d{P142 , il2} and d{P141 , il1}; which gives (iii). 4.4.3 Theorem. Let P be a spectral measure on G with values in 'H. Then, there exists a unitary U in C( G) ® C('H) such that, for all w in C( G)., 1J in 'H:
e,
We shall write:
U=
la
-Xa(s)* ® dP(s) .
4.4 The Kac Algebra K.(G)
143
Proof. Let i' be the non-degenerate representation of Co( G) associated to the spectral measure P, i.e. such that, for any fin C0 (G):
~LU) =
fa f( s )dP( s)
Let us put v = i' o {}; then v is a non-degenerate representation of .C( G). such that, for any win .C(G)., we have:
v(w)
=fa (Aa(s)*,w)dP(s)
By 3.1.4, there exists a unitary U, in £(1-l) ® .C( G) which is the generator of v, and is such that, for all~. 71 in 1-l, win .C(G).:
SoU= uU,u satisfies the theorem. 4.4.4 Proposition. Let G be an abelian locally compact group, G its dual group, in the sense of4.3.7. For any min M 1 (G), let us define the Fourier transform ofm by:
(x E G) Then: (i) For any m in M 1 (G), we have:
U
(fa Aa(s)dm(s)) U* = m
(ii) (Bochner's theorem) Every positive definite function on G is the Fourier transform of a unique positive bounded measure on G. (iii) (Stone's theorem) Every unitary representation i' of G is given by a spectral measure Pp on G, with values in 1-lp, such that, for all s in G, we have:
Proof. For any fin L 1 (G) (which is an ideal of M 1 (G)), (i) has been proved in 4.3. 7 (ii). So, using the non-degeneracy of the representations, (i) is proved for any min M 1 (G). As L 1 (G) has a bounded approximate unit, we see, using 4.4.1 (ii), that every positive definite element in L 00 (G) is of the form U(J0 Aa(s)dm(s))U*, with min M 1 (G)+. So (ii) is proved, using (i) and 1.3.11.
144
4. Duality Theorems for Kac Algebras and Locally Compact Groups
Let us consider the non-degenerate representation of £ 1 (G) obtained from J.L by 1.1.4. By 4.3.7(ii), L 1 (G) is isomorphic to .C(G)., and, by 4.4.2(i) and 4.3.7(i), there is a spectral measure PIJ on G, with values in 1{/J, such that, for all fin L 1 (G), we have: J.LU)
=
Ja /(x)dP~'(x)
from which we get (iii).
4. 5 Characterisation of the Representations and Wendel's Theorem Let lK = (M,F,K,cp) be a Kac algebra. Let A be a von Neumann algebra. In this paragraph, f3 will denote a normal, one-to-one morphism from if to A 0 if such that
(/3 0 i)i' = (i 0 i')/3 f3(1M) = 1A ®1M By 3.2.6, for any non-degenerate representation J.L of M., the algebra AIJ and the morphism :YIJ fulfill these conditions. 4.5.1 Proposition. We have, for all x in if+ and t in JR.:
(i)
(i 0 cp)f3(x) = cp(x)1A
(ii)
(i 0 af)/3 = f3af .
Proof. Let x in if+. We have:
(i 0 cp)f3(x) ®1M= (i 0 i 0 cp)(i 0 F)f3(x)
by 3.7.2
= (i 0
i 0 w)(/3 0 i)F( X) = f3((i 0 cp)i'(x))
by hypothesis
= cp(x)f3(1M)
by 3.7.2 by hypothesis
= cp(x)(lA ®1M)
which brings (i). Lett in JR.. We have:
((i 0 af)/3 0 i)i'
= (i 0
af 0 i)(/3 0 i)i'
4.5 Characterisation of the Representations and Wendel's Theorem
= (i ® uf ® i)(i ® F)f3
145
by hypothesis
= (i ® i ® uf)(i ® F)f3 by 2.7.6 (ii) applied to
= (i ® i ® uf)(f3 ® i)i'
K
by hypothesis
= (f3 ® i)( i ® uf)i'
= (f3 ® i)( uf ® i)i'
by 2. 7.6 (ii) applied to
K
From what follows that (i ® uf)f3 ® i and f3uf ® i coincide on F(M); as it is obvious that they coincide also on C ® M, thanks to 2. 7.6 (iv) applied to K, they will coincide on M ® M, which completes the proof. 4.5.2 Proposition. Let '1/J be a faithful, semi-finite, normal weight on A (to simplify, we shall suppose A C C(Ht!J)). Then: (i} for all x in 'Jtv,, y in IJlcp the operator f3(y )(x ® 1) belongs to 1)11/J®cp, and there is an isometry U in H.,p ® H such that:
U(At!J(x) ® Acp(Y))
= At/J®cp(f3(y)(x ® 1))
(ii) U belongs to A® M. (iii} for all z in M, we have:
f3(z)U
= U(1 ® z)
.
Proof. We have: ('1/J ®
= '1/J(x*x)
by 4.5.1 (i)
and so f3(y)(x ® 1) belongs to 'Ytv>®cf>· Let XI. x2 in 1)11/J, YI, Y2 in !Jlcp; polarizing the preceding equality, we find: ('1/J ®
= tfJ(x2xi)
which can also be written as follows:
(At/J®cp(f3(YI)(xl ® 1)) IAt/J®cp(f3(Y2)(x2 ® 1))) = (At!J(XI) ® Acp(YI) IAt~J(X2) ® Acp(Y2)) which completes the proof of (i). Let be in 2t~' TJ in !it'' X in 'Jtv, and y in IJlcp. We have:
e
(1r'(e) ® 11-'(TJ))U(A.,p(x) ® Acp(Y))
= (1r'(e) 0 ?T'(TJ))A.,p®cf>(f3(y)(x 0 = f3(y)(x 0 1)(e 0 TJ) = f3(y)(1r'(e) 0 1)(A.,p(x) 0 77)
1))
by (i)
146
4. Duality Theorems for Kac Algebras and Locally Compact Groups
For all vector (in Ht/J, by density, we shall have:
And, by having
11'1(e)
converging to 1, we get:
(10 i'(ry))U(( 0 A.p(y)) = ~(y)(( 0 TJ)
Let z in A'. We have: (1 0 i 1 (ry))U(z( 0 A.p(y)) = ~(y)(z( 0 TJ) = (z 01)~(y)(( 0 ry)
by hypothesis
= (z 01)(1 0 i 1(ry))U(( 0 A.p(y))
by the same calculation as above So, by density and linearity, we get: (1 0 i 1(ry))U(z 01) = (z 01)(10 i 1(ry))U
By having i'(ry) converging to 1, we get:
U(z 01) = (z 01)U and soU belongs to A 0 C(H). Let X in S)l,p, y in 'Jl.p n 'Jl~X)/tl
T'J
in 21o, such that IITJII
= 1. We have:
U(At!J(x) 0 J~(w.l-t/4,)J A.p(y))
= U(A
A.p((i 0 w,).f'(y))
= At/J®
= At/J®.p(((i 0
by 4.1.3 (iv) by (i)
i 0 w,)(~ 0 i)i'(y))(x 0 1))
= At/J®.p(((i 0 i 0 w,)((i 0 i')~(y))(x 0 1)))
by hypothesis
= At/J®.p(((i 0 i 0 w,)(i 0 i')~(y))(x 0 1 01))
= At/J®
= (1 0 J~(W .l-1/4,)J)At!J®
by 4.1.3(v) by (i)
4.5 Characterisation of the Representations and Wendel's Theorem
By density of Ll- 114 210 in H, and by polarization, we get, for all win
U(10 J5..(w)J)
147
M.:
= (10 J5..(w)J)U
and, by density of 5..(M.) in M, we can conclude that U belongs to .C(Ht!J)0M, which completes the proof of (ii). Let x in SJ4p, y in 'Jlcp· We have:
,B(z)U(At!J(x) 0 Acp(Y))
= ,B(z)At/J®cp(,B(y)(x 01))
by (i)
= At/J®cp(,B( zy )( x 0 1))
= U(At!J(x) 0 Acp(zy)) = U(10 z)(At!J(x) 0 Acp(y))
by (i)
By density, continuity and linearity we can complete the proof of (iii). 4.5.3 Lemma. With the notations of 4.5.2, let X be in 'Jlcp®cp' Y in 'Jlt/J®cp' x in 94p, and y in 'Jlcp· Then: (i) The operator (.8 0 i)(X)(x 0101) belongs to ')4p®cp®cp and we have:
At/J®cp®cp((.B 0 i)(X)(x 0101)) = (U 01)(At!J(x) 0 Acp®cp(X)) (ii) The operator (.8 0 i)i'(y)(Y 0 1) belongs to 'Jlt/J®cp®cp and we have:
At/J®cp®cp((,80i)i'(y )(Y01 ))
= (10u )(10W*)(U 01 )(10u )(At/J®cp(Y)0Acp(Y )) .
Proof. We have:
(t/; 0 (j; 0 (j;)((x* 0101)(.8 0 i)(X* X)(x 0101)) = tj;(x*(i 0 (j; 0 (j;)(.B 0 i)(.B 0 i)(X* X)x) = tj;( x*(( (j; 0 <j;)(X* X)1 )x) by 4.5.1 (i) = tj;(x*x)((j; 0 (j;)(X* X)< +oo which leads to the first part of (i). By polarization, for any X I, x2 in 'Jlcp®cp and XI, X2 in 'JltP, we shall find:
(At/J®cp®cp((.B 0 i)(XI)(x 0101)) I At/J®cp®cp((.B 0 i)(X2)(x 0101))) = (At!J(XI) 0 Acp®cp(XI) I At~J(x2) 0 Acp®cp(X2)) From what we can deduce the existence of an isometry of H tP 0 H 0 H which sends At~J(x) 0 Acp®cp(X) on ~®cp®cp((.B 0 i)(X)(x 0101)). It comes from 4.5.2 (i) that this isometry coincides with U 01 on the elements of the form
148
4. Duality Theorems for Kac Algebras and Locally Compact Groups
Att>(x)0Acp(YI)0Acp(Y2) (where Y!. Y2 are in !Jlcp); therefore, by linearity and continuity it is equal to U 0 1, which completes the proof of (i). We have: (1/J 0 cj; 0 cj;)((Y* 01)(,8 0 i)F(y*y)(Y 01)) = (1/J 0 cj; )(Y*( i 0 i 0 cj;)(,B 0 i)F(y*y )Y) = (1/J 0 cj;)(Y*,B(i 0 cj;)F(y*y)Y) = (1/J 0 cj;)(Y*Y)cj;(y*y)
by 3.7.2
<+oo which gives the first part of the proof of (ii). Using the same technique, through polarization, we get an isometry of Hti> 0H 0 H sending Att>®®
Ati>®((,B 0 i)(F(y1))(x 0 Y2 01)) = Ati>®cf>®cp((i 0 F)(,B(yl))(x 0 Y2 0 1))
by hypothesis
= At/>®cp®cp((i 0
F)(,B(yi)(x 0 1))(x 0 Y2 0 1)) = (1 0 a)(1 0 W*)At/>®cp(,B(yi)(x 0 1)) 0 A.p(Y2)) by 2.4.9 applied to cj; and 3. 7.3 = (1 0 a)(1 0 W*)(U 01)(Att>(x) 0 A.p(YI) 0 A.p(Y2))
by (i)
This isometry does therefore coincide with (1 0 a)(1 0 W*)(U 0 1)(1 0 a) on those vectors of the form Att>(x) 0 A.p(Y2) 0 A.p(YI), which, by linearity, density and continuity, completes the proof of (ii). 4.5.4 Proposition. With the notation.! of 4.5.2, the i8ometry U 8ati8jie8:
(i 0 F)(U)
= (10 a)(U 01)(10 a)(U 01).
Proof. By 2.6.4, we have:
(i 0 c;F)(U)
= (1 0
a)(1 0 W)(10 a)(U 01)(10 a)(10 W*)(10 a)
Let x in !Jltt>, Y!. Y2 in !Jtcp. We have: (10 W)(10 a)(U 01)(10 a)(10 W*)(Att>(x) 0 Acp(YI) 0 A.p(Y2))
= (10 W)(10 a)(U 01)(Att>(x) 0
A.p0.p(F(y1)(y2 01)))
by 2.4.2(i) applied to][{ and 3.7.3
= (1 0
W)(1 0 a)Att>®
(((,B 0 i)(f'(yi)(Y2 01))(x 01 01)) by 4.5.3(i)
4.5 Characterisation of the Representations and Wendel's Theorem
= = = =
149
(1 ® W)(1 ® a)A.p®ip®,:P((/3 ® i)i'(yi)(f3(Y2)(x ® 1) ® 1)) (U ® 1)(1 ® a)(A.p®,;p(f3(Y2)(x ® 1)) ® Aq,(Yl)) by 4.5.3 (ii) (U ® 1)(1 ® a)(U(A.p(x) ® Aq,(y2)) ® Aq,(YI)) by 4.5.2(i) (U ® 1)(1 ® a)(U ® 1)(1 ® a)(A.p(x) ® Aq,(Yl) ® Acp(Y2))
therefore, we have: (1 ® W)(1 ® a)(U ® 1)(1 ® a)(1 ® W*) = (U ® 1)(1 ® a)(U ® 1)(1 ®a) which completes the proof. 4.5.5 Proposition. With the notation8 of 4.5.2, the isometry U is unitary. Then, by 2.6.5 and 4.5.4, it is the generator of a non-degenerate representation of M* which shall be denoted by 1-'· Moreover, we have:
A = A"' and {3 =
7"' .
Proof. Let P the projection UU*. We have: (i ® F)(P) = (U ® 1)(i ® ~)(P ® 1)(U* ® 1) $ UU*®1 =P®1
by 4.5.4
Applying 2.7.3(i) to IK', we get the existence of a projection Q in A such that P=Q®1 Let z in
M. We have: f3(z)(Q ® 1) = {3(z)UU* = U(1 ® z)U* = UU*U(1 ® z)U* = (Q ® 1)U(1 ® z)U* = (Q ® 1){3( z )( Q ® 1)
by 4.5.2(iii)
by the same calculation
Passing to the adjoint operators, we get:
{3(z)(Q ® 1) = (Q ® 1){3(z) Now let x in 'Jl.p, y in 'Jlcp. We have:
U(Q ® 1)(A.p(x) ® Aq,(y)) = U(A.p(Qx) ® Aq,(y)) = A.p®,;p(f3(y)(Qx ® 1)) = A.p®,;p((Q ® 1){3(y)(x ® 1)) = (Q ® 1)A,p®,;p(f3(y)(x ® 1)) = (Q ® 1)U(A,p(x) ® Aq,(y))
by 4.5.2(i) by the result above by 4.5.2(i)
150
4. Duality Theorems for Kac Algebras and Locally Compact Groups
By linearity and density, we get: UP= U(Q®1)
and:
= (Q®1)U = PU = UU*U = U
uu• = P = u•u P = u•u = 1
Therefore U is unitary. By 2.6.5, U is the generator of a non-degenerate representation of M.; let us note it p.. Then, we have, for z in M: {1(z)
= U(1 ® z)U*
by 4.5.2 (iii) by 3.2.2(i)
=.Y,(z) which completes the proof.
r' ;.,,
= (M' r, tt, cp) be a K ac algebra, i = (M' cj;) the dual Kac algebra, A a von Neumann algebra, {1 an injective normal morphism from M to A® M. Then, the following assertions are equivalent: (i) We have:
4.5.6 Theorem. Let ][{
({1 ® i)i' = (i ® r){1
and {1(1)
=1
{ii) There ezists a non-degenerate representation p. from M. to A such that: {1 0). = ~). X p. Proof. That is clear from 3.2.6 and 4.5.5. 4.5. 7 Corollary. Let G be a locally compact group, A a von Neumann algebra, {1 an injective normal morphism from C( G) to A® C( G). Then, the following assertions are equivalent: {i) We have, for all s in G:
(i ® ra)f1(>.a(s)) = f1(>.a(s)) ® >.a(s) {1(1)
=1
{ii) There ezists a unitary representation P.G of G, such that A is the weak closure of P.a(L 1 (G)), and, for all s in G:
{1(>.a(s))
= P.a(s) ® >.a(s).
Proof. It is just an application of 4.5.6 to Ka(G). 4.5.8 Corollary. Let K be a Kac algebra. Then an element u of C(H) belongs to the intrinsic group G(K) if and only if it is the canonical implementation of an automorphism p of M such that (!1 ® i)i' = rp ([18]).
4.5 Characterisation of the Representations and Wendel's Theorem
151
Proof. Let fJ an automorphism of M such that (fJ®i)i' = i'{J. By 4.5.6, there is a unitary u which is the generator of a one-dimensional representation of M. (so, by 2.6.6 (i), u belongs to G(K)), such that fJ = .:Yu· Moreover, by 4.5.2, we shall have: uAcp(Y) = Acp(fJ(y))
Therefore u is the canonical implementation of fJ. Conversely, let u in G(K). By 2.6.6 (i), u is a one-dimensional representation of M., and then .:Yu is an automorphism of M satisfying:
Using the first part of this proof, we see that the canonical implementation of ~u is an element v of G(K), which satisfies ~u = ~v· Then uv* belongs toM' n M, which means, by 2.7.2, that there exist a complex a such that u = av. As u and v are unitaries, we have lal = 1. As F(u) = u ® u, and r(v) = v ® v, we get a= a 2 , so a= 1 and u = v.
4.5.9 Wendel's Theorem ([199]). Let G be a locally compact group; every automorphism of L 00 ( G) which commutes with the right translations is a left translation. Proof. Let fJ be an automorphism of L 00 ( G) commuting with right translations; we have FafJ = (fJ ® i)Fa, from which, applying 4.5.8 to K 8 (G), we get that there is u in G(K8 (G)) such that fJ(f) = ufu* for all f in L 00 ( G). So, by Eymard's theorem (4.3.2), there exists s in G such that fJ(f) = >..a(s)f>..a(s)*, and so fJ is then the left translation by s- 1 .
4.5.10 Theorem ([163]). Let G be a locally compact group. {i) Let ( M, r, K) be a sub co-involutive H opf-von Neumann algebra of lila (G), such that there ezists a H aar weight r.p on ( M, r, K). Then, there ezists a normal subgroup H of G such that M is the subalgebra of the functions in L 00 (G) invariant by H. Then (M,F,K) is isomorphic to H:Ia(G/H). (ii) Let ][{ be a sub Kac algebra of Ka(G); then there ezists a normal compact subgroup K ofG such that][{ is isomorphic to Ka(G/K). (iii} Let Ks(G)p be a reduced Kac algebra of K 8 (G); then, there ezists a normal compact subgroup K of G such that K 8 (G)p is isomorphic to Ka(G/K). Proof. (i) Let][{= (M,r,,,r.p), K = (M,i',K.,rp) the dual Kac algebra. By 4.2.5(ii), K is isomorphic to Ka(G(K)). As (M,F,K) is a sub co-involutive Hopf-von Neumann algebra of (L 00 (G),Fa,Ka), it is easy to see that M is globally invariant under the automorphisms implemented by >..a(s), for all s in G. Then z -+ >..a(s)z>..a(s)* (z E M) is an automorphism fJ 8 of M satisfying:
152
4. Duality Theorems for Kac Algebras and Locally Compact Groups
Ff3s = ({38 ® i)F We have then a continuous morphism from G to the group of automorphisms = ({3 ® i)F, which, by 4.5.9, is isomorphic to the intrinsic group of i. Let H be the kernel of this morphism; we then get a continuous one-to-one morphism u : GI H -+ G(K). By definition of H, M is included in the subalgebra {! E L00 (G); Ad>..a(s)f = f, '
{3 of M satisfying F{3
see that G(K) - u( GI H) is of measure 0. Let now s be in G(K) - u( GI H); as the set su( G I H) is included in the set G(K) - u( GI H), it is of measure 0, and so is u( GI H), which is impossible. So, u is surjective, and G(K) is isomorphic to GIH, and (i) comes then from 4.3.5. H ][{is a sub-Kac algebra of lK0 (G), its dual K is a reduced Kac algebra of K 8 (G), by 3.7.9 (ii). Let r be the reduction .C(G)-+ M, and let K be the kernel of the morphism r o >.a from G to the unitaries of M. Let us consider the set: {x E .C(G); F8 (x) = x ® x} which is compact and equal to {O}U>..a(G), by 3.6.10 and 4.3.2. Then ..Xa(K) is the intersection of this subset with the kernel of r, and so is compact. So, by 4.3.2, K is a compact normal subgroup of G. Using 4.3.4, we see that the application r o ..Xa from G to G(K) is surjective. As r is open, we see that GIK is then isomorphic and homeomorphic to G(K). So (ii) and (iii) are proved using (i) and 3. 7.9 (ii).
4.6 Heisenberg's Pairing Operator 4.6.1 Lemma. Let][{ = ( M, r, ~, cp) be a K ac algebra. Let p. ( resp. v) be a nondegenerate representation of M. {resp. M.) and Up (resp. U,_.) its generator (we thus have Up E Ap ® M and Uv E Av ® M where Ap (resp. Av) is the von Neumann algebra generated by p. (resp. v )); with the notations of 3.2.2, applied to lK and K, we define i'p for all w in M., and 'Yv for all w in M. by:
i'p(..X(w)) =,(..X x p.)(w) = Up(1 ® ..X(w))u; 7 ,_.(.\(w))
Then, we have:
=,(X x ft.)(w) = u,_.(1 ® X(w))u;
4.6 Heisenberg's Pairing Operator
Proof. Let x in
M,
153
yin M; as, by 3.2.2 (i) and by definition, we have:
and:
"Yv(Y)
= Uv(1 ® y)U;
we can deduce:
(i ® .:Y,)(Uv)(i ® ~)(u; ® 1) = (1 ® u,)(~ ® i)(1 ® Uv)(1 ® u;)(i ® ~)(u; ® 1) = (1 ® u,)(~ ® i)((1 ® Uv)(~ ® i)(1 ® u;)(1 ® u;)) = (1 ® u,)( ~ ® i)( i ® 'Yv )(u;)
which completes the proof.
4.6.2 Corollary and Definition. With the above hypothesis, there exists a unitary element of A 11 ®A,, denoted V,, 11 , such that:
It will be said that tations J.t and v.
v,,v
is the Heisenberg's pairing operator of the represen-
Proof. By the definition of .:y,, the unitary (i ®.:Y,)(U11 )(i ®~)(U; ® 1) belongs to A 11 ®A,® M; it results from 4.6.1 that it belongs as well to A 11 ®A, ®M. By 4.1.5, we then get that it belongs to A 11 ® A, ® C, which completes the proof.
4.6.3 Proposition. With the above notations, let 1-'1 and 1-'2 be two nondegenerate representations of M* and v a non-degenerate representation of M•. We have:
Proof. By definition, we have: Vl'l Xl£2,11
® 1 = (i ® i'l't Xl£2 )(Uv )(i ® (i ® ~)( ~ ® i))(U; ® 1 ® 1)
= (i
i)( ( i @ i @ .:y1'1 )( i @ .:y1'2 )( u11)) (i ® i ® ~)(i ® ~ ® i)(u; ® 1 ® 1)
@ ~@
by 3.2.5
154
4. Duality Theorems for Kac Algebras and Locally Compact Groups
Now, we have also:
(i ® i ® i'~J 1 )(i ® i'~J 2 )(Uv) = (i ® i ® i'~J 1 ((Vv,p 2 1)(i ® ~)(U11 ® 1)) by 4.6.2 = (Vp 2 ,v ® 1 ® 1)(~ ® i ® i)(1 ® (i ® i'p1 )(Uv)) = (Vp 2 ,v ® 1 ® 1)(~ ® i ® i)((1 ® Vp 1 ,v ® 1) (1 ® (i ® ~)(U11 ® 1))) by 4.6.2 = (Vp 2 ,v ® 1 ® 1)(~ ® i ® i)(1 ® Vp1 ,v ® 1) (~ ® i ® i)(i ® i ® ~)(1 ® U11 ® 1) From what we can deduce that the operator Vp 1 xp 2 , 11 ® 1 is equal to:
(i ® ~ ® i)((V~J 2 , 11 ® 1)(~ ® i ® i)(1 ® Vp1 , 11 ® 1)(~ ® i ® i) (i ® i ® ~)(1 ® U11 ® 1))(i ® i ® ~)(i ® ~ ® i)(u: ® 1 ® 1) which may be written:
which completes the proof. 4.6.4 Proposition. With the notations of 4.6.2, let V11 ,p be the Heisenberg's pairing operator of the representations v and i' (the background Kac algebra being then K). We have:
Proof By definition, we have: Vv,p ® 1 = (i ® 'Yv)(UIJ)(i ® ~)(u; ® 1) = ((i ®~)(Up® 1)(1 ® 'Yv)(u;n· = (~ ® i)((1 ®Up)(~® i)(i ® 'Yv)(U;))*
= c~ ® i)(Vp,ll ® 1)*
by 4.6.2
which completes the proof. 4.6.5 Proposition. Let i'l' 1'2 be non-degenerate representations of M* and v a non-degenerate representation of M*. {i) Lett in Hom(i't, i'2)i we have:
{ii) ~ o f.ll
4.6 Heisenberg's Pairing Operator
155
~(1)
= 1 and
Let~
be a normal morphism from Ap 1 to AJI 2 such that = Jl2; we have: ( i ® ~) VJ1 1 ,v = Vp2,v
{iii) In particular, with the notations of 1.6.1 (ii}, we have:
Proof. The assertion (i) results immediately of 3.2.3 (i), (ii) results of 3.2.3 (ii) and (ii) implies (iii). 4.6.6 Corollaries. {i} Let u, u1, u2 be elements of G(:OC), i.e. one-dimensional representations of M*. Let v a representation of M*, by definition, the unitary Vu,v belongs to Av and moreover, by 2.6.6 {ii} and 4.6.3, we have:
Therefore the mapping u -+ Vu,v is a unitary representation of the intrinsic group G(:OC) in Av. We have, in particular, V1,v = 1. Let us remark that:
{ii) Let u in G(:OC) and v in G(OC). The unitary Vu,v is then a complex number of modulus one which shall be denoted x( u, v ). By 3.2.2 (i}, we get: x(u,v) = i'u(v)v* = uvu*v* Therefore, we have: uv = x(u,v)vu and by (i), x is a bicharacter on the product G(:OC) x G(OC) which will be called the Heisenberg's bicharacter associated to :OC. {iii} Let .A be the Fourier representation of M*, v a representation of M*. The unitary VA v thus belongs to Av ® M; moreover, by definition, we have: '
VA,v ® 1 = (i ® i'A)(Uv)(i ® c;)(U: ® 1) = (i ® i')(Uv )(i ® c;)(U: ® 1)
= Uv®1
by 3.2.2 (iv) by 1.5.1 {i)
Then, VA,v is equal to the generator Uv ofv. In particular, for v in G(OC), VA,v is equal to v. We see also that VA~ is nothing but the fundamental operator W. Finally, by the results above, .:Oe have:
156
4. Duality Theorems for Kac Algebras and Locally Compact Groups
4.6.1 Examples. Let lK = lKa(G), lK = lK8 (G). To every representation v of C(G)*, we can associate, by 4.4.2(i), a spectral measure Pv on G with values
in 1iv such that for all w in £(G)*, we have:
v(w)
= fa(>..(s)*,w)dPv(s)
It is then easy to check that, for all unitary representations p, of G:
Vl',v =fa p,(s)dPv(s) and in particular
W =fa >..(s)dP5.(s) Now let 'Y be a continuous character on G, i.e., by 3.6.12, 'Yin G(lKa(G)); for in G, fin L 1 (G), we have:
8
Therefore: and:
x( 'Y, >..a( s)) when G is abelian, for
8
in G and 'Yin
= 1( 8)
G, it
can be expressed as follows:
and therefore: which is the classical Heisenberg's commutation relation. 4.6.8 Proposition. Let v be a non-degenerate representation of M*. For all 0 in B(JK), let us put (where 1r denotes the universal representation of M*):
il( 0)
= (i Q9 O)(V?r,V)
Then il is the unique non-degenerate representation of B(JK) on 1iv such that:
il(sA)*
=v
If, using 3.3.4, one identifies M* with A(JK) = (sA)*(M*), then il extends v. The representation il will be called the extention of v to B(JK).
4.6 Heisenberg's Pairing Operator
157
Proof. Let us recall that (W*('K), ~s1rx1r, s;r) is a co-involutive Hopf-von Neumann algebra, the predual of which is B(K) (3.1.5 (ii)). We have:
(i ® ~S1rX1r )(V1r,v) = ( i ® ~)(V1rX1r,v) by 4.6.5 (iii) = (V1r,v ® 1)(1 ® u)(V1r,v ® 1)(1 ® u) by 4.6.3 And using 1.5.1 (i), we see that ii is a multiplicative mapping from B(:K) to Av. Let win M•. We have:
ii(s,x).(w) = (i ® (s,x).(w))(V7r,v) = (i ®w)(i ® s,x)(V1r,v) = (i ®w)(Uv) by 4.6.5(iii) and 4.6.6(iii) = v(w) by the definition of Uv By 3.4.4, we know that A(K) is a bilateral ideal of B(K). The restriction to A(K) of ii being a non-degenerate representation, it follows easily that ii is non-degenerate and involutive. As A(K) is an ideal of B(K), we easily get the unicity of ii: let v 1 be another extension of v, for all (J in B(K) and win A(K), we have:
v1(9)v(w) = v'(fJ * (s,x).(w)) = ii(fJ * (s,x).(w)) = ii(fJ)v(w)
by hypothesis
which completes the proof, because v is non-degenerate. 4.6.9 Remarks. (i) The extension of the Fourier representation 5. to B(:K) is the Fourier-Stieltjes representation K.7r* defined in 1.6.9. (ii) In the group case, the algebra B(Ka(G)) identifies itself with the classical Fourier-Stieltjes algebra B(G) (1.6.3 (iii)). Moreover, with the notations of 4.4.2, it is clear that, for every f in B( G), and any non-degenerate representation v of A( G) (or of C0 ( G)), we get:
ii(f)
=fa f(s)dP (s) 11
(iii) Still in the group case, if K = IK8 (G), M* is identified with L1 (G) and B(IK8 (G)) with the involutive Banach algebra M 1 (G). To each nondegenerate representation v of L 1 (G), one associates a representation ii of M 1 (G). By the unicity of 4.6.8, it is clear that it is just the usual procedure for locally compact groups, as described in 1.1.4. 4.6.10 Proposition. Let 1r {resp. 7r) be the universal representation of M* {resp. M.}, if {resp. }rJ be its eztension to B(K) (reap. B(IK)} as defined in 4.6.8, and if* (reap. *•) the transposed mapping of the latter. We have:
158
4. Duality Theorems for Kac Algebras and Locally Compact Groups
{i) s;r~* =if
{ii) the representations if and ~ are faithful
{iii) s_xif = ~**· Proof. Let (}in B(IK). By definition of~. for every iJ in B(lK), we have:
{s;r~*(iJ),9)
= {B,~(9os;r)) = {0,~(9°os;r)*) =
{iJ, (i ® 9° 0 s;r )(v: *))
{iJ, (i ® 9)(v; *)) ' = {iJ ® 9, v11',1r)
=
'
by 4.6.8 by 1.2.s by 4.6.4
= {if( B), 9)
by 4.6.8
which completes the proof of (i). By transposing the equality in 4.6.8, we find, using (i):
** = s_x(~)* = s_xs;rif as, thanks to 1.6.9 applied to K, ** is faithful, it yields (ii). By using the above equality, we get:
~**
= ~s_x(~)*
= s_xs;r(~)*
by 3.3.3 by (i)
= s_xif which completes the proof.
4. 7 A Tatsuuma Type Theorem for Kac Algebras 4.7.1 Lemma. Let lK be a Kac algebra. Let x in W*(IK); x is a character on B(IK) = (W* (IK))* (i.e. s11'x11'( x) = x ® x) if and only if, for every nondegenerate representations 1-' and v of M*, we have:
s 11 x 11 (x)
= s 11 (x) ® s,.,(x).
Proof. Let w in M*; we have:
s 11 xv(w) = (JJ X v)(w) = (s 11 ® s 11 )(1r X 1r)(w) = (s 11 ® sv)s11'X11'1r(w)
by 1.6.1 (ii) by 1.6.4 (iii) by 1.6.1 (ii)
4.7 A Tatsuuma Type Theorem for Kac Algebras
159
So, by continuity, for all x in W*(:K), we have:
81'xv(x)
= (81' ® 8v)811'x11'(x)
Let us assume that x defines a character on B(K). We have then:
81'xv(x)
= (81' ® 8v)(x ® x) = 8,(x) ® 8v(x)
Conversely, let us assume that this last equality is satisfied for every pair
(JL, v) of non-degenerate representations. In the case of (1r, 1r ), we get: 81rx11'(x) = 81r(x) ® 81r(x) = x ® x which completes the proof.
4.7.2 Theorem. Let K be a Kac algebra. {i) An element x of W*(K) belongs to G(W*(K)) if and only if 8,\(x) and, for any non-degenerate representations JL and v of M., we have:
81'xv(x)
#0
= 8,(x) ® 8v(x)
{ii) The restriction of 8,\ to G(W*(K)) is an isomorphism and a homeomorphism from this group to G(JK). Proof. Let x be a character on B(K) such that 8.\ ( x)
# 0. We have:
f(8,\(x)) = (8,\ ® 8,\)((811'x11'(x)) = (8,\ ® 8,\)(x ® x) = 8,\(x) ® 8,\(x)
by 3.3.3
Thus, by 3.6.10, 8,\(x) belongs to G(K), and is therefore unitary. Also, we find, with the definitions of 3.2.2:
.:Y1r8,\(x) = (8.\x11'(x) = ((8,\ ® i)811'x11'(x) =(i®8,\)(x®x) = x ® 8,\(x)
by 3.2.2 (iii) by 1.6.4(i)
As 711'(1) = 1®1, it is easily deduced from this that xis unitary and therefore belongs to G(W*(K)). As we have 8,\(1) = 1, the converse is immediate and (i) follows. Let Xl and x2 be in G(W*(K)) such that 8,\(xl) = 8,\(x2). It results from the above computation that: x1 ® 8,\(xl) = 711'(8,\(xl)) = .:Y1r(8,\(x2)) = X2 ® 8,\(X2)
by hypothesis
160
4. Duality Theorems for Kac Algebras and Locally Compact Groups
which implies x1 = x2 and the injectivity of the restriction of s_x to G(W*(K)). Let y in G(K). Then, with the definitions of 4.6.2, the Heisenberg's pairing operator Vy,'ll' belongs to W*(K) and satisfies the following:
s'll'x'II'(Vy,'ll')
= Vy,'II'X'II' = Vy,'II'®Vy,'ll'
by 4.6.5 (iii) by 4.6.3
Therefore, Vy,'ll' belongs to G(W*(K)). Finally, we have:
s_x(Vy,'ll')
= Vy,.X =y
by 4.6.5 (iii) by 4.6.6 (iii) and 2.6.6 (i)
which proves that the restriction of s .x to G(W* (K)) is a bijective application from G(W*(K)) to G(K); it is clearly a group homomorphism. The continuity of s .x is trivial. Conversely, as, by 4.6.2, we have:
we get the continuity of the application y-+ Vy,'ll'·
4.7.3 Corollary (Ernest's Theorem [44]). Let G be a locally compact group. The mapping 1ra : G -+ W*(G) implements an isomorphism and a homeomorphism from G onto G(W*(G)).
Proof. By 4.7.2, s_x implements an isomorphism and a homeomorphism from G(W*(G)) onto G(K8 (G)) and, by Eymard's theorem (4.3.2), >.a implements an isomorphism and a homeomorphism from G onto G(K8 (G)). The results come from the formula >.a= s_x1ra. 4.7.4 Corollary (Tatsuuma's Theorem [168]). Let G be a locally compact group. Let x in W*(G). The two following asllertions are equivalent: {i) there exilltll llome s in G such that x = 1ra(s) {ii) for every continuous unitary representation" p., v of G, we have: {a) sl'®11 (x) = s,(x) ® s 11 (x) (b) s_x0 (x) "f 0.
Proof. By 4.7.2 and 4.7.1, the second assertion is equivalent to x belonging to G(W*(G)), the corollary then results from 4.7.3.
Chapter 5 The Category of Kac Algebras
In what follows, K1 = (Mt.n,~~:t.'P1) and K2 = (M2,r2,~~:2,cp2) are two Kac algebras, K1 = (Mt.f1J'-t.
162
5.
The Category of Kac Algebras
As a corollary, similar results occur for isomorphisms of the FourierStieltjes algebras B(K1) and B(K2) ( 5. 6. 6 ), and, in the case of two locally compact groups, we recover Johnson's theorem ([65]) about isomorphisms between the Banach algebras M 1(G1) and M 1(G2) (5.6.9), and Walter's theorem ([194]) about isomorphisms between the Fourier-Stieltjes algebras B(G1) and B(G2). Thus, these four different results, which used different types of proof, are there shown, as M. Walter guessed (see the introduction of [194]), to be actually the same property.
5.1 K ac Algebra M orphisms 5.1.1 Definitions. We shall call K-morphism from ][{1 to ][{2 an IH!-morphism a from W*(JK1) to W*(JK2). By transposition, we get an involutive Banach algebras a* from B(JK2) to B(JK1). The class of the Kac algebras, equipped with these morphisms, thereby becomes a category. It shall be denoted by K. 5.1.2 Theorem. With the above notations and those of 4.6.8, there is a unique normal morphism a from W*(K2) to W*(K1) such that:
or also, and equivalently, such that:
Moreover, find:
a is
a K-morphism from
We shall say that Proof. Let
w2
a is
OC2
to
OC1.
By iterating the process, we
the dual morphism of a.
in M2*· By 4.6.8, we have: ?r1a*(s~ 2 )*(w2)
= (i 0
a*(s~ 2 h(w2))(V'Ir1 ,n-1 )
= (i 0w2)(i 0 s~ 2 )(i 0 a)(V'Ir1 ,n-1 ) As a(1) = 1 and s ~ 2 (1) = 1, the operator ( i 0 s ~ 2 )( i 0 a )(V'/1"1 ,n-1 ) is a unitary of W*(KI) 0 M2, and by 1.5.3, it implies that ?rla*(s>.)* is non-degenerate. Thus, if we put:
S.t Kac Algebra Morphisms
t63
we have: &(1) = 1
In another way, & is defined as to make the following diagram commute:
We shall have: &i2(s_x 2 ).(w2) = &1r2(w2) = ita.(s_x2 ).(w2)
by 4.6.8 by definition of &
Therefore &i2 and ita• coincide over A(K2); as A(K2) is a an ideal of B(K2) and as the restriction of ita• to A(K2) is non-degenerate, we easily get:
By definition of it and i2, this equality can be equivalently written as follows, for all() in B(K2):
which is also equivalent to:
or, thanks to 4.6.4, to:
Let {3 beanormalmorphismfrom W*(K2) to W*(Kt), such that f3i2 =ita•. For all w2 in M2*, we have: {31r2(w2) = f3i2(s_x 2 ).(w2) = ita.(s_x 2 ).(w2) = &i2(s_x2 ).(w2) = &1r2(w2) therefore {3 =&,which yields the unicity of&.
by 4.6.8 by assumption by definition by 4.6.8
164
5. The Category of Kac Algebras
Let &* be the transposed of & and B1 in B('Kl). We have:
*1(&*(81))
= (i ® &*81)(V?r2,71'2)
by 4.6.8
= (i ® 81)(i ® &)(V?r2,71'2)
= (i®B1)(a®i)(V?r1 ,71'1 ) = a(i ® 81)(V1r1 ,71'1 )
= a*1(B1) Therefore, we get:
by 4.6.8
-
-
-ir1&* = a?r1 and, as, by 4.6.10 (ii), *2 is faithful, we obtain that, as a*1, &* is an involutive algebra morphism. By transposing, recalling that &(1) = 1, we get the fact that & is a morphism of Kac algebras. Finally, it is clear, by(*), that a··= a, which completes the proof. 5.1.3 Theorem. The correspondance which associates to any Kac algebra its dual Kac algebra (as defined in 3.1.4}, and to any morphism the dual'Kmorphism (as defined in 5.1.2}, is a duality functor of JC into itself. It shall be denoted by D.
Proof. Let 'K1, 'K2 and 'Ka be three Kac algebras, a be a morphism from lK1 to 'K2 and /3 be a morphism from 'K2 to 'Ka. Let us consider the morphism /3a from 'K1 to lKa. By using 5.1.2 repeatedly, we get:
(/3a ® i)(V1r1 ,71'1 )
=
(/3 ® i)( i ® &)(V1r2,71'2 )
= (i ® &)(/3 ® i)(V1r2 ,71'2 ) = (i ® &,8)(V?ra,71'a) Therefore, we have (/3at = completes the proof.
&,8, which, because of the already known results,
5.1.4 Theorem. Let G1 and G2 be two locally compact groups and m be a continuous morphism from G1 to G2. Then: (i) There exists a unique K-morphism, denoted by 'Ka(m), from 'Ka(G2) to 'Ka(G1) (i.e., here, an ]8[-morphism from M 1(G2)* to M 1(G1)* }, the transposed of which (it is an involutive Banach algebras morphism} is the mapping from M 1(G1) to M 1(G2) that sends every measure of M 1(G1) on its image by m. (ii} There exists a unique 'K-morphism, denoted by 'K,(m), from 'K8 (G1) to 'K8 (G2) (i.e. an ]8[-morphism from W*(G1) to W*(G2)) such that, for all g in G1:
5.1 Kac Algebra Morphisms
165
where 1r1 and 1r2 8tand re8pectively for the univer8al repre8entation8 of G1 and G2. (iii} Let U8 denote ICa (re8p. IC 8 } the full 8ub-category of IC made up of the abelian (re8p. 8ymmetric} Kac algebra8. The mapping which a88ociate8 to a locally compact group G the Kac algebra lKa(G) (re8p. lK8 (G)), and to a continuou8 morphi8m of group8 the morphi8m 1Ka(m) (re8p. K8 (m)) a8 above defined, i8 a duality (re8p. an equivalence) functor between the category of locally compact group8 equipped with the continuou8 morphi8m8 and the category ICa (re8p. IC 8 }; it 8hall be denoted by Ka (re8p. 1K8 ). (iv) We have: K 8 =Do Ka. Proof. The mapping g -+ 1r2(m(g)) is a continuous representation of G1 in W*(G2); thus there exists a normal morphism, denoted by lK8 (m), from W*(G1) to W*(G2) such that, for all 9 in G1:
In particular, we have:
1Ka(m)(l) = 1 by using 1.6.8, we immediately check that lKa(m) is an Ill-morphism; the unicity is trivial, which completes the proof of (ii). The transposed of the dual morphism, i.e. (K8 (mn*, is an involutive Banach algebra morphism from M 1(G1) to M 1(G2) such that, for all p. in M 1(G1), we have:
7r2((Ka(mth(p.)) = 1Ka(m)(7r1(JJ)) =
f
la1
by 5.1.2
1r2(m(9))dp.(g)
From where we immediately get that (JK8 (mn*(JJ) is the image measure m(p. ); starting from this equality, by transposing and dualizing, the unicity in (ii) implies the unicity of this morphism 1Ka(m), which yields (i). Let f3 be a morphism from lK8 (G1) to K8 (G2)· It is clear that f3 will map the intrinsic group of W*(G1) in the intrinsic group of W*(G2). As, by 4.7.3, these groups are isomorphic both algebraically and topologically to G1 and G2 respectively, there is a continuous morphism m from G1 to G2 such that, for every 9 in G1, we have:
1r1(9) = 1r2(m(9)) which is nothing but to say that:
f3 = lKa(m)
166
5. The Category of Kac Algebras
As each symmetric Kac algebra is of the form IK8 (G) (4.2.5(i)), it follows that IK8 is an equivalence between the category of locally compact groups and K'- 8 • The ends of (iii) and (iv) are straightforward. 5.1.5 Remark. With the above notations, and identifying B(Ka(Gi)) to B(Gi), fori= 1,2, it is easily checked that for all fin B(G2), we have: K 8 (m).(J) =
f om .
5.2 IHJ.-Morphisms of Kac Algebras 5.2.1 Lemma. Let K1 and IK2 be two Kac algebras, 6 be an lll-morphism from (MI. H, 11:1) to (M2, F2, 11:2) and J1. be a non-degenerate representation of Mt.. Then J1. o 6* is a non-degenerate representation of M2* on 1-tJJ, the generator of which is equal to ( i ® 6)( UJJ).
Proof. Let w2 in M2•· By definition of the generator UJJ, we have:
As we have 6(1) = 1, the operator (i®6)(UJJ) is a unitary of AJJ®M2; by 1.5.2, it is the generator of the representation J1. o 6., which is then non-degenerate. 5.2.2 Theorem. Let K1 and K2 be two Kac algebras, 6 be an 111-morphism from (M1, F1, 11:1) to (M2, F2, 11:2), 6* be the involutive Banach algebras morphism from M2* to Mh obtained by transposing 6, W*(6.) be the homomorphism from W*(K2) to W*(IK1) obtained by applying the functor W* to 6* (i.e. such that W*(6.)11'2 = 11'16.). Then, we have: (i} W*(6.) is a K-morphism from :i2 to :i1; (ii) the involutive algebra morphism W*(6*)*, from B(K1) to B(K2), obtained by transposing W*(6.), is the unique Banach space morphism which makes the following diagram commute:
B(K1) - - - + B(K2) IC111h
l
M1
6
l
11:271'2•
M2
where 11'i stands for the universal representation of Mi*' and Fov.rier-Stieltjes representation of B(Ki) (i = 1, 2);
ll:i11'i
for the
5.2 H-Morphisms of Kac Algebras
167
{iii) The mapping W*(6.t is the unique morphism from Kt to K2 which makes the following diagram commute:
w•(o.r W*(Kt) - - - . W*(K2) 8 1 X
1
1
8 2 X
Proof. By definition, W*(S.) is a normal morphism from W*(K2) to W*(Kt) such that: W*(8.)1r2 = 1r18* As, by 5.2.1, 1r18* is non-degenerate, we get:
W*(8.){1)
=1
and by transposing the above equality, we find:
and, by hypothesis:
11:21r2* W*(S.).
= 8~~:11rt.
As, by 1.6.9, 11:21r2* is faithful, we get the unicity of W*(S.).; we may show, in the same way, that W*(S.). is an involutive Banach algebras morphism; by transposing, we get that W* (8*) is a Kac algebra morphism, which completes the proof of (i) and (ii). Let w1 in Mt.. We have:
s..\ 2 W*(S.t?rt(wt)
= s..\ 2 W*(S.th(s,x 1 ).(wt) = s..\2 *-2 W*(6.).(s,x 1 ).(w1) = 11:21r2* W*( 8.).( s,x1 ).(wt) = 8~~: 1 7rt.{s,x 1 ).(wi) = s.Xt{wt)
= Ss..\ 1 11-t(wt) Therefore, we have:
s_x 2 W*(S.t = 6s_x 1 Conversely, let a beaK-morphism from K1 to K2 such that:
s·.x2 a= 8s·.xl
by 4.6.8 by 5.1.2 by 4.6.10{iii) by (ii) by 4.6.9{i)
168
5. The Category of Kac Algebras
by transposing, it comes:
a.(s_x)•
= (s_x)•o* = (W*(o.n.(s_x 2 )*
And thus, a* and (W*(o*n* coincide over A(K2); for A(K2) being an ideal of B(K2) and for the restriction of (W*(o.n. being non-degenerate, we get the unicity and therefore (iii).
5.2.3 Definition. With the above notations, we shall say that the K:-morphism
W*(o.r from K:1 to K2 is the extension of the llll-morphi8m 8; it will be denoted by 8. The dual ][{-morphism W*(o.) from K2 to K1 will be called the coexten8ion of 8. The llll-morphism 8 being given, the K-morphisms 8 and SA are respectively characterized by the equalities:
s·~2 8 = os·~1 SA7r2 = 7r10* •
5.2.4 Proposition. Let a be a K-morphi8m from K1 to K:2. The following a88ertion8 are equivalent: (i) The representation of(M1). in M2 defined by s_x 2 a?r1 i8 qua8i-equivalent
>.1.
to a 8ub-repre8entation of the Fourier repre8entation (ii} There exi8t8 an llll-morphi8m 0 from (Mt, Ft, K1) to (M2, r2, K2) 8Uch that a i8 the exten8ion of 8. Then the llll-morphi8m 8 i8 unique. Proof Let us assume (i). It means that there is a normal morphism 8 from M1 to M2 such that 8(1) = 1 and:
which gives:
s·~2 a= os·~1 Thanks to 3.3.3 it can easily be checked that 8 is an .llll-morphism; with 5.2.2 (ii) we then get (ii). Let us assume (ii). By 5.2.2 (iii) and 5.2.3, it implies:
therefore: which is nothing but (i).
5.2 llll-Morphisms of Kac Algebras
169
5.2.5 Proposition. Let a be a K-morphism from ][{1 to ][{2. The following assertions are equivalent: (i) The representation of M2* in M1 defined by K17rha.(s~ 2 ). is quasiequivalent to a sub-representation of the Fourier representation >.2. (ii) There exists an IHI-morphism o from (M2, f2, K2) to (MI. .i\, Kt) such that a is the co-extension of 6. Then, the IHI-morphism 6 is unique. Proof. Let us assume (i). It means that there is a normal morphism 6 from M2 to M1, such that 6(1) = 1, and that we have:
6>.2
= K17rha.(s_\2 ). by 4.6.10 (iii)
= s.\ 1 1i'ta.(s_\2 ).
= s .\1 &1i'2( s ~2 ).
by 5.1.2
= s.\1 a7r2
by 4.6.8
Thanks to 5.2.4, we then see that 6 is an IHI-morphism from (M2, f2, K2) to (MI.ft,Kt) such that a= 8, which yields (ii). Let us assume (ii). By 5.2.3 and 5.2.2 (ii), we have:
and therefore: Ktiha•(s_\ 2 )* = 6K27r2•(s_\ 2 ) = 6>.2
by 4.6.9 (i) applied to K2
which completes the proof. 5.2.6 Proposition. Let Gt and G2 be two locally compact groups, u be a continuous morphism from G1 to G2 and Ka(u) and K 8 (u) be the morphisms defined in 5.1.4 (i) and (ii). Then, we have: (a) The following assertions are equivalent: (i) Ka( u) is an extension. (ii) K 8 (u) is a co-extension. (iii) There exists an IHI-morphism lHla(u) 1Hia(G2) --t IHia(Gt) such that, for all f in L 00 (G2):
lHla(u)(f) = f o u (iv) The image of the left Haar measure on G1 by u is absolutely continuou.s with respect to the left Haar measure on G2. ( v) The morphism u is strict, and has an open range.
170
5. The Category of Kac Algebras
(b) The (i) {ii) {iii)
following assertions are equivalent: lK8 (u) is an extension. Ka(u) is a co-extension. There exists an JII-morphism 1II8 (u) that, for all s in G2:
lii8 (G1)
-t
1IIs(G2) such
(iv) The representation >.a 2 o u of G1 is quasi sub-equivalent to the representation >.a1 • ( v) The morphism u is strict, and has a compact kernel. Proof. It is clear that (a)(i) and (ii) (resp. (b)(i) and (ii)) are equivalent. Let us assume (a)(i). From 5.2.5, it follows that there is an JII-morphism 6 from L=( G2) to L=( Gl) such that 6(!) = f o u for all f in B( G2). This equality can be extended, by norm continuity, to all continuous bounded functions f on G2, and then, by ultraweak continuity, to all f in L=(G2)· So, we have (a)(iii). Let us suppose now (a)(iii) and call '-Pi the Haar weight on L=(Gi) {for i = 1,2). It is immediate that '-Pl oliia(u) is a semi-finite, normal trace on L=( G2); therefore, there exists a positive element g, affiliated to L=( G2) such that '-PlCf o u) = '-P2Cfg) for all fin L=(G2), which implies (a)(iv). Let us suppose now (a)(iv); it is then clear that the application f - t f o u defined from L=(G2) to L=(Gl) is a normal morphism; it is easy to check that it is an !II- morphism, whose extension is Ka ( u ). So we have proved that (a)(i)-a(iv) are equivalent. Let us assume these properties. Coming back to the JII-morphism liia(u), we see that this morphism may be decomposed into a reduction (which, by 4.3.6 (ii), is the restriction L=( G2) - t L=( G'), where G' is an open subgroup of G2), and an JII-isomorphism from L=(G') to a sub co-involutive Hopf-von Neumann algebra of L=(Gl)· By 4.5.10(i), there is a normal subgroup H of G1 such that this sub von Neumann algebra of L=(Gl) is isomorphic to L=(GI/H), and, using 4.3.5, we get that G' is isomorphic to GI/H. It is easy to see that we have just got the canonical decomposition of u, which is then strict and with an open range, which is (a)(v). Let us now assume (a)(v). Let w be in .C(G2)*, f,g in K(GI)· Then, we have (where, for s1 in G1, 81 means its class in GI/Keru):
f Qw o u(sl)f(sl)g(sl)dsl Ja1 = ( w, f >.a 2 (u(sl)*)f(si)g(sl)dsl) la1 = (w, f ds11 >.a (u(t)*)f(t)g(t)dt) 2
JGt/Keru
s1Keru
5.2 l!ll-Morphisms of Kac Algebras
171
(w, JG1/Keru { Aa (u(8t)*)dst1 f(t)g(t)dt) s1Keru = (w, f Aa (82)*h(82)d82) la2
=
2
2
by changing the variable u(8t)
{
= 82 and defining the function h by:
h(u(8t)) = 1
f(t)g(t)dt s1Keru
h(82) = 0 if 82 does not belong to u(Gt) It is then clear that h is continuous on a compact, and is null outside it. So fa 2 AG 2 ( 82) h( 82 )d82 belongs to the definition ideal of the Haa.r weight on £(G2) (3.6.11). By 3.1.3, the representation w-+ (iw o u of £(G2)• is quasiequivalent to a sub-representation of the Fourier representation of K 8 (G2), that is of the Gelfand representation of £(G2) •. So we get (a)(i), by 5.2.5. The equivalence of (b)(i), (b)(iii) and (b)(iv) is a corollary of 5.2.4. Let R = supportE8 (u); from 1.2.7 and 4.5.10(iii), we see that Ks(Gt)R is isomorphic to K8 ( Gtf K), where K is a compact normal subgroup of Gt, and, from 4.3.6(i), we see that E 8 (u)(£(Gt)) is generated by all Aa2 (8), where 8 runs into a closed subgroup of G2. It is then easy to see that K = Ker u, and that G' = 1m u; as nll8 ( Gtf K) is isomorphic to E 8 ( G'), we see, by 4.3.5, that u is strict, which is (v). Let us now assume (b)(v). Let f be in L 1(Gt), g,h continuous functions on G2, with compact supports. Then, we have:
la (fa 2
1
Aa2 (u(8t))f(8t)d8t)9(82)h(82)d82 = =
la (la 1
2
g(u(8! 1 )82)h(82)d82) f(8t)d8t
r f(8t)k(8t)d8
la1
with:
k(8t) =
r
la2
g(u(8! 1 )82)h(82)d82
It is clear that k is continuous with compact support; so k belongs to the definition ideal of the Haa.r weight on L 00 (Gt). So, using 3.1.3, we see that the representation f-+ fa 1 Aa2 (u(8t))j(8t)d8t of L 1 (Gt) is quasi-equivalent to a subrepresentation of the Fourier representation of Ka(G1 ), that is the left regular representation AG1 • So we get (b )(iv).
172
5. The Category of Kac Algebras
5.3 Strict IHI-Morphisms 5.3.1 Theorem. Let OC1 and OC2 be two Kac algebras, u be an llli-morphism from (M1,F1,11:1) to (M2,r2,11:2) and ii the extension of u. The following assertions are equivalent: (i) The morphism ii is the co-extension of an llli-morphism from the Hopf-von Neumann algebra (M2, f2, 1\:2) to (M1, f1, 1\:1); (ii) There exists a von Neumann algebra morphism v from M2 to M1 such that, for every w in M2*' we have:
u
Then, the morphisms v and
u are
equal.
Proof. By definition, the mapping ii: W*(JK1)-+ W*(JK2) satifies:
therefore, we have: and, also:
K111'h(uh(s_x)*
= K111'h(s)q )*u* =..\1u*
by 4.6.9(i)
the theorem is then a direct consequence of 5.2.5.
5.3.2 Definition. Every llli-morphism verifying the conditions of 5.3.1 will be said to be strict. Given a pair of Kac algebras, the extension operation is clearly putting strict llli-morphisms, and OC-morphisms which are both an extension and a co-extension, into a bijective correspondance. Equipped with the class of strict llli-morphisms, the Kac algebras form then a category with a duality, the dual of a strict llli-morphism u being then defined as the lllimorphism u which verify, for all win (M2k
We have already met some strict llli-morphisms, namely the llli-isomorphisms (3.7.6), the reduction of a Kac algebra (3.7.10) and the injection of a Kac sub-algebra into a Kac algebra (3.3.8).
5.3.3 Proposition. Let G1 and G2 be two locally compact groups and m be a continuous morphism from G1 to G2. The following assertions are equivalent:
5.3 Strict llll-Morphisms
I73
{i) The morphism Ka(m) is both an extension and a co-extension. (ii) The morphism K 8 (m) is both an extension and a co-extension. (iii) The morphism m is strict, has an open range and a compact kernel. If they are satisfied, it is possible to define the following two strict lllimorphisms:
lllia(m): L 00 (G2)-+ L 00 (GI) such that lllia(m)f = f om for all f in L 00 ( G2) llli 8 (m): .C(GI)-+ .C(G2) such that llli8 (m).Xa1 (s) = Aa2 (m(s)) for all s in GI . Proof. It is a straightforward consequence of 5.2.6(a) and (b) and 5.3.2. 5.3.4 Theorem. Let ]({I and K2 be two Kac algebras and u be an llli-morphism from (MI. rl. KI) to (M2, r2, 11:2)· The following two assertions are equivalent: (i) The llli-morphism u is a strict llli-morphism. (ii) The subalgebra u(MI) is o'f2 -invariant and the restriction of 1.()2 to u(MI) is semi-finite. Proof. Let us assume (ii). Let Ru be the support of u. The morphism u can be decomposed into three components, u = i o a o r, where i the injection of u(MI) in M2, a is an llli-isomorphism of Min,. onto u(MI) and r is the reduction MI -+ Min,.. We know that Min,. and u(MI) can be equipped with Kac algebra structures and that r, i and a are strict llli-morphisms; therefore, by composition, so is u, which brings (i). Conversely, let us assume that u is strict. With the same definitions as above, let us decompose u into the product j or where j = i o a, i.e. j is the llli-morphism from Min,. to M2 such that j(xR,.) = u(x) for all x in MI. The dual strict llli-morphism r is injective and allows the identification of (Min,.r and r((Min,.ni it is defined, by 5.3.2, by rAin= Air*, where Ain is the Fourier representation of Min,.. We have:
r Aini*
= AI r*j* = AI u* = UA2
by definition of the dual strict llli-morphism u. As r is an injective homomorphism, we get, for every W2 in M2*:
Let WI in
Mh.
Thanks to Kaplansky's theorem, we have:
llu*(wl)ll = sup{l(u*(wl),A2(w2))1, W2 E M2 •• li.A2(w2)11:::; 1} = sup{l(uA2(w2),wi)I, w2 E M2*• IIA2(w2)ll :::; 1}
174
5. The Category of Kac Algebras
= sup{l(rA1Rj.(w2),w1}1, w2 E M2., IIA2(w2)ll ~ 1}
by(*)
= sup{I(A1Rj.(w2), r.(w1)}1, W2 E M2•· IIA2(w2)11 ~ 1} ~ llr.(w1)11 sup{IIA1Rj.(w2)11, w2 E M2., IIA2(w2)ll ~ 1} ~ 11r.(w1)1i
by(**)
The mapping r. bein~ surjective, we can define a continuous linear mapping e from (M1R,. ); to (M2)• such that er* = u•. It is a simple exercise to prove that e is an involutive Banach algebras morphism. By transposing, one gets that the normal linear mapping e* from M2 to (M1Ru satisfies re* = u. By r being injective, we get that e* is mutiplicative and that e*(1) = 1, therefore, e* is an lHf.-morphism. Let us assume that u.(w1) = 0; by the above calculation, it implies:
r
for all w2 in M2• such that IIA2(w2)ll ~ 1, and thus for all w2 in M2•· As j is injective, j. is surjective and then we have, for all win (M1R)*:
which, by density, implies r.(wl) = 0. It ensures that e is injective and therefore that e* is surjective, we then have:
By 3.7.10, we know that r((M1Ru
uf
n is a sub-Kac algebra of MI. therefore
1 -invariant and the restriction
a
5.3.5 Corollary. Let lK1 and lK2 be two Kac algebraJJ and u be a surjective JHf.-morphism from (MI. H, ~~:1) to (M2, F2, 11:2); then u is strict.
5.4 Preliminaries About Jordan Homomorphisms 5.4.1 Definition. Let M1 and M2 be two von Neumann algebras and i be a linear bijective isometry from M1 to M2. For any projection Pin the centre of M2, we define a linear mapping lp: M1-+ M2 by writing, for all x in M1:
lp(x) = i(x)£(1)* P
5.4 Preliminaries About Jordan Homomorphisms
In particular, we shall have £1 (x) sets:
Ph
= f( x )£(1 )*.
175
We define the two following
= {P;
P projection in the centre of M2 such that lp is a homomorphism of algebras from M1 to M2}.
Pa = {P; P projection in the centre of M2 such that lp is an anti-homomorphism of algebras from M1 to M2}. Let us remark that for any P in Ph (resp. Pa), fp is an involutive homomorphism (resp. anti-homomorphism), for all Jordan homomorphisms are involutive, by definition. 5.4.2 Theorem ([76], [149]). With the above notation.s, we have: (i) £(1) i& unitary; (ii) £1 i& a Jordan homomorphi&m from M1 to M2; (iii) There exi&t& a projection R &uch that R belong& to Ph and 1 - R belongs to P a. Proof. By ([76], th. 7), we have (i) and the mapping l from M1 to M2 defined by l(x) = f(1)*f(x), for all x in M1. is a Jordan isomorphism. Using (i), we get (ii). The last assertion then results from ([149], th. 3.3).
5.4.3 Lemma. With the above definition&, let P in Ph (resp. Pa) and Q a central projection of M2 &uch that Q ~ P; then Q belongs to Ph (re&p. Pa)· Proof. It is a trivial consequence of the following equality:
fq(x) =·fp(x)Q
for all x in M1 .
5.4.4 Lemma. With the above definitions, the set Ph (resp. Pa) is a lattice. Proof It is enough to check that, if P and Q belong to Ph, so does P+Q-PQ. Let x,y be in M1. We have:
fP+Q-PQ(x)fP+Q-PQ(Y)
= (fp(x) + fq-Pq(x))(fp(y) + fQ-PQ(Y)) = lp(x)fp(y) + fq-Pq(x)fQ-PQ(Y) = fp(xy)
+ fq-Pq(xy)
because P belongs to Ph, as well as Q- PQ, thanks to 5.4.3. Therefore:
and P
+Q -
PQ belongs to Ph. The proof for P a is identical.
176
5. The Category of Kac Algebras
5.4.5 Lemma. With the above notations, the set Ph (resp. Pa) has a greatest element. Proof. Thanks to 5.4.4, the set Ph, which is bounded, does have an upperbound Sh which belongs to the weak closure of Ph· Let us prove that Sh belongs to Ph Let x, y be in M1. We have:
lsh(xy) = l(xy)l(l)* sh = =
=
lim l(xy)i(l)* p
PE'Ph
lim l(x)l(l)* Pl(y)l(l)* P
PE'Ph
lim l(x)l(l)*l(y)l(l)* P
PE'Ph
= l(x)l(l)*l(y)l(l)* sh = lsh(x)lsh(y) which completes the proof for Ph. The proof for P a is identical. 5.4.6 Definition. With the above definitions, let us put:
Sh
= ma:x.Ph
and
Sa
= ma:x.Pa
Let us remark that 5.4.2 (iii) implies Sh +Sa
~
.
1.
5.5 Isometries of the Preduals of Kac Algebras In what follows, we consider two Kac algebras lK1 = (M1, Ft, 11:1,
l(l)* ·w 0
= (i(l)* ·w)
0
{iii) The mapping l p being ultra weakly continuous, we can consider its tran8posed (lp h : M2* -+ Mh. We have, then, for all projections P in the centre of M2 and for all w in M2*:
(lp)*(w)=T(l(l)*P·w).
5.5 lsometries of the Preduals of Kac Algebras
177
Proof. Let w,w1 in M2*· We have:
(F2(i(l)),w ® w1 }
* w1 } = (1, T(w * w = (1, T(w) * T(w 1 )} = (i(I),w
1 )}
by hypothesis
= (Ft(I),T(w) ®T(w = (I® 1, T(w) ® T(w 1 )}
1 )}
= (I,T(w)}(I,T(w')} = (i(l),w)(i(l),w1} = (i(I) ® i(l),w ® w1 } therefore, by linearity, density and continuity, we have:
r2(i(I))
= i(I) ® i(I)
since, by 5.4.2(i), we have i(l) :/: 0, it completes the proof of (i), using 2.6.6 (i). Let now x be in M2. We have:
(x,i(l)* ·w 0 )
= (xi(l)*,w = (~~:2(xi(I)*)*,w}0 )
= (~~:2(x*)~~:2(i(I)),w)-
= (~~:2(x*)i(l)*,w)= (~~:2(x*),i(I)* · w)-
by (i)
= (x, (i(l)* · w) 0 }
which completes the proof of (ii). Let now y be in M1. We have:
(y,(ip)*(w))
= (ip(y),w) = (i(y)i(l)*P,w) = (i(y),i(l)*P·w} = (y,T(i(l)*P·w))
which completes the proof. 5.5.2 Proposition. With the notations of 5.4, for all x in M1, we have:
(i) (ii} (iii) (iv)
(ish® ish)(Ft(x)) (is,.® is..)(F1(x))
= F2(i1(x))(Sh ® Sh) = F2(i1(x))(Sa ®Sa)
r2(Sh) ~ sh ® sh F2(Sa) ~ Sa® Sa .
178
5. The Category of Kac Algebras
Proof. Let w, w1 in M2•· We have:
((ish ®ish)(rt(x)),w ®w')
= (n(x),(ish).(w) ® (ish).(w')) = (n(x), T(i(1)* Sh · w) ® T(i(1)* Sh · w')) = (x, T(i(1)*Sh · w) * T(i(1)* Sh · w')) = (x, T((i(1)* Sh · w) * (i(1)* Sh · w'))} = (i(x), (i(1)* sh. w) * (i(1)* sh. w'))
by 5.5.1 (iii) by hypothesis
= (r2(i(x)), (i(1)* sh. w) ® (i(1)* sh. w')) = (r2(i(x)), (i(1)* Sh ® i(1)* Sh) · (w ® w')) = (r2(i(x ))(i(1)* Sh ® i(1)* Sh),w ® w')
= (r2(i(x)i(1)*)(Sh ® Sh),w ® w')
by 5.5.1 (i)
= (r2(it(x))(Sh ® Sh),w ®w1 )
by linearity, continuity and density, we get (i). The proof of (ii) is identical. Let Q be the projection in the centre of M2 such that r2(Q) is the central support of Sh ® Sh in the commutant of r2(M2) (Sh ® Sh belongs indeed toM~® M~ which is included in (r2(M2))'). Let us call~ the isomorphism between the two involutive algebras r2(M2)(Sh ® Sh) and r2(M2Q). Let X in Mt; we have:
r2(iq(x))
= r2(i(x)i(1)*Q) = r2(it(x)Q) = r2(it(x))r2(Q) = ~(r2(it(x))(Sh ® Sh)) =~((ish® ish)(n(x)))
by (i). Therefore the mapping r2iq is multiplicative, r2 being injective, iq is multiplicative, hence Q ~ Sh, by 5.4.6. Therefore, we have:
which brings (iii). The proof of (iv) is identical. 5.5.3 Lemma. Let R be a projection 8ati8fying the condition 5.4.2 (iii). Then the operator:
U = ((iR ® i)(Wi) + (it-Rt~:t ® i)(Wt))(i(1) ® 1) i8 a unitary of M2 ® kft, and, for all w in M2* and() in
{U,w ® fJ)
(Mt)•,
we have:
= (.\t(Tw),fJ)
Let w remark that iR and it-RII:l are, both, normal homomorphi8m8 from
Mt to M2.
5.5 Isometries of the Preduals of Kac Algebras
179
Proof. We have:
U* = (l(1)* ® 1)((iR ® i)(W1) + (l1-RII:1 ® i)(Wi)) hence:
U*U = (i(1)* ® 1)((lR ® i)(W1 Wi) + (l1-RII:1 ® i)(WiW1)(l(1) ® 1)) = (i(1)* ® 1)((iR ® i)(1 ® 1) + (l1-RII:1 ® i)(1 ® 1)(£(1) ® 1)) = (i(1)* ® 1)(R ® 1 + (1- R) ® 1)(£(1) ® 1) =1®1 because l(1) is unitary by 5.5.1 (i). The same kind of calculation would yield UU* = 1 ® 1. Since W1 belongs to M 1 ® M11 it is clear that U belongs to M2 ® M1. Now, we have:
(U,w ® 9} = ((iR ® i)(Wi),l(1) · w ® 9} + ((l1-RII:1 ® i)(W1),l(1) · w ® 9} = (Wi, (lR).(l(1) · w) ® 9} + (W1, (l1-R).(l(1) · w) o 11:1 ® 9} = (Wi, (lR).(l(1) · w) ® 9} + (Wi, (l1-R).(l(1) · w) ® 9} by 2.6.1 (i) by 5.5.1 (iii) and 5.4.2 = (Wi, T(R · w) ® 9 + T((1- R) · w) ® 9) = (Wi, T(w) ® 9} = (.\1(T(w)),8}
by 2.6.1(i)
which completes the proof. 5.5.4 Proposition. With the above notations: (i} the mapping T is involutive.
We have also: (ii} (iii} (iv)
11:2£1 = i111:1 11:2(Sh) = sh 11:2(Sa) =Sa.
Proof. We have, with the notations of 5.5.3:
.\1(T(w)) = (w ® i)(U) = (i ®w)(~U) The operator ~U being unitary in M1 ® M2 and the mapping .\1T being multiplicative, it follows from 1.5.1 (i) and 2.6.5 that T is an involutive representation of M2•i as .\1 is involutive and injective, the proof of (i) is completed.
180
5. The Category of Kac Algebras
Let now x be in M1, w in M2•· We have: {K2l1(x),w)
= {K2l1(x*)*,w) = {l1 ( x* ),
by 5.4.2 (ii)
w 0 )-
= {x*,(l1).(w = {x*, T( l(1 )* · w
by 5.5.1 (iii)
= {x*,T((l(1)* ·w) 0 ) ) -
by 5.5.1 (ii)
0 ))0 ))-
= {x*,T(l(1)* ·w)
0 )-
by (i)
= {K1(x),T(l(1)* ·w))
= {K1(x), (l1).(w))
by 5.5.1 (iii)
= {l1K1(x),w)
which completes the proof of (ii). Let Pin 'Ph, x, yin M1. We have:
l" 2(P)(xy)
= l(xy)l(1)* 1£2(P) = l1(xy)K2(P) = "2(.e1"1(xy)P)
by (ii)
= K2(lpK1(xy)) = l£2(lp(K1(Y)K1(x)))
= "2(lpK1(y)lpK1(x)) = l£2lPl£1 (X )K2lPl£1 (y)
by hypothesis
= 1£2(.e1 ( 1£1 (X) )P)K2( .e1 ( 1£1 (y ))P) = l£2.e1 1£1 (X )1£2( P)l£2.e1 "1 (y )K2( P) = l1(x)K2(P)l1(y)K2(P)
by (ii)
= l~t2(P)(x)l"2(P)(y)
therefore "2(P) belongs to 'Ph· So, "2(Sh) belongs to 'Ph and "2(Sh) ~ Sh by definition; since 1£2 is involutive, we get (iii). The proof of (iv) is identical. 5.5.5 Theorem. Let :K1 = (MI.F!,"I.Cf'1) and lK2 = (M2,r2,1£2,Cf'2) be two Kac algebras. Let us assume that there exists a linear, multiplicative, bijective isometry T from the Banach algebra M2* to the Banach algebra Mt.. Then :K1 is isomorphic either to lK2 or K~. More precisely, if .e stands for the transposed mapping ofT, we have: (i) The operator l(1) belongs to the intrinsic group oflK2. (ii) At least one of the two following assertions is true: (a) The mapping l1 : M1 --+ M2 defined for all x in M1 by:
l1(x) = l(x)l(1)*
5.5 Isometries of the Preduals of Kac Algebras
(b)
tSt
i.5 an JEll-isomorphism from IKt to K2 (it implies that it as a homomorphism of von Neumann algebras). The mapping it is an antihomomorphism of von Neumann algebras and the mapping i! : Mt -+ M~ defined, for all x in Mt, by:
is an JEll-isomorphism from Kt to II{~. Proof. By 5.4.6, 5.5.2 {iii) and (iv), 5.5.4 (iii) and (iv) the couple of projections (Sh, Sa) as defined in 5.4.6 satisfies the hypothesis of the lemma 2.6.8 applied to K2. One of the two projections is therefore equal to 1. Let us assume that Sh = 1. Then, the mapping it is involutive and multiplicative by 5.4.2 {ii), normal and bijective by construction; we have:
it{1) = 1 r2it =(it® it)r2 11:2it = itll:t
by 5.4.2 {i) by 5.5.2{i) by 5.5.4 {ii)
So, it is an JEll-isomorphism from IKt to K2. Let us assume that Sa = 1. Then the mapping i! is involutive, multiplicative and linear by 5.4.2 {ii), normal and bijective from Mt to M~ by construction. Finally, it verifies:
ll{1) = 1 r~l! = ctt ® il)n ~~:~i1 = i'tll:t
by 5.4.2{i) by 5.5.2 {ii) and 1.2.10 {i) by 5.5.4{ii) and 1.2.10{i)
So, i! is an JEll-isomorphism from IKt to ~- Thus, {ii) is proved and {i) was proved in 5.5.1 {i). 5.5.6 Theorem. Let Kt = (Mt.Ft,ll:t.C,Ot) and K2 = (M2,r2,~~:2,cp2) be two Kac algebras. Let u be a normal isomorphism from Mt to M2 such that:
u(1) r2u
=1 = (u®u)n
Proof. Let us apply 5.5.5 (ii) to the transposed mapping u* : M 2* -+ Mt.; with u being multiplicative, we are in the first case of 5.5.5 (ii); as u(1) = 1, we get that u is an 111-isomorphism.
t82
5. The Category of Kac Algebras
r,
5.5. 7 Corollary. Let K = (M, K-, cp) be a Kac algebra. If the quadruple (M, K- 0 , cp 0 ) is also a Kac algebra, then we have K- 0 = K- and cp 0 is proportional to cp.
r,
Proof. We just apply 5.5.6 to the identity map of M. 5.5.8 Lemma. With the hypothesis and notations of the second case of theorem 5.5.5 {ii), we have, for all w in M2*:
(lt).(w1) = T(i(1)* · w) . Proof. Let x in Mt. We have:
(x,(i't).(w'))
= (ll(x),w') = (it(x),w) = (x,(it).(w)) = (x,T(i(1)* ·w))
by 5.5.1 (iii), which completes the proof. 5.5.9 Corollary. Let Kt = (Mt,Tt,K-t,CfJt) and K2 = (M2,r2,K-2,CfJ2) be two Kac algebras. Let us assume that there is a linear multiplicative, bijective isometry T, from the Banach algebra M2* on the Banach algebra Mt.. Then i2 is E-isomorphic to either it or to i~. More precisely, if i denotes the transposed mapping ofT, we have: {i} The operator i(1) belongs to the intrinsic group of K2; {ii} There is an E-isomorphism ~ from i2 onto it or to i~ {in the first case it is a von Neumann algebras homomorphism from Mt to M2, in the second case it is an anti-homomorphism} such that, for all w in M2*' we have: ~(i'l(t)(.\2(w))) = .\t(Tw)
where i'l(t) denotes the automorphism of M2 implemented by i(1) (cf. 3.2.2}. Proof. Let us assume that we are in the first situation of 5.5.5 (ii). Then it is E-isomorphism from Kt on K2. The dual E-isomorphism (cf. 5.3.2) it from i2 to it is defined, for all win M2., by:
it(.\2(w))
= .\t((it).(w)) = .\t(T(i(1)* · w))
by 5.5.1 (iii)
Now, we have: it(i'l(t)(A2(w))) = it(.\2(i(1) · w)) = .\t(T(w)) The theorem stands for ~ = it.
by 3.2.2(v) by what is above
5.5 Isometries of the Preduals of Kac Algebras
183
Let us assume now that we are in the second situation of 5.5.5 (ii). Then
.ei is an 1!1-isomorphism from lK1 on K2. The duall!l-isomorphism ii from~ on lK1 will also be, straightforwardly, an 1!1-isomorphism from 5.3.2, it is defined for all w' in (M~)*, by:
lK2 on lKl. By
with the help of lemma 5.5.8, it can as well be written:
ii(A2(w))
= A1(T(.e(1)* · w))
As before, we can deduce from it that the theorem stands with cp
= ii.
5.5.10 Corollary. Let K1 = (M1,Ft,~~:1.
Proof. It is nothing but corollary 5.5.9 applied to the pair (JK2, lKt)· 5.5.11 Wendel's Theorem ([198]). Let G1. G2 be two locally compact groups, T a linear, multiplicative isometry from L 1 (G1) to L 1 ( G2). Then, there exist: {a) a character X on G2 {b) a bicontinuous isomorphism a from G2 to G1 such that, for all f in L 1(G1) and almost all s in G2:
(Tf)(s)
= x(s)f(a(s))
.
Proof. Let us apply 5.5.5 to Ka(Gt) and Ka(G2)i as lK0 (G2) is commutative, there is only one case; the intrinsic group of !Ka(G2) is the set of characters on G2 (3.6.12), and the l!l-isomorphism between !Ka(G2) and Ka(G1) comes from an isomorphism of Gt and G2 (4.3.5).
184
5. The Category of Kac Algebras
5.5.12 Walter's Theorem ((194]). Let G1. G2 be two locally compact group8, T a linear, multiplicative i!Jometry from A( G1) to A( G2 ). Then there exi!Jt: {a) an element s in G1 (b) a bicontinuou!J isomorphism o: from G2 on G1 or G~PP such that, for all tin G2 and fin A(G1): (Tf)(t) = f(s- 1 o:(t)) .
Proof. Let us apply 5.5.10 to lKa(G1) and lKa(G2)i the intrinsic group of lKs(Gl) is the set Pa(s), s E GI} (4.3.2), and the IH!-isomorphism between Ka(G1) and lKa(G2) comes from an isomorphism of G1 and G2 (4.3.5). The result comes then from the fact that Ka(G2)' = lKa(G~PP) (2.2.5).
5. 6 Isometries of Fourier-Stieltjes Algebras Let lK1 and lK2 be two Kac algebras. In what follows, we consider a multiplicative, isometric, linear bijection T from B(lK2) to B(lK1) (as defined in 1.6). Its transposed mapping f which s~nds W*(lK1) into W*(lK2) is then an ultraweakly continuous, isometric, linear bijection. 5.6.1 Lemma. (i) The operator£(1) belong!J to the intrin!Jic group ofW*(lK2)· (ii) The operator s,x 2 (f(l)) belongs to the intrinsic group ofK2. (iii) As £1 is ultraweakly continuous, it is legitimate to con!Jider it!J transposed (f1h : B(lK2) - t B(lK1). Then, with the notations of 5.4, we have, for all(} in B(lK2): (f1h(B) = T(f(l)* ·B) Moreover, if() is positive, so will be T(f(l)* ·B). Proof. The proof of (i) is strictly analogous to 5.5.1 (i), because it results from 5.4.2 that £(1) is invertible; (ii) is just a corollary of (i). The proof of the first part of (iii) is absolutely identical to the one of 5.5.l(iii). Let us assume(} to be positive. Let x be a positive element of W*(lKI). We have
because, by 5.4.2 (ii), £1 is a Jordan isomorphism, which completes the proof. 5.6.2 Lemma. {i) Let][{ be a Kac algebra. We define a !Jet Q by: Q
= {Q E W*(lK);
Q projection and Q =/;1, s11'x11'(Q) :5 Q ® Q}
5.6 lsometries of Fourier-Stieltjes Algebras
185
Then Q has a greatest element and: max Q
= 1- supps_x
(ii} With the constructions and notations of (i} associated to the two Kac algebras IK1 and IK2, we have:
Proof. Because ). '::F 0, it is clear that the projection 1 - supps_x is different from 1. Moreover, we have: (s.x ® i)s11"x11"(1- supps_x)
= s.xx11"(1- supps_x) = ~..Y11"s.x(1- supps_x) = 0
by 1.6.4(i) by 3.2.2(iii)
Therefore s11"X11"(1- supps_x) belongs to Ker(s.x ® i) and:
s11"x11"(1- supps_x):::; (1- supps_x) ® 1 By, 3.3.3, we also have:
s_xs;r(1- supps_x)
= Ks_x(1- supps_x) = 0
Therefore:
s;r(1- supps_x):::; 1- supps_x by s;r being involutive, we get in fact:
s;r(1 - supp s .x) = 1 - supp s .x We then can write down:
s11"X11"(1- supps_x) = s11"x11"s;r(1- supps_x) = ~( s;r ® s;r )s11"x11"(1 - supp s .x) :::; ~(s;r(1- supps_x) ® 1) = 1 ® (1- supps_x) Finally, by using again ( *), it comes:
s11"x11"(1- supps_x):::; (1- supps_x) ® (1- supps_x) therefore 1 - supp s .x belongs to Q.
by 1.6.6 by(*) by(**)
186
5. The Category of Kac Algebras
Let Q be in Q. We have: by 3.2.2 (iv) by 3.2.2 (iii) by 1.6.4 (iii) by hypothesis
Fs>.(Q) = .Y>.s>.(Q) = ~S>.x>.(Q) = ~(s>. ® S>.)s'll"x'II"(Q) ~ ~(S>. ® S>.)(Q ® Q) = S>.(Q) ® S>.(Q) ~
S>.(Q) ® 1
It then results from 2.7.3(ii) that S>.(Q) is either equal to 0 or 1. Let us assume S>.(Q) = 1; it is equivalent to Q ~ supps>,, which implies: Q
+ (1- supps>,) ~ 1
and:
s"II"X"II"(Q) + s7rx7r(1- supps>.) ~ 1 ® 1 and then:
Q ® Q + (1- supps>,) ® (1- supps>.)
~
1®1
And: ((1- Q)®supps>.)(Q®Q + (1- supps>,)®(1- supps>,))((1- Q)®supps>,) ~ (1- Q)®supps>. which leads to: (1- Q) ® supps>, = 0 which is impossible, Q being different from 1, and supps>. different from 0. Therefore we must have S>.(Q) = 0, which is Q ~ 1-supps>. and it completes the proof of (i). Let now Q be in 'h· As £1 is a Jordan isomorphism, i1(Q) is a projector of W*(OC2). Moreover it is not equal to 1, because i1(Q) = 1 would obviously be equivalent to Q = 1. Now, let 6 and 61 be two positive elements of B(OC2). We have: {s7r2X7r2(il(Q)),6 ® 6') = {s71"2x71"2(i(Q)i(1)*),6 ® 6} = {s7r2 x7r2 (i(Q))(i(1)* ® £(1)*), 6 ® 61} = {s7r2 x7r2 (i(Q)),i(1)* · 6 ® £(1)* · 6') = {.i(Q), (£(1)*. 6) * (£(1)*. 61 )} = {Q, T((£(1)* · 6 * (£(1)* · 6'))) = (Q, T(£(1)* · 6) * T(£(1)* · 61 ))
by 5.4.1 by 5.6.1 (i)
by hypothesis
5.6 Isometries of Fourier-Stieltjes Algebras
187
= (s1r2 x1r2 (Q), T(£(1)* ·B)® T(£(1)* · B')) ::; (Q ® Q, T(£(1)* ·B)® T(£(1)* · B'))
by asswnption
= (Q, T(£(1)* · B))(Q, T(£(1)* · B'))
== (Q, (il).(B))(Q, (£1).(8'))
by 5.6.1 (iii)
= (il(Q),B)(il(Q),B')
= (il(Q)®il(Q),B®B') So, we get:
and therefore el(Q) belongs to Q2. Therefore we get i1(Q1) C Q2. As £1 is bijective, we could prove i1 1(Q2) C Ql the same way, which completes the proof of (ii). 5.6.3 Proposition. With the above notations, we have:
(i) (ii)
(iii)
£1(1-supps_x 1 ) = 1-supps_x2 i(Ker s _x 1 ) = Ker s _x 2
T(A(K2)) = A(JK1) .
Proof. Let us apply 5.6.2 to prove (i), considering that £1 preserves the order. The ideal Ker s _x 1 is generated by the projection 1 - supp s _x 1 • Let x in W*(JKI)· As £1 is a Jordan isomorphism (5.4.2), we have: i1(x(1- supps_x1 )
= !C£1(x)i1(1- supps_x1 ) = i1(x)(1- supps_x 2 )
+ £1(1- supps_x1 )i1(x)) by (i)
As £1 is bijective, we get £1 (Ker s _x 1 ) = Ker s _x 2 • And £(1) being unitary and Kers_x 2 a bilateral ideal, it completes the proof of (ii). Let Bin B(JK2); by (ii), T(B) vanishes over Kers_x 2 if and only if B vanishes over i(Ker s _x 1 ) = Ker s _x 2 , which gives the result, thanks to 3.3.4. 5.6.4 Notations. The restriction of T to A(lK2) satisfies the hypothesis of 5.5.10. There is thus an element u of the intrinsic group of OC2 and an llllisomorphism ~from lK2 to lK1 or lKl such that, for all Bin A(JK2), we have:
Let us determine u more accurately; the mapping ( s _x 1 ) ; 1T( s _x 2 ). is an isometric linear bijection from (M2)• to (MI)• which shall be denoted by T.
188
5. The Category of Kac Algebras
Let l : M1 -+ M2 its transposed. By 5.5.9, we get u = l(l ); and by transposing the relation T( s .x 2 ). = (s .x1 ).T which defines T, we get that s.x 2 f = ls.x 1 , by definition off and l. Then, we have:
5.6.5 Lemma. With the above notationll, we have, for all() in B(K2):
Proof. To simplify, we shall put 'Y
=7
8 >. 2
(l( 1)). In 5.6.4, the above relation has
been proved for() in A(K2). Now let win M2•· Let us recall that (s.x 2 )*(w) is the generic element of A(K2) and that A(K2) is a bilateral ideal of B(K2) (3.4.4). By applying 5.6.4, we then find:
or:
and, by using 5.6.4 again:
which, by 4.6.9 (ii), can also be written:
by having .X2(w) converging to 1, we complete the proof. 5.6.6 Theorem. Let K1 and K2 be two Kac algebrall. We allllume that there ezilltll a multiplicative, illometric, linear, bijective mapping T from the FourierStieltjell algebra B(K2) on B(K1). Then, there ezilltll an ]8[-illomorphillm from K2 onto K1 or rrq. More precillely, iff lltandll for the tranllpolled ofT, we have: (i) The operator s.x 2 (f(l)) belongll to the intriruic group ofK2. (ii) There ill an 18[-illomorphillm iP from K2 onto K1 or K} (in the jirllt calle £1 ill a von Neumann algebra homomorphillm from W*(K1) to W*(K2), in the llecond calle it is an anti-homomorphism) such that, for all() in B(K2), we have:
5.6 Isometries of Fourier-Stieltjes Algebras
189
Proof. It is enough to put 5.6.1 (i) and 5.6.5 together.
5.6.7 Corollary. Let :K1 and :K2 be two Kac algebras. Let IJi be a normal isomorphism from W*(:K1) onto W*(:K2) such that:
(i.e. such that IJi respects the canonical coproduct of W*(:Kl) and W*(K2)). Then, there exists an lEn-isomorphism !I from :K2 onto :K1 such that, for all w in M1., we have:
We have also:
Proof. Let us apply 5.6.6 to the transposed mapping .P. = B(:K2) -+ B(:Kl). As IJi is multiplicative, we are in the first case, furthermore, as .P(1) = 1, there is an lEn-isomorphism !I from :K2 onto :K1 such that, for all 8 in B(:K2), we have: Because !JI'i:2
= l'i:l !J, it can also be written:
Therefore, for all win Mh, we have: {1Ji('11"1(w)),8}
= {w,'ll"h'I/J•(8)} = {w,!J'II"2•(8)}
= (w o !1, '~~"2•( 8)} = {'11"2(W 0 !J), 8} which gives the first result. We can see that: s;r2 1Ji'11"1(w)
= s;r2 '11"2(w o !I)= 1i"2(w o !I)= '11"2(w o !I o l'i:2) = '~~"2(w o l'i:l o !J) = !Ji11"1(w o l'i:l)
= !Ji1i"1(w) = !Jis;r1 '~~"l(w) which gives the second result, by the ultraweak density of '11"1(M1.) in W*(:K1).
190
5. The Category of Kac Algebras
5.6.8 Corollary. Let IK1 and IK2 be two Kac-algebra,, tJi a IK-isomorphism from IK1 to IK2 (i.e. an lH!-isomorphism from W*(JK1) to W*(K2)). Then there exists an lH!-isomorphism u from (Mt. Ft.K1) to (M2, F2, 11:2) such that tJi is the extension of u. Therefore, IK-isomorphisms are lH!-isomorphisms.
Proof. It is a particular case of 5.6. 7. 5.6.9 Corollary (Johnson's Theorem [65]). Let G1 and G2 be two locally compact groups. Let T a multiplicative, linear, bijective mapping from M 1( G1) to M 1 (G2). Then there exists: (i) a character x on G2 (ii) a bi-continuous isomorphism a from G2 to G1 such that for all measure p. of M 1(G1) we have:
Proof. By 4.4.1 {ii), the algebra M 1(G1) is the Fourier-Stieltjes algebra associated to the Kac algebra IK8 {G1)· Let us recall that, by 3.6.12, the intrinsic group of 1Ka(G1) is composed of the characters on G1. Therefore, by using 5.6.6, we see that there is a character x' on G1 and an lH!-isomorphism ~ from IK8 {G1) to IK8 {G2) (because IK8 (G2)~ = IK8 {G2)) such that, for all p. in M 1(G1), we have: >.a2 (T P.) = ~(f3x' >.al (p.)) We easily compute that for all p. in M 1(G1) we have:
(**) On the other hand, by 4.3.5, there is a bicontinuous isomorphism a 1 from G1 to G2 such that, for all s in G1:
By integrating, we find, for all p. in M 1(G1):
(***) Going back to ( *), we have:
>.a2 (T P.) = ~( >.al (x' P.)) = >.a2 (a'(x' P.) and therefore:
Tp. = a 1 (~~:1 p.) = (~~:' o a 1 - 1 )(a1 (p.))
We finally reach the result by writing
x = x' o a'- 1 and a
=
a'- 1 .
5.6 Isometries of Fourier-Stieltjes Algebras
191
5.6.10 Corollary (Walter's Theorem [194]). Let G1 and G2 be two locally compact groups. Let T be a multiplicative, isometric, linear, bijective mapping from B(G1) to B(G2)· Then there ezists: {i) an elements in G1 (ii) a bicontinuous isomorphism a: from G2 to G1 or to ct;PP such that, for all t in G2 and f in B(G1), we have: (Tf)(t)
= f(s- 1 a:(t))
.
Proof. By 1.6.3 (iii), up to the Fourier-Stieltjes representations, we have B(Gi) = B(Ka(Gi)) (i = 1,2). Therefore, applying 5.6.6, we get the existence of an element u in G(K8 (G1)) and an E-isomorphism iP from Ea(Gl) to E 0 (G2) or Ea(G2)~ = E0 (G~PP), such that for all fin B(G1), we have:
By 4.3.2, there exists sin G1 such that u in L 00 (Gt) and almost all tin G1:
= Aa1 (s).
Then, we have for all
f
on the other hand, by 4.3.5, it exists a bicontinuous isomorphism a: from G2 to G1 or G~PP such that: iP(f) = f 0 0: (***) Going back to(*), we finally find, for all tin G1 and fin B(G1), that: (Tf)(t) = (f3>.a 1 (s)U))(a:(t))
= f(s- 1a:(t)) which completes the proof.
Chapter 6 Special Cases: Unimodular, Compact, Discrete and Finite-Dimensional Kac Algebras
Let :K = (M,r,,,cp) be a Kac algebra, K = (M,i',k,cj;) the dual Kac algebra. We have seen that the modular operator ..1 = L1c,0 is the RadonNikodym derivative of the weight cp with respect to the weight cp o K ( 9.6. 7). So, it is just a straightforward remark to notice that cp is invariant under K if and only if cj; is a trace. Moreover, the class of Kac algebras whose Haar weight is a trace invariant under K is closed under duality ( 6.1..4). These Kac algebras are called "unimodular" because, for any locally compact group G, the Kac algebra :Ka( G) is unimodular if and only if the group G is unimodular. Unimodular Kac algebras are the objects studied by Kac in 1961 ([66], [70]). We show later another analogy with the group case, namely that if cp a finite weight, then (M,T,K,cp) is a unimodular Kac algebra (6.2.1); it is called "of compact type", because :Ka (G) is of compact type if and only if G is compact. We prove then, after Kac ([67]), that every representation of the involutive Banach algebra M. is the sum of irreducible representations (this leads, for compact groups, to the Peter-Weyl theorem) and that the Fourier representation is the sum of all (equivalent classes) of irreducible representations of M. ( 6.2. 5 ). With the help of Eymard's theorem, this leads to Tannaka's duality theorem for compact groups (6.2.6). If :K is such that the Banach algebra M. has a unit, then :K is a unimodular algebra and K is of compact type (6.9.9). So, such Kac algebras will be called "of discrete type". Moreover ([67]), the von Neumann algebra M is then the sum of finite-dimensional matrix algebras: with di = dim Hi < oo
M = ffiC(Hi) i
and the trace cp is then given by:
cp ( E¥xi)
= ~diTri(xi) I
where
Xi
belongs to C(Hi), and
Tri
is the canonical trace on C(Hi)·
6.1 Unimodular Kac Algebras
193
Moreover, we get, following Ocneanu ((109]), an existence theorem for a Haar trace in this case; let (M, r, ~~:) be a co-involutive Hopf-von Neumann algebra, such that M = $i£(Hi) with di = dim Hi < oo and some Hio equal to C; let p be the one-dimensional projector associated to Hio; if p gives a unity of the Banach algebra M*, and if F(p) satisfies a certain (quite natural) condition involving ~~:, then, there is a Haar trace t.p and ( M, r, ~~:, t.p) is a Kac algebra of discrete type (6.9.5). This result appears, then, to be, in the non-commutative case, the analog of Krel'n's matrix block algebras (6.4.5), and, so leads to Krel'n's duality theorem (6.4.6). More generally (6.5.2), we can associate to each co-involutive Hopfvon Neumann algebra a discrete type Kac algebra (or, by duality, a compact type Kac algebra); in the group case, we recover Bohr compactification of locally compact groups ( 6.5.4 ). We then also get an existence theorem of a Haar state in the compact type case (6.5.8). At last, we get, after Kac ((69]), an easy result ( 6.6.1) which strengthen the analogy with locally compact groups: Kac algebras which are both of compact and discrete type are finite dimensional (and vice versa). Then, following Kac and Paljutkin ((75]), we give an existence theorem for a Haar state on a finitedimensional co-involutive Hopf-von Neumann algebra ( 6. 6.4 ). This last result makes the link ((110]) with the algebraic Hopf algebra theory, as exposed in (1] or (154]. For other specific results about finite-dimensional Kac algebras, we refer to (74], (75], (110], (71], (72], (4].
6.1 Unimodular Kac Algebras 6.1.1 Lemma. Let ( M, r, 11:) be a co-involutive Hopf-von Neumann algebra, t.p be a faithful, semi-finite, normal trace on M (we consider elements of M as operators on Hcp)· For all x in 'Jtp, we shall consider the element Wz of M* defined for all y of M by: Wz(Y)
= t.p(xy) = t.p(yx)
Then: (i) For all XI. x2 in 'Jtp, we have:
Therefore the set {wz, x E 'Jtp} is dense in M*. (ii) For all x in IJtcp, the element Wz belongs to lcp and we have, with the notations of 2.1.6:
194
6. Special Cases
(iii} For all
X
in mcp, we have with the notations of 1.1.1 (ii):
(iv) For all w in
M.,
we have, with the notations of 1.1.1 (ii) and 2.1.6:
(v) If
Wz
= W~e(z•)
and Wz belongs then to Icp n I;. (vi) Let us suppose that M = ffiieJ.C.(Hi), with di = dimHi < oo, and
where the {ejh~j~d; form a basis of Hi, and units.
ej,k
are the associated matrix
Proof. We have, for ally in M:
which gives (i). Hz is in mcp, we have:
Wz(z*)
= cp(z*x) = (Acp(x) I Acp(z))
which immediately yields (ii), by 2.1.6 (i) and (ii). By definition, we have:
which is (iii). We have: llwllcp = sup{l(x*,w}l, X E mcp, cp(x*x) :51} = sup{l(x,w}l,x E ~. cp(x*x) ~ 1} = sup{l(x*,w}l, X E mcp. cp(x*x) :51} = llwllcp
by 2.1.6(i) because
6.1 Unimodular Kac Algebras
195
which gives (iv ). Let us suppose that cp = cp o ~~:; we have: w~(y)
= Wz(~~:(y*)) =
by 1.2.5
cp(x~~:(y*))
= cp(x*~~:(y)) = cp(~~:(x*)y)
by hypothesis
= W~t(z•)(Y)
which gives the first part of (v), the second part being trivial then. For x = EBiXi in M+ (with Xi in .C(Hi)+), we have, with the notations of 1.2.11 (ii):
cp o ~~:(x)
= L~Tr;{Vix~vn = LdiTrix~ = cp(x) i
Moreover, we have then:
= diTri(x;e~,k)
= di L(Xie~,kei Iej) i
which completes the proof, thanks to (v).
r,
6.1.2 Theorem. Let ][{ = (M, K, cp) be a Kac algebra and lK = (M, f, be the dual K ac algebra. The following assertions are equivalent: (i) The Haar weight cp is a trace. {ii) The dual Haar weight~ satisfies:
;;., f;)
Proof. It is a straightforward consequence of 3.6.7 and 4.1.1. 6.1.3 Definition. A Kac algebra ][{ = (M' r, K, cp) will be called unimodular if ~~:-invariant trace.
cp is a
6.1.4 Proposition. A K ac algebra is unimodular if and only if its dual is unimodular.
196
6. Special Cases
Proof. It is an immediate corollary of 6.1.2.
6.1.5 Proposition. Let lK = ( M, r, "'• cp) be a unimodular K ac algebra and = ( M, f, 11:, ~) be the dual K ac algebra. Then, for any x, y in 'Jtp the element w Atp(z),A'P(y) belongs to I,;p and we have:
K.
Proof. By 6.1.1 (ii), Wz and Wy belong to I.p, and by 3.5.2 (i), .X(wz) and .X(wy) to m,;p. By 6.1.4, ~is a trace and therefore .X(wz).X(wy)* belongs to VR,;p. So, by 6.1.1 (ii) applied to ~. W>.(w,.)>.(w 11 )• belongs to I,;p. We have also: W>.(w,.)>.(w 11 )• = WAV>(>.(w,.)),AV>(>.(w11 ))
by 6.1.1 (i)
= Wa(w,.),a(w 11 )
by 3.5.4 (ii)
= wAtp(z),Atp(y)
by 6.1.1 (ii)
Therefore we have: a(wA'P(z),A'P(y))
= a(W>.(w,.)>.(w11 )•) = A,;p(.X(wz).X(wy)*)
= a(wz*W~) = a(wz*W~~:(y)•)
by 6.1.1 (ii) applied to~ by 3.5.4 (ii) by 6.1.1 (v)
and it completes the proof. 6.1.6 Proposition. Let G be a locally compact group. The following assertions are equivalent: (i} The group G is unimodular. (ii} The Haar weight 'Pa ofKa(G) is Ka-invariant. (iii} The Kac algebra lKa(G) is unimodular. (iv) The Kac algebra lK8 (G) is unimodular. (v) The Haar weight cp 8 of lK8 ( G) is a trace. Proof. The equivalence of (i) and (ii) results from the definitions of the Haar weight on Ka(G) and of the unimodularity of G. The equivalence of (ii) and (iii) and of (iv) and (v) respectively are mere applications of the definition 6.1.3. Finally the equivalence of (iii) and (iv) is a corollary of 6.1.4.
6.1. 7 Proposition. Let lK = (M, r, "'• cp) be a K ac algebra. The following assertions are equivalent: (i) The weight cp ill 8trictly 8emi-finite (in the 8en8e of [15]}.
6.2 Compact Type Kac Algebras
197
(ii} There is a sub-Kac algebra of :K which is a trace Kac algebra. (iii) There is a reduced Kac algebra ofJK which is an invariant weight Kac algebra. (iv) The sub-algebra M'P is a sub-Kac algebra of :K. Proof. For every Kac algebra, it is clear from (HWiii) that l'i.(M"') is equal toM"' and from 2.7.6(ii) that F(M"') is included in M"' ® M"'. Moreover, we know, by (15), that r.p is strictly semi-finite if and only if the restriction of r.p to M'P is a semi-finite trace. Therefore, we see, by using 2.7.7, that r.p is strictly semi-finite if and only if M'P is a trace sub-Kac algebra of lK. Thus, (i) implies (iv) which implies (ii). Conversely, let us assume (ii) and denote by CM, K;, cp) the trace sub-Kac algebra of lK. We have, for any X in Mandt in R, ui(x) = ur(x) = X and therefore M is included in M'P, which implies that the restriction of r.p to M'P is semi-finite, so, r.p is strictly semi-finite. The equivalence between (ii) and (iii) immediately results from 6.1.2 and 3.7.9(ii) and 3.7.10.
r,
6.1.8 Corollary. Let G be a locally compact group. The weight r.p 8 on C( G) is strictly semi-finite if and only if there exists an open subgroup of G which is unimodular.
Proof. By 6.1.7, the weight r.p 8 will be strictly semi-finite if and only if there exists a reduced Kac algebra of :K8 (Gr (i.e. of :Ka(G) by 4.1.2) admitting an invariant weight. As :Ka(G) is abelian, it is a trace Kac algebra; then the assumption is equivalent to the existence of a reduced Kac algebra of :Ka(G) being unimodular. By 4.3.6 (ii), it is equivalent to the existence of an open subgroup H of G such that :Ka(H) is unimodular which is, in turn, by 6.1.6 equivalent to H being unimodular.
6.2 Compact Type Kac Algebras
r, "')
6.2.1 Theorem. Let (M, be a co-involutive Hopf von Neumann algebra. Let r.p be a finite faithful normal weight on M such that, for all x, y in M, we have: ( i ® r.p )((1 ® y*)r(x )) = "'( i ® r.p )(F(y*)(1 ® x ))
Then, ( M, r, "'' r.p) is a unimodular K ac algebra. Such a K ac algebra will be called of compact type. We have then: W*C()
C()*W = w(1)r.p r.po = r.p •
=
198
6. Special Cases
= 1 in the above formula,
Proof. Putting y
it comes:
(i ® cp)(r(x)) = cp(x)1 applied to
~~:( x ),
it gives:
cp o ~~:(x)1 = (i ®
cp)F(~~:(x)) =
(i ® cp),(~~: ®
~~:)F(x) =
(cp o 11: ® i)F(x)
and:
cp o ~~:(x)cp(1) = (cp o 11: ® cp)r(x) = cp(x)cp o ~~:(1) = cp(x)cp(1) therefore cp o 11: = cp and cp 0 = cp. Let tin lR, we have:
rar = (i ® af)r rar = (af ® i)r
by 2.5.6 by 2.7.5 (i) because cp 0 II:= cp
therefore, we get: ra~
= (af ® af)r
On the other hand, it results from(*) that (cp®cp)F(x) = cp(x)cp(1) for all x in M, and as F(M) is aj®r.p invariant, by 2.7.6 (iii), we have:
a~r.pr=rar we finally get ra~ = rar' and, r being injective, it implies a~ = af and then, for all t in JR, af = id. Therefore cp is a trace, the axiom (HWiii) obviously holds, and ( M' r, K, cp) is a Kac algebra, it is unimodular because cp is a ~~:-invariant trace. The formula W*C;? = w(1)cp is given by(*); using the involution and the fact that cp = cp 0 , we get C;?*W = w(1 )cp. 6.2.2 Theorem. Let G be a locally compact group. The following allllertionll are equivalent: {i) The group G ill compact. {ii) The Kac algebra Ka(G) ill of compact type. Proof. It is trivial.
r,
6.2.3 Lemma. Let ][{ = (M' K, cp) a compact type K ac algebra llUCh that = 1. Then there exilltll an illometry I from Hr.p to Hr.p ® Hr.p lluch that, for all x, y, z in M, all w in Ir.p, all e, TJ in H, we have:
cp(1) {i) (ii) {iii)
IAr.p(x) = Ar.p®r.p(F(x)) I*(Ar.p(Y) ® Ar.p(z)) = a(wy*Wz) (.A(w)e ITJ) = (Ia(w) ITJ ® Je).
6.2 Compact Type Kac Algebras
199
Proof. As, by 6.2.1 (•), we have (cp®cp)F(x) = cp(x) for all x in M, by density and polarization, we can define a unique isometry I from Hcp to Hcp ® Hcp verifying (i). Then, we get: (I*(Acp(y) ® Acp(z)) I Acp(x)) = (Acp(y) ® Acp(z) I IAcp(x)) = (Acp(Y) ® Acp( z) 1Acp®cp(r( x))) = (cp ® cp)(F(x*)(y ® z)) = (F(x*),wy ®wz) by 6.1.1 = (x*,wy*Wz} by 2.1.6 (ii) = (a(wy*Wz) IAcp(x)) which gives (ii). We have:
(A(w )Acp(Y) I Acp(z)) = (A(w )*, wA
by 3.5.2(i)
= (Ia(w) I Acp(Y) ® iAcp(z))
and (iii) follows, by density of Acp(M) in Hcp. 6.2.4 Proposition. Let lK = (M,r,~~:,cp) be a compact type Kac algebra such that cp(1) = 1. The Hilbert algebra Aq,('Jlq,) is then equal to H; moreover, it
exists a continuous algebra morphism b from H to M. such that, for all w in lcp: b(a(w)) = w. Proof. By 6.2.3(iii), for all win lcp and e in H, we have: IIA(w)ell ~ lla(w)ll 11e11
e
therefore, every vector in H belongs to the achieved Hilbert algebra of a(Icp n ~), i.e. to Aq,('Jlq,). Also, we have, for all w in lcp:
lla(w)ll = llwllcp = sup{l(x*,w)l, x EM, cp(x*x) ~ 1} ~ sup{l(x*,w)l, x EM, llxll ~ 1} because cp(1) = 1 =llwll Thus, it exists a linear mapping b from a(Icp) to lcp such that llbll ~ 1 and b(a(w)) = w for all win lcp. By density, it is possible to extend b to the whole
200
6. Special Cases
of H; moreover b is clearly an involutive algebra morphism from a(IV' n to IV' n I~, which, by continuity, completes the proof. 6.2.5 Theorem. Let K
r;)
= (M' r, K, t.p) be a compact type K ac algebra, such that
t.p(l) = 1. Then we have:
(i) The Banach algebra M* admits wA'P(l) as unit. Therefore we have A(K) = B(K), and s,x is a lf:n-isomorphismfrom W*(:K) to (M,F,K.). (ii) The element >.( t.p) is a one-dimensional projection p of the centre of M, and we have, for all x in M: px = xp = wA'P(l)(x)p
F( X )(p @ 1) = p @ X i'(x)(l ®p) = x ®p Moreover, the unit wA'P(l) is a homomorphism from M to C. (iii) For all w in M., >.(w) is a compact operator on H, and, if w belongs to IV', >.(w) is a Hilbert-Schmidt operator on H. The Fourier representation >. can be decomposed into a direct sum of irreducible finite-dimensional representations {>.diei· Therefore, for all i in I, there exist Hilbert spaces Hi, with di = dim Hi < +oo, such that the algebra M is isomorphic to
EI1iei.C(Hi)· (iv) We have, with the above notations:
where {e),kh$;j,k$;d; is a system of matrix units of .C(Hi)· (v) The Haar weight rj; is equal to Ei diTri, where Tri is the canonical trace on .C(Hi)· It is also equal to the restriction toM of the canonical trace on .C(H), and we have, for all x in M:
(vi) For all i in I and integers 1, m such that 0 ~ 1, m ~ di, we have: d;
~(ild ~;) 6 *ild ~i
L..J
k=l
"'k•"l
"'k•"'m
= 61 mWA 'P (1) 1
where 61,m is the Kronecker symbol and {eLh=l, ... ,d; i& the orthogonal basis of Hi corresponding to the matrix units e~,k·
6.2 Compact Type Kac Algebras
201
Proof. For all win M*, we have: (-\(wA 10 ( 1 )),w) = (-\*(w)Acp(1) I Acp(1)
= (.A(w o K)Acp(1) IAcp(1))
by 3.7.3
= (Acp((w ® i)F(1)) I Acp(1))
by 2.3.5
= w(1)
therefore -\(wA 10 ( 1 )) = 1, and, thanks to .A being injective, we see that wA 10 ( 1 ) is the unit of M*, which gives (i). The relations 'P*'P = r.p and r.p 0 = r.p (6.2.1) imply that .A( r.p) is a projection. As we have W*'P = 'P*W = w(1)r.p for all w in M*, we get that this projection lies in the centre of M. Moreover, for all x in M, we have:
.A(r.p)Acp(x) = Acp((r.p o K ® i)F(x)) = r.p(x)Acp(1) = (Acp(x) I Acp(1))Acp(1)
by 2.3.5
therefore .A( r.p) = p is the one-dimensional projection on CAcp(1 ). Now, for X in M, in H, we have:
e,.,
(xpe
1"1) =
ce IAcp(1))(ry IAcp(1))-(xAcp(1) I Acp(1))
= (pe
lry)w A10 (1)(x)
therefore pxp = wA 10 ( 1 )(x)p, and, asp is central, we get the first formula of (ii), from which it is easy to get that wA10 ( 1) is multiplicative. Then, for all win
M*
we have:
w(xp) = w(px) = wA 10 ( 1 )(x)w(p) Then for all w1.w2 in
M*
and x in
M,
we have:
therefore we get the second formula of (ii). The last one can be proved the same way, therefore we have (ii). As, by 6.2.4, His a complete Hilbert algebra, we know by ([24], 1 §8.5) that H = $ieiPiH, w~ere the family {Pi he I is the set of the minimal projections of the centre of M. We know that PiH, up to a constant c7, is isomorphic to the Hilbert-Schmidt operators algebra on a Hilbert space Hi, that M is isomorphic to $iEJ.C(Hi) and then that cp = Eiei ciTq. By using the standard representation of M on $iei(Hi ®Hi), we get that cp is equal to
202
6. Special Cases
the restriction of the canonical trace on C($i(Hi ®.iii)). The same holds for every standard representation of M, and in particular on H. The representation A is therefore the sum of irreducible representations w-+ A(w)pi, equivalent to representations Ai of M. on spaces Hi such that C(Hi) = Ai(M.)". If {e;};eJ is an orthonormal basis of H, for all win IV', we can compute the Hilbert-Schmidt norm of A(w):
IIA(w)llhs =
L
I(A(w)e; I e~c)l 2
j,kEJ
=L
I(Ia(w) I e~c ® ie;)l 2
by 6.2.3 (iii)
j,lc
llla(w)ll 2 = lla(w)ll 2 = llwll~ < +oo.
=
Therefore, A(w) is a Hilbert-Schmidt operator. By density it implies that for all win M., A(w) is a compact operator on H. And, for all i, PiA(w) is also a compact operator on PiH. Now, the isomorphism which maps (up to ci) PiH onto the Hilbert-Schmidt operators on Hi (i.e. onto Hi® .iii) is actually mapping Pi A(w) on Ai(w) ® 1; thus, the compactness of these operators imply:
which completes the proof of (iii). Let now {ej}j=l,... ,d; be an orthonormal basis of Hi and e~,lc the associated matrix units. Let Yl, Y2 in rot
yf
(w111 o i£ ®w112 )(i'(p))
= (w112 ®w111 o k)(A x A)(cp) = (A.(w112 )A.(w111 o k),cp) = cp(-\(w112 o k)X(w111 )) = cp(X(w11; )* X(w111 ))
= (A"'(X(w111 )) IAtp(X(w11;))) = (A
d;
= L: Ci L:CY~Yi ej 1ej) i
j=l
by 3.2.2 (iv) by 1.4.3 by 3.7.3 by 1.2.5 and 6.1.1 (iii) by 6.1.1 (ii)
6.2 Compact Type Kac Algebras d;
= I: c; i
I: (yfe;
d)(y~·e; 1eL)-
1
j,k=1 d;
I: c; I: CYte;
=
203
i
e;)
d)CY~d 1
1
j,k=1 d;
=
L c; L i
Tr;( e~,kYl)'I'r;( eL,jY~)
j,k=1 d;
=
L c; L
(Tr; ® Tri)(e~,1cYl ® eL,;y~)
j,k=1
i
Thanks to 6.1.1 (i), by linearity and density, this implies: d;
rA(p ) =
L c. L -1
I
i
A( ei· k ) ® eki · :J, ,:J
K
j,k=1
eL,j
So, the operator c; 1 r;;:k= 1 it(e~,k) ® is equal to F(p)(~t(Pi) ®Pi) and, therefore, is a projection; by 1.2.11 (ii), it implies c; = d;, which completes the proof of (iv). For all x in M, we have:
cp(xp) = WAcp(1)( X )cp(p)
by (ii)
= wAcp(1)(x)
by (iv)
which completes the proof of (v). We also have, for 0 :::; I, m :::; d;:
rA(P )(1
JO,.
'DI
i ) el,m
A( i ) i i = d-1"" i L.,;"' ej,k ® ek,jel,m j,k
A( i ) ® eki m = di-1 I: "'elk lc
'
'
by (iv)
204
6. Special Cases
Now, we can write down, thanks to 6.1.1 (vi):
=di 2 L)~it(ei )•®we; k
k,l
= di 2 L:C
k,m
,f(x))
(f?)(F(x)(K(4, 1)* 0 e~,m))
k
= di 1 ((f? 0 = di 1 ((f? 0
(f?)(f(x)F(p)(1® ei,m))
by(***)
(f?)(f(xp)(1® ef,m))
= wAcp(l)(x)di 1 ((f? 0 (f?)(F(p)(1 0
= di 1wAcp(I)(x)(f?((i 0
ei,m))
by (ii)
(f?)(F(p)(10 ei,m)))
= di 1wAcp(l)(
= di 1wAcp(l)(f?(( i 0 (f?)((1 0 P )f( e1,m))) by 6.2.1 and 6.1.4 1 by (ii) = di wAcp(l) (x )(f?(( i 0
by (v) by (v)
which completes the proof. 6.2.6 Theorem. Let lK = ( M, F, "'• 'P) be a compact type K ac algebra .mch that = 1. Then we have: (i) Every non-degenerate representation of M* can be decomposed in a direct sum of finite-dimensional irreducible representations. (ii) Every irreducible representation of M* is finite-dimensional and is equivalent to a component of the Fourier representation >.. So, the Fourier representation is the sum of all the (equivalence classes of} irreducible representations of M*. Let us note Irr this set. (iii) For J.L in Irr, let HJJ be a Hilbert space such that J.L(M*) = .C(HJJ), dJJ = dimHJJ < +oo. The Hilbert space H is then isomorphic to the Hilbert sum (J)JJEirrHJJ 0 RJJ, where the norm on HJJ is multiplied by dw (iv) Let {ej}j=l, ... ,dp be an orthonormal basis of Hw The vectors
'P(1)
{ d1/ 2 A'P(J.L*( ile; .e: )*)} (J.L
E Irr, 0
:5 j, k :5 dJJ) form an orthonormal basis
6.2 Compact Type Kac Algebras
of H and for all
205
ein H, we have: dp
e=
I: dl-' I: (JL(b(e))ej iJEirr
1
e~)Ac,o(JL.(ne~ .e: )*) .
j,k=l
1
Proof. Let JL be a non-degenerate representation of M. on a Hilbert space HI-'. With the notations of 6.2.5, we put:
If i i= j, it is easy to check that Ki is orthogonal to K;. As H = $ieiPiH and b(H) is dense in M., we get that HI-'= $ieiKi. Let now "' be in Ki. It is clear that the subspace JL(M.)TJ contains JL(b(piH))TJ which is finite-dimensional, as dense subspace. Therefore the space K = {JL(M. )TJ}- is finite-dimensional; it is also JL(M.) invariant; therefore K has a subspace K' which is JL(M.) invariant and such that the corresponding representation of M. is irreducible. H we consider a maximal family KOt of two by two orthogonal, finitedimensional and JL( M. )-invariant subspaces of H 1-' such that the corresponding representations of M. are irreducible, we have necessarily HI-' = $OtKOt which completes the proof of (i). Let us assume that JL is irreducible and let us use the above arguments. There exists i in I such that HI-' = Ki. Let .,, TJ' be in HI-'. We have, for all w in lc,o: By continuity, we have, for all
ein H, thanks to 6.2.4:
We then have:
<e IPiAc,o(JL.(nfl,fl' )*))
= (JL(b(Pien., ITJ')
by(*)
= (JL(b(e))TJ ITJ') because"' belongs to Ki = <e I Ac,o(JL.(nfl,fl' )*))
Therefore, we get that:
Ac,o(JL•(Qfl,rt )*) = PiAc,o(JL•(Qfl,fl' )*) E PiH ~Hi® Hi and then A(wJJ.(n.,,.,,)•) (which is equal to 1r(Ac,o(JL•(Qf1,f1' )*)), thanks to 6.1.1 (ii) and 3.5.4 (i)) belongs to .C(Hi)·
206
6. Special Cases
For w in lcp n ~' we have:
(ii'(Acp(JL.(il'l)*)a(w) Ia(w)) = (Acp(JL•(!1'1)*) Ia(w*w 0 ))
= (JL(W*W 0 )711711 ) = IIJL(w )7711 2 2:: 0 0
So, 1i'(Acp(JL•(!1'1)*)) belongs to £(Hi)+, and therefore there exists e in PiH such that ii'(e) = 1i'(Acp(JL•(!1'1)*)) 112 and for any win lcp, we have:
(.\i(w )e Ie)= (.\(w)e Ie)= (a(w) I*'Ce)*e) = (a(w) IAcp(JL•(!1'1)*))
= (JL(w)71171)
As Ai and JL are irreducible, we see that JL is unitarily equivalent to Ai, which completes the proof of (ii). Thanks to (ii) it is possible to put a bijection between Irr and the set I defined in 6.2.5 (i); then (iii) can be deduced from 6.2.5 (iii). For p. E Irr, let e'j,k be the matrix units associated to the basis {ej}j=l, ... ,dl'. We have:
where Trp is the canonical trace on C(Hp)· Therefore, by (iii), an orthonormal basis of H is made of the following elements {d; 1/ 2 Acp(e~,j), p. E Irr, 1 5 j,k 5 dp}. For win lcp, we then have:
(a(w) IAcp(JL.(ile;.e:)*)) = {p.(w),ile;.e:} = (JL(w)ejl e:)
= Trp( e'j,kp.(w )) = d-;; 1 cj;(e~J, k.\(w)) r
through the identification of p. to a component of.\, by (iii) 1 (a(w) IAcp(e~ .)) = d-;; r ,J An orthonormal basis of H is thus made of:
and, using the formula ( *), we get:
which completes the proof of (iv) and of the theorem.
6.2 Compact Type Ka.c Algebras
207
6.2.7 Corollary. Let G be a compact group, equipped with a normalized left Haar measure. We have: (i) The Hilbert space L 2 (G) is a subalgebra of L 1 (G) (ii) The Fourier algebra A( G) has a unit, A( G) is equal to B(G), and S>.a is an lHl-isomorphism from W*( G) to lK8 ( G). (iii) Every representation of G can be decomposed into a direct sum of finite-dimensional irreducible representations. Moreover, every irreducible representation of G is finite-dimensional and is equivalent to a component of the left regular representation >..a. So, the left regular representation >..a is the sum of all (classes of) irreducible representations of G. Let us note this set IrrG. (iv) (Peter- Weyl theorem) For all p. in IrrG, let {ejh~j~dl' be an orthonormal basis of the space H 11 such that p.(L 1 (G)) = C(H11 ). Then if we put, for s in G, 1-'j,k(s) = (p.(s)ej Ien, for every f in L 2 (G), we have: f =
L 11Eirra
dl'
:E (1
j,k=1
f(s)P.j,k(s)ds) Jl.j,k.
a
Proof. Through the use of 6.2.2, it is the translation of 6.2.4, 6.2.5 (i) and 6.2.6 applied to !Ka(G).
6.2.8 Corollary (Tannaka's Theorem (166]). Let G a compact group, Irr G the set of all (classes of) irreducibles representations of G, and, for each v in Irr G, H v a Hilbert space such that v( L 1 (G)) = C( H v); let us now choose x = EBirr axv an element of EBirr a( H v) different from 0. Then, the two following assertions are equivalent: (i) There exists a unique s in G such that, for all v in Irra: Xv
= v(s)
(ii) For any pair p, v in Irr G, and 1r1, ... , 7rn in Irr G, if V is a unitary in H 11 0 Hv, and mk integers such that:
then:
Proof. For any p, v in lrr G, there exist 1r1, ••• , 1rn in Irr G, a unitary U11 ,v in C(H11 0 Hv), and integers m 11 ,v,1r~c such that, for all sinG, we have: ul',v(J.t(s) 0 v(s))u;,v
= ~(l.C(cml' ....... lc) 0
7rk(s ))
208
6. Special Cases
By hypothesis, this implies:
u,..,v(x,.. ®xv)u;,v = ffi(lcccmp,v,r") ®x7r,) k
By 6.2.7, we have £(G)= EBirra£(Hv), and, for any sinG: rs(.\a(s)) = .\a(8) ® .\a(8) = ffi(Jl(8) ® v(8)) p,v
and therefore, for any y = EBvYv in £(G), we have:
and then, thanks to ( *): p,v
=x®x As, by hypothesis, x is not equal to 0, x belongs then to the intrinsic group of 8 E G} by 4.3.2. As .\a = EBirrGV by 6.2. 7 (iii), we get (i). As the implication (i) => (ii) is obvious, the result is proved.
JK.,( G), i.e. to the set {Aa( 8 ),
6.3 Discrete Type Kac Algebras 6.3.1 Definition. Let K = (M,r, ,;,,cp) be a Kac algebra. It shall be said of discrete type when the algebra M* is unital. 6.3.2 Theorem. Let G be a locally compact group. The following assertions are equivalent: {i) The group G is discrete. {ii) The Kac algebra Ka(G) is a discrete type Kac algebra. Proof. It is well known that G is discrete if and only if the algebra £ 1 ( G) has a unit (1.1.3). 6.3.3 Theorem. Let]({ be a Kac algebra. The following assertions are equivalent: (i) The Kac algebra]({ is of discrete type. {ii) The Kac algebra K is of compact type.
6.3 Discrete Type Kac Algebras
209
Proof. Let us assmne (i). Because of M* being unital, so is .A(M.); let e be the unit of .A(M.). We have xe = x for all x in .A(M.); by having x strongly converging to 1, we get 1 = e, and so 1 belongs to .A(M.). And, as Icp n I~ is dense in M. and .A norm-continuous, we get that 1 belongs to the norm closure of .A( Icp n I~) and therefore to the norm closure of 'Jl
6.3.4 Corollary. Let][{ be a discrete type Kac algebra. Then we have: (i) The Kac algebra ][{ is unimodular. (ii) There exist Hilbert spaces Hi, with di = dimHi < +oo, such that the algebra M is isomorphic to $iei.C(Hi)· (iii) There exists p, one-dimensional projection of the centre of M such that, for all x in M, we have:
T(x)(p® 1) = p® X T(x)(1 ®p) =X ®p T(p) = Ldi 1 LK(e~,k) ®
e1,;
j,k
iEI
where e~,k are matrix units of .C(Hi)· (iv) We have:
cp = :L:diTri iEl
where Tri is the canonical trace on .C(Hi), and the unite of M* is defined, for all x in M, by: e(x) = cp(px) The H aar weight cp is also equal to the restriction to M of the canonical trace on .C(H). (v) We have, for all i in I:
where S1,m is the Kronecker symbol and {e1h=l, ... ,d; is the orthogonal basis of Hi corresponding to the matrix units ei.J, k" Proof. The assertion (i) results from 6.2.1 and 6.1.4, (ii) from 6.2.5 (iii), (iii) from 6.2.5(ii) and (iv), (iv) from 6.2.5(v), and finally (v) from 6.2.5(vi).
210
6. Special Cases
6.3.5 Theorem ([109]). Let (M, r, II':) be a co-involutive Hopf-von Neumann algebra such that M = $iei.C.(Hi), with di =dim Hi < co. Then the following assertions are equivalent: {i} There exists a weight cp on M such that (M,F,II':,cp) is a discrete type K ac algebra. (ii) There exists a one-dimensional projection p in the centre of M such that, for all x in M, we have: r(x)(p® 1)
= p®
X
r(x)(1®p)=x®p r(p)
=L
dilL 1\':<e~.k) ® eL,;
iEJ
j,k
where the e~,k are matrix unit of C(Hi)· Moreover, cp is then equal to EieJ dll'q, where 'I'ri is the canonical trace on C.( Hi). The unit of c of M* is a homomorphism and satisfies c( x) = cp( xp) for all x in M. If we use the standard representation of M on H = $ieJ(Hi®Hi), then cp is equal to the restriction to M of the canonical trace on C.( H). Proof. By 6.3.4 (iii), we know that (i) implies (ii). Let us assume (ii), and let us put cp = EieJdiTri· We have then cp(p) = 1. Let us put e:(x) = cp(xp) for all x in M. Then, for all win M., we have:
(x,c*w} = (F(x),c ®w} = (cp ®w)(F(x)(p® 1)) = ( cp ® W)(p ® X)
by assumption
= (x,w} Therefore c*W = w for all win M •. We show that W*c = w the very same way. Let {ejh:S;;:S;d; be the orthonormal basis of Hi associated to e~,k· We have:
cp(eL,;)
= di L<4.;efl ef) = okJdi I
Therefore, using ( * ), we find:
(i ® cp)F(p) =
Ldi 1 Lcp(e1,;)11':(e~,k) = L iEJ
j,k
ll':(e~,;) = 11':(1) = 1
iEl
From ( *) we can also get:
(ne;.et o 1\': ® i)r(p) = di~eL.; (i ® ne;.et)r(p) = dill\':(4,;)
6.3 Discrete Type Kac Algebras
211
And then: r(et,;)
= di(ne}.e~ o ~~: ® i ® i)(i ® r)r(p) = di(nt:~ t:i o ~~: ® i ® i)(r ® i)r(p)
..,,. "
(***)
which yields: (i ® c,o)r(4 3·) = di(nt:~ t:i o ~~: ® i)r((i ® c,o)r(p)) =dint:~ t:i (1)1 , , ,. . '"" = 6;,kdi = c,o(ei,;)1
.,.
,
By linearity, we get (i ® c,o)r(x) = c,o(x)1, for alliin I and all x in £(Hi), and by linearity and normality, for all x in M+. Therefore c,o is left-invariant, and r(IJltp) c IJli®tp· Similarly, using (•), we find, for i1 in I, and 0:5 m,Z :5 di': (1 ®
e~, 1 )r(p) = (1 ® e~,1 ) (~ di L ~~:(e~,q) ® e~,q) 1
= d"f
aei .,
1~
p,q
.,
L...J ~~:( e~,l) ® e~,p p
and then: (1 ® ef:m)* rc4,;) =
di(nej.e~ 0 II:® i ® i)((1 ® 1 ® ef::n)(r ®i)r(p))
=
di(ne}.e~ 0 II:® i ® i)(r ® i)((1 ® e~,l)r(p))
= did"f 1 E
by (••)
by the above
p
This implies: (i ® c,o)((1 ® ef:m)* r(et,;)) = di(ne;.e~
0
II:® i)r(~~:(e~,l))
.,
= di~~:(i ® ne}.et)r(e:n,1)
and, using(***): (i®c,o)((1®efm)*r(4 3·)) = didi'~~:(nt:i' t:i'o~~:®i®Dt:i t:i )(r®i)r(p) (****) '
'
'im>'il
"'"'"
Using once more (*), we get: r(p)(1 ® 4,;) = (Ed"f 1 i'
E~~:ce~,q) ® e~,p) c1 ® 4,;) p,q
212
6. Special Cases
and then:
r(ef:n)(1 ® 4,j)
= r(e::,r)(1 ® ei,j)
= di'(D,;;,
ci'
o K ® i ® i)((F ® i)F(p)(1 ® 1 ® ei;)) by ( ***)
=di'(D,;;,
ci'
o~~:®i®i)(F®i)(F(p)(1®eL 3·))
C.,m,~l
'-m'''
=
di,di 1
'
,
L(De~,ei' 0 K ® i)F(~~:(eL,q)) ® e~,j q
by the above computation This implies:
(i ®
OK®i)F(~~:(eL 3·)) , = didi'(Dc;, ci' o ~~: ® i)F((i ®De~
=di'(De;'
ei'
"m'''
= didi'(Dci, ci' <,m•"l = (i 0
ci
"J'"k
<>m•"l
o ~~:
® i ®De~
ci
"J'"k
)F(p))
by ( **)
)(r ® i)F(p)
by(****)
By linearity, we get, for all Xi' in C(Hi') and Yi in C(Hi):
(i ®
= ~~:((i ®
Let W be the fundamental operator constructed in 2.4.2 (i), thanks to
(A
let ( M' r, K) be a co-involutive H opfabelian and i8omorphic to l 00 (I). The a 8tructure of di8crete group and then 8uch that, for all x in M:
F(x)(be: QSl 1) =be; QSl X F(x)(1 QSl be;)= X QSl be; F(be:)
=L
K(bi) ® bj
iEI
(where Iii 8tand8 for the characteri8tic function of {i} over I).
6.4 Krel'n's Duality Theorem
213
Proof. It is a consequence of 6.3.5, 4.2.5 (ii) and 6.3.2. 6.3.7 Corollary. Let (M, r, "') be a co-involutive Hopf-von Neumann algebra, such that M = tBieiC(Hi) with di = dimHi < +oo, and is symmetric. Then, the following assertions are equivalent: {i) There exists a compact group G such that (M,T,K.) ~ llli8 (G). (ii) There ezists a one-dimensional projection p in the centre of M such that for all z in M, we have:
r
T(z)(p®1) =p®z T(p) =
L di L K.(e~,k) ® 4,; 1
iEI
where the
j,k
4,; are matriz units for C(Hi)·
Proof. It is a consequence of 6.3.5, 4.2.5 (i), 6.3.3 and 6.2.2.
6.4
Kreltn's Duality Theorem
6.4.1 Preliminaries and Notations. Let I be a set, and, for all i in I, let di be inN, Hi be an Hilbert space of dimension di, {e~h~i:~d; an orthonormal basis of Hi, ei.J, k the matrix units associated to these basis. We shall write H = tBieiHi, D the von Neumann algebra tBieiC(Hi), Pi the projection on Hi, which is in the centre of D, A the vector subspace generated in C(H) by the C(Hi), Tq the canonical trace on £(Hi), and r.p the trace :Ei di'I'ri on D. We may, as well (and shall often) consider A as a subspace of the predual D., via the linear injection which sends e3i. k to ne~ e; (that we shall note, to
n;
•
'
J' lc
simplify, k). Then, fo; it, ... , ik being two by two different elements of I, and ()k in C(Hi~c)*, we have, in D., II :Ek Okll = :Ek I!Okll and, Pi· (:Ek Ok) = 0. Using the Hahn-Banach theorem, we see that A is then dense in D •.
6.4.2 Definition. With the notations of 6.4.1, we shall say that A is a Krein algebra if: (i) there is a product *and an involution ° on A, such that A is then an involutive algebra and that there is i 0 in I with di = 1, such that the unit element of C(Hi0 ) (which is isomorphic to C), noted eio, is a unit in A. (ii) for every i,j in I, there exist k1, ... , kn in I such that, for any Yi in .C(Hi) and Yj in .C(H;), the product Yi*Yj belongs to 1 £(Hk,.)· More precisely, there exists mi,j,k,. inN such that didj = EP mi,j,k,.dk,. (so that we
$;=
214
6o Special Cases
may identify .C(Hi) ® .C(H;) with EB;= 1 (Cmi,j,kp ® .C(Hkp))), and a unitary Ui,j in .C(Hi) ® .C(H;) such that, for any Yi in .C(Hi) and Yj in .C(H;) and Zkp in .C(Hkp) such as Yi*Yj = ffipZkp' we have:
ui,j(Yi ® Y;)Ui~j =
n
E9 (lc(cmi,j,kp) ® Zkp)
0
p=l
(iii) For every i in I, there exists i 1 in I such that, for any Yi in .C(Hi), yf belongs to .C(Hi' )o More precisely, we have di = di' (so that we may identify .C(Hi) with .C(Hi' )) and that there exists a unitary Vi in .C(Hi) such that, for any Yi in .C(Hi), we have:
yf = Vi*(yi)tlti where (Yi)t means the element of .C(Hi) whose matrix in the basis {c~} is the transposed matrix of Yi (iv) In the decomposition described in axiom (ii), the space Hio defined in (i) appears if and only if j is equal to the element i 1 defined in (iii); moreover, we have then mi,i' ,io = 1. (v) For alliin I, we have: o
6.4.3 Theorem. With the notationa of 6.4.1, let U8 8uppo8e that there exi8t on D a coproduct Fn and a co-involution "'D' 8uch that (D, Fn, "'D• cp) i8 a di8crete type Kac algebra. Then, A (conaidered a8 a 8Ub8pace of D*) i8 a den8e 8ub-involutive algebra of D*, which i8 a Krein algebra. Proof. By 1.2.11 (i), for all i,j in I, there exist k1, ... , kn in I, mi,j,kp ... , in N, such that didj = L:P mi,j,kpdkp (so that we may identify .C(Hi) ® .C(H;) with 1 (Cmi,i,kp ® .C(Hkp))) and, and a unitary Ui,j in .C(Hi) ® .C(H;) such that, for all x = ffikXk in D, we have: mi,j,k,.
EB;=
Therefore, for any gi in .C(Hi)* and
{li
in .C(H;h, we get:
which gives that A is a subalgebra of D*, which satisfies 6.4.2 (ii).
6.4 Kran's Duality Theorem
215
Using 1.2.11 (ii), we get that A is invariant under the involution of D., and satisfies 6.4.2 (iii). Let e be the unit of D.; as e is a homomorphism, its support p~ is a dimension-one projection in the centre of D; therefore, there exists i 0 in I such that dio = 1, p~ = Pio and e = fleo• where is a unit vector of the one-dimensional space Hio; therefore A satisfies 6.4.2 (i). Moreover, in the decomposition ( *), the index i 0 appears if and only if there exists a dimension-one projection pin £(Hi® H;) such that, for all x in D, we have r( X )p = e( X )p. But then, we have:
eo
But, by 6.3.4, we have:
TD(P~)(Pi ® P;)
=0
if j "# i 1 TD(P~)P(Pi1 ®Pi) = Pi
where the dimension-one projection Pi has been defined in 1.2.11 (ii). So, such a projection p does not exist if j "# i 1, and is equal to Pi (and therefore unique) if j = i 1• Therefore A satifies 6.4.2(iv); as 6.4.2(v) is given by 6.3.4(v), the result is proved. 6.4.4 Theorem [109]. With the notations of 6.4.1, let us suppose that A is a Krean algebra. Then, there exist on D a coproduct rD and a co-involution "D' such that (D,rD,"D•
Proof. Let us consider that A is a subspace of D.; by 6.4.2(ii) we have then, for all x = E&kxk in D, i,j in I, 0 ~ l,m ~ di, 0 ~ r,s ~ d;:
or:
When taking for j the element i 0 defined in 6.4.2 (i), we see that, for any i in I, we have mi,io,i = 1. Therefore, if we put: n·=~m·L· I L.J :J,t<,l j,k
we shall have 1
~ ni ~
oo.
216
6. Special Cases
Let us put U us put:
= tFJi,jUi,j;
it belongs to .C(H 0 H); for x
rD(ffi xk)
= U* ( EB (I.cccn~c) 0 kEI
in D, let
xk)) U
Then, clearly, rD is a normal one-to-one morphism, rD(l) IIFD(x)ll = llxll for all x in D. Moreover, using(*): {FD(ffi Xk), ilf,m 0 nt,s)
= tfJkxk
= 1, and we have
= {(ffi Xk), ill,m*Qt,s)
So, by linearity, we get for all X in D and n, Q'' iln, Q~ in A:
{FD(x), Q 0 Q') = {x, Q*Q') (rD(x), LQn 0
n~)
=
n
(ii)
(x, 2:nn*n~) n
and then:
112: iln*Q~~~ ~ 112: iln ° Q~~~ n
n
l(rD(x),~iln0il~)~ ~ llxiiii~Qn0il~~~ Therefore, by density of A in D*, we see that FD(x) belongs to D 0 D. Moreover, the product * being associative, we get that
(FD 0 i)FD
= (i 0
and so FD is a coproduct over D. By 6.4.2 (iii), we have, for all i in I, 0
( nf,m t or, for all x
= tfJkXk
~
FD)FD 1, m ~ di:
= Vi* . nf,m . Vi
in D: (iii)
Let us now put V = t£Ji V/; it belongs to .C(H), and, for x in D, we have:
KD(x) We have !!KD(x)!!
=
= V(ffi 4)V*
llxl!, and, using(***):
{KD(x)*,njm),
= (Vit(xD*Vih,nfm)= (x,(nfmt) , ,
6.4 Krein's Duality Theorem
and so, by linearity, for all
217
n in A, we have: (iv)
which leads to uno II ::; liD II, and, then, to l(~n(x), D) I::; llxllllilll; by density of A in D*, we get that ~n(x) belongs to D. Then, by transposing (ii) and (iv), we get that (D, Fn, ~D) is a co-involutive Hopf-von Neumann algebra; by the same arguments, we see that A is a sub-involutive algebra of D*; let Qio be the unit of A; just by density of A in D*, we see that it is a unit for all D*, which by definition, satisfies, for all x in D:
(v) We have, then:
Fn(x)(1 ®PiJ = (i ® nio)(F(x))(1 ®Pio) =X ®Pio
(vi)
Fn(x)(Pio ® 1) = (nio ® i)(F(x))(Pio ® 1) = Pio ® x
(vii)
In (i), the one-dimensional space Hio appears if and only if there exists a one-dimensional projection Pi,j in .C(Hi ® Hj) such that, for all x in D, we have:
Pi,jU~j ( ~(1 C.(Cm;,; ,kp) ® Xkp)) ui,j =
ui~j ( ~(lC.(Cm;,j,lcp) ® Xkp)) ui,jPi,j =
XioPi,j
or, thanks to the definition of Fn:
By 6.4.2 (iv ), this happens only if j is equal to the index i 1 defined in 6.4.2 (iii), and, moreover, this projection Pi i' is unique. Let us now consider the proj~ction Fn(Pio )(Pi ® Pj ); we have, thanks to (v), and because Pi® Pj is in the centre of D ® D:
Fn(Pio )(Pi ® Pj )Fn (X) = Fn( XPio )(Pi ® Pj) = nio (X )Fn(Pio )(Pi ® Pj) Fn(x)Fn(Pio)(Pi ® Pj)
= Fn(xpiJ(Pi ® Pj) = nio(x)Fn(PiJ(Pi ® Pj)
So, it implies that Fn(Pio )(Pi ® Pj) = 0 if j is different from i 1 , defined in 6.4.2 (iii), and that, for all i, Fn(Pi 0 )(Pi ®Pi') is a one-dimensional projection in .C(Hi ®Hi').
218
6. Special Cases
On the other hand, by 1.2.11 (ii), we get that the operator: Pi= di 1
L ~~:n(e~,s) ® e!,q q,s
is a projection in C(Hi') ® C(Hi), which satisfies, by 6.1.1 (vi), for any i1, i2 in I:
(PiFn(x),f1\ ;1 ®f1~; 2 ~; 2 ) =di1 1 di2 1 (Pirn(x),w ( ;1 ep ,eq
.. , ... m
K.
eq,p
)
®w ;2
el,m
}
= di36i,ilh"i,i2 ( Fn(x), ~wK.(e~,j) ® we},m) 3
= di 1 h"i,ilh"i,i2
(x, ~ nej.e~ * ne;.e:r.) 3
by 6.4.2(v) By putting x
= 1 in the preceding calculation, we get:
and therefore:
which, by linearity and continuity, leads to:
Therefore we get, by the unicity of the dimension-one projection Pi' i: I
And, as Fn(PiJ(Pj ®Pi)
Fn(PiJ
= 0 if j
is different from i 1 , we have:
= LPi = Ldi 1 I:~~:n(e~, 8 )®e!,q
(viii)
q,s
and, thanks to (vi), (vii), (viii), Pio satisfies the conditions of 6.3.5, which completes the proof. 6.4.5 Corollary. With the notation, of 6.4.1, the following propo8itiom are equivalent: (i) A i8 a Krein algebra.
6.5 Characterisation of Compact Type Kac Algebras
(ii) There exist on D a coproduct
rv
219
and a co-involution "'D' such that
(D, Fv, "'D• cp) is a Kac algebra of discrete type. Then, A (considered as a subspace of D.) is a dense sub-involutive algebra of D •. Proof. We have proved in 6.4.3 that (ii) implies (i), and in 6.4.4 that (i) implies (ii).
6.4.6 Corollary (Krem's Theorem [83]). With the notations of 6.4.1, the following propositions are equivalent: (i) A is an abelian Krein algebra. (ii) The set of the characters on A which are continuous with respect to the norm of D., is, for the weak topology of D, a compact group G. The set I may then be identified to the set of (classes of) irreducible representations of G. For all v in I, let H11 be the finite-dimensional Hilbert space such that C(H11 ) is the algebra generated by v(G); then A may be identified to the algebra of functions s-+ (v(s !11), for all v in I, 77 in H 11 •
)e
e,
Proof. Using 6.4.5, 4.2.4, 6.3.3 and 6.2.2, we see that (i) is equivalent to D being isomorphic to some£( G), with G compact; then, the involutive algebra D* is isomorphic to A( G), and, by 4.3.3, G is isomorphic and homeomorphic to the spectrum of D.; as >.a = E9IrrGV by 6.2.7(iii), we see that the set I is equal to Irr G, and, for all v in I, H11 is the finite-dimensional Hilbert space such that C(H11 ) is the algebra generated by v(G). Moreover, via the isomorphism between D* and A( G), A may be considered as a subspace of A( G), precisely the space generated by the functions s -+ ( v( s )ef Iej), for all v in IrrG, {en being a basis for H 11 , which completes the proof.
6.5 Characterisation of Compact Type Kac Algebras 6.5.1 Notations. Let (M, r, ~t) be a co-involutive Hopf-von Neumann algebra. Let J be the set of (equivalence classes of) finite-dimensional representations of M., with a unitary generator in the sense of 1.5.2. Let v1 be in J and V2 C VI. then, by 1.5.4 (ii ), V2 belongs to J. Let J' the subset of J formed by irreducible representations. As the trivial representation belongs to J', this set is not empty. Let v be in J, the representation il defined, for any w in M., by il(w) = v(w o ~t)t (where t stands for the transposition), belongs to J, by 1.5.9. It is also clear that, if vis in J', so is il. Let us consider J' as equivalent classes, and let us pick up, in each class, a representation v which operates on a Hilbert space H11 such that C(H11 ) is the von Neumann algebra generated by v. Let us write I for the set of such v's. We shall denote by p the representation E911eJV.
220
6. Special Cases
By 1.5.4, p has a unitary generator and the elements of I being two by two disjoint, the von Neumann algebra D generated by pis ffiveiC(H,_.). We shall then use all the notations of 6.4.1. Each element(},_. of C(Hv )*can be isometrically identified with an element of D* which shall still be denoted by 8,_.. As we have, through this identification, for all win M.: (p(w),8,_.)
= (v(w),8,_.)
we get, in M, for all(},_. in C(H,_,).:
Moreover, let us recall that, for all v in I, as vis non-degenerate, the mappings v* are one-to-one. 6.5.2 Theorem. With the notations of 6.5.1, we have: {i) the algebra D can be equipped with a coproduct rd and a co-involution "'d such that, for all w in M*:
rd(p(w)) = r;(p X p)(w) "'d(p(w)) = p(w o K) and (D, rd. Kd) is a co-involutive Hopf-von Neumann algebra. {ii) the subspace A is a Krezn algebra; if considered as a subspace of D., it is a dense sub-involutive algebra of D •. (iii) the quadruple (D, rd, "'d• cp) is a discrete type Kac algebra. Proof. Let p., v be in I. By 1.5.5, p. x v belongs to J'. Decomposing this representation into irreducible components, we find a unitary U11 ,,_. belonging to £(H11 ® H,_.), integers m 11 ,v,7rk and elements 'Irk in I, such that, for any w in M., we have:
It implies, for any w in M.:
II(P.
X
v)(w)ll ~sup 117rk(w)ll ~sup llv(w)ll k
vEJ
= llp(w)ll
As (p x p)(w) is a direct sum of elements of the form (p.
II(P X p)(w)ll
X
v)(w), we have:
~ llp(w)ll
As (v X 1)(w) = v(w) for all win M* and all v in I, we have therefore if we put n1r = E 11 ,,_. m 11 ,v,1r 1 we have 1 ~ n1r ~ oo.
mv,l,v
= 1;
6.5 Characterisation of Compact Type Kac Algebras
Let us put U define rd by:
= ffip,velUp,v
221
in C(Hp ® Hp) 1 and, for x1r in C(H1r ), let us
For any w in M., we have:
c;rd(p(w))
= c;rd( EB 1r(w)) 7rEJ
= U* ( EB (l.c(C" .. ) ® 1r(w))) U 1rEl
=
=
EB u;,v( EB (l.cccm,.,v, .. ) ® 7r(w)))uJI,V p,v 1rEl ffi(JL x v)(w)
JI,V
= (p x p)(w)
Therefore, by continuity, we find that rd( X) belongs to D ® D for all X in D. Using the definition it is immediate to check that (rd ® i)rd = (i ® rd)rd. Finally it is clear by ( **) that rd(l) = 1 and that rd is injective. Let v be in J. The representation ii(w) = v(w o ~~:)tis in J 1; so there exists an element p. in I and there exists a unitary Vv in C(Hv) such that, through the natural identification of Hv and Hp, we have, for all win M.:
p.(w) = Vvii(w)V; Let us put V
= ffivelVv
in C(Hp) and, for Xv in .C(Hv), let us define:
~~:d(EB xv) = vel
vf EB xt)v*
~El
(****)
For all w in M*, we have:
Kd(p(w)) = Kd(EB v(w)) vEl = v ( ,1 ii(w o
= =
~~:))v*
EB Vvii(w o ~~:)v;
vel
E9 v(w o ~~:)
vEl = p(w o ~~:)
by(****)
222
6. Special Cases
Therefore, by continuity ~~:d(x) belongs to D for all x in D and it is clearly an involution. By(****) we have ~~:d(l) = 1. Let 8 be in D*, win M*, we have: (p*(8o~~:d),w)
= (p(w),8o~~:d)
= (~~:d(p(w)),8) = (p(w o ~~:),8) = (p*(8),w o ~~:) = (~~:p*(8),w) and, thus, we have, for all 8 in D*:
p*(8o ~~:d)= ~~:p*(8) From this result we can get, for 81,82 in D*: ((~~:d ® ~~:d)rd(p(w)),81 ®
82)
= ('(~~:d ® ~~:d)(p x p)(w),81 ® 82} = ((p x p)(w), 82 o ll:d ® 81 o Kd} = (p*(82 o ~~:d)P*(81 o ~~:d),w) = (~~:p*(82)~~:p*(OI),w} = (~~:(p*(01)P*(02)),w) = (p*(01)P*(02),w o ~~:)
= ((p X
p)(w),81 ® 82} = (,rd(p(w o ~~:)), 81 ® 82) = (,rd(~~:d(p(w)),01 ®82) and so, we get: (~~:d 0 ~~:d)rd
= ,rd~~:d
which completes the proof of (i). Moreover, for x = ffi1rx11' in D, we get, using(*) and(**):
( u;,v ( ~(lc(Cm#,.,,>
18)
X11'A,))Up,v, Q~,s
18)
nr,m)
= ( ,rd(~x1r), n~,s ® nr,m) = ( E9 x1r, nr,m *n~,s) 11' and so, A is a subalgebra of D* which satisfies 6.4.2 (ii). Moreover, the element ill, associated to the trivial representation 1, belongs to A, and is a unit for D*. Moreover, we have, for all v in I, using(****):
(Vvx~v:, n~,r) = (Vvx~tv:, n~,s)- = ( Kd( ~ x11') *, n~,s) = (~x1r,(n~,st)
6.5 Characterisation of Compact Type Kac Algebras
223
and: which shows that A is globally invariant under the involution of D., and satisfies 6.4.2 (iii). By 1.5.7, we have, for all v in I, 0 ~ l,m,p ~ dv:
L v.(nr,m)*v.(nr,p) = c5m,p1 k
and then, thanks to the injectivity of v., we get that A satifies 6.4.2 (v). The trivial representation 1 appears in (*) if and only if it exists a onedimensional projection P~J,v in .C(H#J ® Hv) such that: P~J,v(JJ X
= (JJ X
v)(w)
v)(w)piJ,v
= w(1)p1J,v
Let {e~h:5q:5n a basis of HIJ, {e~h:5a:5m a basis of Hv. Every one-dimensional projection on H#J ® H v is given by a vector = Eq,B aq,ae~ ® e~ in H #J ® Hv such that Eq,alaq,al 2 = 1 and, for all TJ in H#J ® Hv:
e
As in 1.5.8, let us put X~q = l'•(n~,. rwq , ....~" r ), XtvB , = v.(nf:" a Then, for all w in M., we have, using 1.4.3: ~
J
e"C ).
and, thus:
(p(JJ
X
v)(w )(e~ ® e~) Ie~ ®en = ({JJ X v)(w)(e~ ® e~) lp(e~ ®en) = ((JJ
X
v)(w)(e~ ® e~) I L iir,taq',a'e~, ® e~,) q',a'
= ar,t{L
q',a'
iiq',a'X~,,qx~,, 8 ,w)
The same way, we get:
({JJ
X
v)(w )p(e~ ® e~) Ie~ ®en =
(
X
v)(w)
(E iiq,all!q',a'e~, q',a'
®
e~,) I e~ ® er)
224
6. Special Cases
and: (w(1)p(e~
® e~) Ie~ ®en= w(1)aq,s0r,t
The relation ( *) is therefore equivalent to the existence of mn complex numbers {aq,s} 1s11 s.. such that: lS•Sm
Lq,s laq,sl 2 = 1 ""Qql 8 , , B = aq ,s1 L....t ,8 1XI', q ,q X"' q',s' aq' ,8 1X I'r,q,xt, ' s' -- Or ,t 1 q',s'
L
VqVs} VrVt
and:
L aq,s(x~,,q)* = L aq',s' (L:<x~,,q)*x~,,q) x~',s q q',s' q = L aq',s'Dq',q"X~',s
by 1.5.7
q',s'
and also:
ar,t(x~,q")*
=L
aq',s'(x~,q,)*x~,q,xf. 8 ,
q',s'
Lr Or,t(x~,q" )* = q',s' L Oq',s' (L:<x~,q" )*x~,q) xr,., r =
L aq',s'Dq',q"xf.
by 1.5.7
8,
q',s'
therefore ( **) implies:
L aq,s(x~,,q)* = L q
Lar,t(x~,q,)* r
aq',s'X~',s}
•'
= Laq',s'xf. s'
(***) 8,
6.5 Characterisation of Compact Type Kac Algebras
225
Conversely, let us assume ( *** ). We have:
= Lar, 8 x~,q,(x~,q')*
Laq',ax~,q'x~,,s s'
and:
L a 91 if~
r
,s'x~, 9,x~,s' = Lr
ar,s
(2: x~,q' x~,q' )*) (
= a 9 ,al
r
The same way, we have:
L a 9',s'X~1 , 9,x~,,s
=
s'
L a 9 , 8 x~,, 9,(x~1 , 9 )* q
and:
L ~~
aq',a'x~,,q,x~,,s = Laq,s(Lx~,,q,(x~,,q)*) = a u, 1 9
8
~
q
And then we see that ( **) and ( ***) are equivalent. It implies that ( *) is, in fact, equivalent to the existence of complex numbers {aq,a} lSqSn satisfying: lS•Sm
q,s 'I:a 9 ,,(x~,, 9 )* = 'Laq',a'x~,,s q
•'
L ar ,t( x~,q' )* = L aq' ,a' xf,a' r
s1
The projection Pp,v of ( *) being then associated to the vector Eq,a a 9 , 8 e~ ®e~. Let then U be the mapping from HJ.' to H v defined by (U e~ I e~) = a 9 ,8 • We have:
(,u(w)U*e~ Ie~,)
= L a 9 ,,(,u(w)e~ Ie~,) q
=
La q
9 ,8(ji.(!1e::,e:P,),w) q
= I:aq,a{(x~,,q)*,w) q
=L •'
=
Ciq',a'(x~,,a,w)
(v(w)e~ I ~aq',a'e~)
= (v(w)e~ IU e~,)
226
6. Special Cases
Therefore, we have ;;.(w)U* = U*v(w) for all win M •. Taking the adjoints, we get U;;.(w) = v(w)U for all win M •. Therefore:
U*Ujt(w)
= U*v(w)U = ;;.(w)U*U
And, jl being irreducible, we get that there is c inC such that U*U = clH. Then m = n. But, we have:
UU*c:~
=L
ilq,s
q
L ctq,8 te~, = L ilq,sctq,s'e~, s1
q,s'
Therefore, for all s, we have Eq !aq,sl 2 = c and then: 1=
L !aq,s! 2 = nc q,s
which implies c = n -l. Let us put V = n- 112 U. Then V is a unitary which makes v and jl equivalent. Conversely, let us assume that there exists a unitary V making v and jl equivalent. Let [aq,sh$q,s$n be the matrix, with respect to the basis {c:~} and {c:~} of the operator U = n 112 v. We check that:
Furthemore, it is clear that Ujl(w)
= v(w)U implies:
L ctr,t ( x~,q' )* = L ctq' ,s' xf, r
8t
s1
and that ;;.(w)U* = U*v(w) implies:
and, therefore, we get ( *). Let us assume that the multiplicity mp,v is greater than 1. Then, there exists two orthogonal projections p and q satisfying ( *), that is two orthogonal vectors = Eq,s aq,sc:~ ® c:~ and ., = Eq,s /3q,sc:~ ® c:~ which are of norm one and satisfying ( *** ). Therefore there exists two matrices U1 and U2 satisfying, for all win M.:
e
;;.(w)Ui = Uiv(w) jl(w)U2 = U2v(w)
and and
U1i1(w) = v(w)U1 U2jl(w) = v(w)U2
6.5 Characterisation of Compact Type Kac Algebras
227
It implies UiU1jJ.(w) = U2v(w)Ul = U2UliJ.(w). And jJ. being irreducible, there exists d inC such that UiU1 = dl. But, we have:
u;ulc:~ = :2::: fJq,s' :2::: aq,s'C:~, = :2::: {Jq,saq,s'C:~, q
s1
q,s 1
which implies, for all s: ~ -
L...J /3q,saq,s
I
=c
q
and: 0
= (e 177) = L
aq,s{Jq,s
= nc'
q,s
Therefore we have c1 = 0, and then UiU1 = 0, which is impossible because = n1, by the above, which completes the proof of {ii). Then {iii) is just a corollary of 6.4.3, because we have, with the notations of 6.4.3, FD = <;Fd.
u2u;
r,
6.5.3 Definition. Let JH[ = (M, K) a co-involutive Hopf-von Neumann algebra. We shall denote by D(JH[) = (D, rd, "'d• cpd) the discrete type Kac algebra defined in 6.5.2. If lK = ( M, F, "'• cp) is a Kac algebra, the discrete type Kac algebra associated to the triple {M, K) will be denoted by D(JK). By definition, using 4.6.8, it is clear that D(JK) = D(W*(K)).
r,
6.5.4 Proposition. Let lK be a compact type Kac algebra. Then, the discrete type Kac algebra D(JK) associated by 6.5.2 is the dual Kac algebra K. Proof. By 6.2.6 {ii), it is clear that the representation p defined in 6.4.4 is then nothing but the Fourier representation. The definitions of rd and "'d given in 6.5.2 are then the same as the definition of i' {3.2.2{iv)) and K {3.3.1). The result comes then from 2.7.7.
6.5.5 Theorem. Let G be a locally compact group. We shall denote by Gd the discrete topological group having the same underlying group. Then, we have:
D(JH[s(G))
= D{W*(G)) = lKa(Gd)
.
Proof. By construction, D(JH[8 (G)) = (D(W*(G)) is built up with finitedimensional irreducible representations of £(G). = A( G). By 4.3.3, they are nothing but the points of the group G. The representation p, as defined in 6.5.1, is therefore, for fin A(G) or in B(G), given by:
p(J) =
EB sEG
f(s)
228
6. Special Cases
It is then clear that the von Neumann algebra generated by p is t=(G). Also, we have seen in 2.6.6(ii) that the Kronecker product of the characters s and t of A( G) is equal to the product ts (s, t E G). Therefore, for f in A(G) or in B(G), we have ~(p X p)(f) = Fa(!). By 5.5.7, we then have D(JH[8 (G)) = lKa(Gd), which completes the proof.
6.5.6 Theorem. Let G be a locally compact group. There ezists a compact group bG, unique up to an isormorphism, and a continuous morphism p. : G -+ bG such that for every compact group K, and every morphism a : G -+ K, there is a morphism f3: bG-+ K such that f3 o p. =a. In that situation, we have:
D(JH[a( G)) = IKs(bG) and p.(G) is dense in bG. Proof. Let us compute D(JH[a(G)). By construction, it is the von Neumann algebra generated by the representation p = EBveJV, where J is the set of (equivalence classes) of finite-dimensional irreducible representations of G. Let VI. v2 be in J. For any fin LI(G), fh in (A 111 )*, 82 in (AV2)*, we have:
((vi
X
v2)(!), 9I ® 82}
= (vh( 9I)ZI2*(82), f} = (v2*(92)vh(9I).J}
by 1.4.3
because L 00 (G) is abelian
= ((112 X vl)(f),82 ® 9I} = (~( Zl2 X VI)Cf), 9I ® 82} Therefore, by linearity and continuity, we have VI X v2 = ~(v2 X vi)· It implies that ~(p X p) = p X p and, therefore, that the Kac algebra D(JH[a(G)) is symmetric. By 4.2.5, there exists a locally compact group bG such that D(JH[a(G)) is equal to lK8 (bG); moreover, as it is of discrete type, bG is actually compact by 6.3.3 and 6.2.2. Finally, bG being the intrinsic group of D(JH[a(G)), it contains the subgroup {pa(s), s E G}, where PG is the unitary representation of G deduced from the representation p associated to Ea(G) by 6.5.1. Thus, PG is a continuous morphism from G to bG, from what we get by 5.1.4(i) the existence of an JH[-morphism lK8 (p) from W*(G) to W*(bG) such that:
Ks(p)(7ra(s)) = 'lrbG(p(s)) Now, let a : G defined by:
-+
K and lK8 (a) the JH[-morphism from W*(G) to W*(K)
6.5 Characterisation of Compact Type Kac Algebras
229
Because of K being compact, by 6.2.7{ii), S>.K is an H-isomorphism from W*(K) to H 8 (K), and AK is the sum of finite-dimensional irreducible representations Vi, by 6.2.7 {iii). For every i, Vi o a is a finite-dimensional representation of G, then there exists 'lri: Ap = .C(bG))--+ .C(Hv;) such that: 1ri(>.ba(p(s))) =Vi o a(s).
There exists therefore 1r: D(Ha(G))--+ .C(K) such that:
As bG is compact, S>. 6a is a H-isomorphism from W*(bG) to K 8(bG) which, by definition, is equal to D{lllla{G)). As 1r is an llll-morphism, we get, using 5.1.4{iii), that there exists a continuous morphism {3 : bG --+ K such that s >.K o 1K8({3) = 1r o s >.ba, and we have, for all s in G: s>.K o 1K8([3) o K8(p)7ra(s)
= 1r o s>.6a
o 7rba(p(s))
= 1r o >.ba(p(s)) = S>.K o 1K8(a)7ra(s)
or, as s >.K is injective: that is, by 5.1.4 {iii): f3op=a
We can also deduce p( G) = bG which completes the proof. 6.5. 7 Proposition. Let ( M, r, "') be a co-involutive H opf-von Neumann algebra. Let p be the representation defined in 6.5.1, D be the von Neumann algebra generated by p, H be a Hilbert space on which D has a standard representation, (D, rd, ltd, 'Pd) be the discrete type Kac algebra associated as in 6.4.5 {iii). Let us recall that, by 6.9.4, we have 'Pd = tra I D. Let us assume that the set of elements w in M. such that p(w) be a Hilbert-Schmidt operator is dense in M •. Then, there ezists a morphism~ from b toM such that ~(>.(9)) = ~tp.(9) for all9 in D. when>. is the Fourier representation of the Kac algebra D. Moreover,~ is an H-morphismfrom (D,i'd,K.d) to (M,F,~t). Proof. For all win M., lh,82 in M., we have:
(p.(91*92),w) = (p(w),91*92) = (Fdp(w),91 ® 82)
= ((p X p)(w), 82 ® 81) = (p.(92)p.(9I),w)
by 6.4.5 (iii) by 1.4.3
230
6. Special Cases
then:
p.(ih *92)
= p.(92)p.(91)
The mapping K.p* from D. to M is then multiplicative. We have also, with 9 inM.: (p.(IP),w} = (p(w),9°}
= (itdp(w)*,9}= (p(w o K-),9}0
= (p(w),9)-
by 6.4.2(i) by 1.2.5
= (p.(9),w)-
= (p.(9)*,w)
by 1.2.5
then: And so K.p* is a representation of D •. Moreover, we have: ((K.p.).(w),9)
= (K.p.(9),w) = (p.(9),w o K.) = (p(w o K-),9} = (K-d1J(w),9}
Therefore we have for all win M.:
As, by 6.3.3 and 6.1.4, 'Pd is a K.d-invariant trace, we see that we have: {wE M.; p(w) E me,}= {wE M.; (K-p.).(w) E 'Jt~} and, then, by 3.1.3, K.p* is quasi-equivalent to a subrepresentation of ,\. So, there exists iP: b-+ M such that, for all9 in D., we have:
iP(,\(9)) = K.p.(9)
= 1. Then, we have, for w1,w2 in M.: (riP(-\(9)),wl ®w2} = (rK.p.(9),wl ®w2} = (p.(9),w2 o K.*Wl o K.}
and we have, too, iP(l)
= = = =
(p(w2 o K.*Wl o K-),9} (p(w2 o K-)p(wl o K-), 9} ((K.p.).(w2)(K.p.).(w1),9} ((K.p*
X
= ((iP o ,\
K.p.)(9),w2 ®wl} iP o -\)(6),w2 ® w1}
by 1.4.3
X
= {(iP®iP)(,\
-\)(6),w2 ®w1} = {(iP ® iP)i'd,\(O),wl ® w2) X
by 1.4.5 (ii) by 6.4.5(i)
6.5 Characterisation of Compact Type Kac Algebras
231
therefore, by continuity, we have:
We have, for any 9 in D., by 3.3.1 and 6.3.2:
Therefore ~;;;,d =
K-~,
which completes the proof.
6.5.8 Theorem. Let ( M, r, K-) be a co-involutive H opf-von Neumann algebra. Let p be the representation defined in 6.5.1, H be a Hilbert space on which p(M.)" has a standard representation. Then, the following assertions are equivalent: {i) The representation p is faithful and the set of elements w in M., such that p(w) is a Hilbert-Schmidt operator in H, is dense in M •. {ii) There exists a state cp on M such that (M, r, K-, cp) is a compact type K ac algebra. Then, p is the Fourier representation of (M, r, K-, cp ), and H is isomorphic to Hcp. Proof. Let us assume (i). We have the morphism~ as defined in 6.5.7. Asp is faithful, the representation K-P• generates M and ~ is surjective. Let P be the support of~. We have (M,F,K-) = (D,/'d,K.d)p. Now, (D,i'd,K.d,rh) is a compact type Kac algebra and so is (D, rd, K.d, ch)P which is (ii). Let us assume (ii). Let][{ be the compact type Kac algebra (M,F,K-,cp). By 6.2.6 (ii), the representation p is equal to the Fourier representation of lK and, by 4.1.3 (ii) and 6.2.5 (iii), we have (i), which completes the proof.
6.5.9 Corollary. Let (M, r, K-) be an abelian co-involutive Hopf-von Neumann algebra. Let p be the representation defined in 6.5.1, H be a Hilbert space on which p(M.)" has a standard representation. Then, the following assertions are equivalent: {i) The representation p is faithful and the set of elements w in M* such that p(w) is a Hilbert-Schmidt operator on H is dense in M •. (ii) There exists a compact group G such that (M,F,K-) ~ IEila(G). Proof. It is a consequence of 6.5.8, 4.2.5 (ii) and 6.2.2.
6.5.10 Corollary. Let ( M, r, K-) be a symmetric co-involutive H opf-von N eumann algebra. Let p be the representation defined in 6.5.1, H be a Hilbert space on which p(M.)" has a standard representation. Then the following assertions are equivalent: {i) The representation p is faithful and the set of elements w in M* such that p(w) is a Hilbert-Schmidt operator on H is dense in M •.
232
6. Special Cases
{ii) There exists a discrete group G such that
(M,F,~t) ~
lHl8 (G).
Proof. It is a consequence of 6.5.8, 4.2.5 (i), 6.3.3 and 6.3.2.
6. 6 Finite Dimensional K ac Algebras 6.6.1 Theorem. Let :K = ( M, r, ~t, 'P) be a K ac algebra. The following assertions are equivalent: {i} The Kac algebra :K is both of compact and discrete type. {ii) The von Neumann algebra M is finite-dimensional. In this case, if H is a Hilbert space on which M has a standard representation, we have 'P = tr H I M. Proof Let us assume (i). By 6.3.4 (iv), we have 'P = trH I M. As, by definition
'P(l) is finite, we get dimH < oo and (ii) follows. Let us assume (ii). Then, we have dim..\(M.) = dimM. Therefore, ..\(M.) is equal to its closure M, which implies that ..\( M.) contains the unity, and then that M. is unital. Thus, we get that dimM = dimM and that :K is of discrete type. By iterating the argument we find that lK is also of discrete type and using 6.3.3 we get (i). 6.6.2 Lemma. Let ( M, r, ~t) be a co-involutive H opf-von Neumann algebra. Let us assume that M is finite-dimensional, therefore equal to Ef)~=l C(Hi) with di =dim Hi < oo. Let Pi be the projection on Hi. The multiplication on M, induces a linear mapping m : M ® M -+ M such that m( a® b) = ab for all a, b in M. Moreover It® i is a linear mapping from M ® M toM® M. For e.71 in H = E9~=1Hi, X in M, we have: n
d;
(m(~t ® i)F(x)e 177) =(I: L n~.e~ *ne,e~,x) i=l k=l n
(m(i ®
d;
~t)F(x)e 177) =(I: L ne,e~ *n~.e~ ,x) i=l k=l
where the {eLh~k~d; form an orthogonal basis of Hi. Proof. The vectors {eLh~k~d;,l~i~n form an orthogonal basis of H. Let X = L:P ap ® bp an element of M ® M. We have: ((mX)e I77) = (
L apbpe I77) p
6.6 Finite Dimensional Kac Algebras
233
p
p
=
i,k
L: L((ap ® bp)(eL ®e) 111 ® eL> p
i,k
= L:(x(eL ® e> 111 ® ei> i,k
It implies that:
and we have:
ne, 11 om o ("' ® i) or= L:U1eL 11 o,., ® ne.e~) or= i,k
L(n~.e~ ® ne.e~) or i,k
The same way, we get:
ne.'1 0 m 0 (i ® K) 0 r = L
ne.e~ *n~.e~
i,k
which completes the proof.
6.6.3 Lemma. To the hypothesis and notations of 6.6.2, we add up the assumption that it exists a homomorphism e from M to C which, considered as an element of M., is a unit of the algebra M. and verifies also, for all x in
M:
m(K ® i)F(x) = e(x)1 {So, e is a co-unit, and K an antipode, in the sense of [1] or [154].) Then, we have, for all x in M:
{i) (ii)
eK(x) = e(x) m(i ® K)F(x) = e(x)1 .
Proof. Since e is the unit of M., we have e 0 =e. As e is a homomorphism, it is positive, therefore: and we have (i).
234
6. Special Cases
It is clear that
m~
= 11:m(11: ® 11:), therefore we have for all x
m(i ®
~~:)T(x)
=
in M:
m(~~: ® ~~:)(~~: ®
i)T(x) = 11:m~(11: ® i)T(x) = ~~:m(i ® ~~:)~r(x) = ~~:m( i ® ")(" ® ")r( ~~:( x))
= ~~:m(11: ® i)T(~~:(x)) = ~~:(e(x)1) = e(x)1
by 1.2.5 by assumption
which completes the proof of (ii).
r,
6.6.4 Proposition. Let (M, ~~:) satisfying the hypothesis and notations of 6.6.3. Then M. is a Krein algebra. Proof. Let Pe: be the support of e. As Pe: belongs to the centre of M, there exists i 0 , 1 :$; i 0 :$; n, such that Hio = C and Pe: = Pio; so M. satisfies 6.4.2 (i). By 1.2.11 (i) and (ii), we get that M. satisfies 6.4.2 (ii) and (iii). H in H;, for j '# i, we have Dei ,e = 0 for alii (1 :$; 1:$; di' ); therefore we get:
e
and then:
"n~i
L.., p
i 1ef) =59 1c:(x) t:i *n~:i t:i (x) = n~:i t:i om o (~~: ® i) o r(x) = (c:(x)e9 "I'"" .. q ..., "11 '"I I
which means that M. satisfies 6.4.2(v). Let e be a unit vector of Hi ® H;' decomposed into e = Eq,B aq,se~ ® and such that T(x)e = c:(x)e. We have:
or:
"aq,B (r( X), Dt:i t:i ® nt:i t:i) = e( X)apr L-1 ~,,.. p ~·~r , q,B
d
6.6 Finite Dimensional Kac Algebras
then: ""'O:q,ail.:i .:i L..J ._, ''-P q,B
* il.:i.. , ,~r.:i = ap,re
and also:
Then we have:
Thanks to 6.4.2 (v), we have:
Let be the linear mapping Hi-+ H; defined by eve~ I e~) = O:q,B· For all x = EDxi in M, r,s such that 1 :=::; r :=::; d;, 1 :=::; s :=::; di, we have:
v
(x~v*et lei)= (x~ (Lap,re~)
=L
lei)
ap,r(x~e~ Ie/)
p
=
L ap,r(xiell e~) p
= (
x, L ap,rilei ,e~) p
= (
x, L a,,an;~,et) B
B
B
(~t(x)et I I:at,ae~) = (~t(x)et 1vej) =
B
= (V*~t(x)et Iej)
235
236
6. Special Cases
and so we get, by linearity:
from which we infer that ~~:(pi)= Pi and that the operator Vis the operator defined in 1.2.11 (ii); therefore, the vector is equal to Eq e~ ®Vie~ and, so, there is a one dimensional projection pin Hi®Hj such that F(x)p = e:(x)p for all x in M if and only if j = i 1 , and this projection is then unique. So, M. satisfies 6.4.2 (iv ), and the result is proved.
Vi
e
6.6.5 Kac-PaJjutkin's Theorem ([75]). Let ( M, r, ~~:) be a co-involutive H opfvon Neumann algebra such that M is finite dimensional. Then the following assertions are equivalent: {i) There exists a semi-finite, faithful, normal weight on M+ such that (M,r,~~:,cp) is a Kac algebra. {ii) There exists a homomorphism c from M to C which, considered as an element of M., is a unit of M., and which satisfies, for x in M: m(~~: ®
i)F(x) = e:(x)1
The quadruple (M,r,e:,~~:) is then a Hopf-algebra in the sense o/[1] or [154]. In that situation, the Haar weight is then equal to trn I M, where H is a Hilbert space on which M has a standard representation; it is therefore a finite trace. Proof. Let us assume (i). Let A be the Fourier representation of JK, let us put Xi,j = A.(wej,eJ, where {eih is a basis of the finite-dimensional Hilbert space Hcp. By 1.4.2, we have:
and then:
m(~~: ® i)F(xi,j) = .L:>Lxk,j k
As A is a finite-dimensional representation with a unitary generator, by 1.5.7, we have: m(~~: ® i)F(x·I,J·) = 6·1,)·1 And, by 6.6.1, we obtain that lK is of discrete type. Therefore, by 6.3.5 there exists a homomorphism c in M. which is a unit of M •. Then, we have:
6.6 Finite Dimensional Kac Algebras
237
Then, for all i, j, we have:
and, every element of .A*(M*) being a linear combination of Xi,j, by linearity we get (ii). The implication (ii) =? (i) is given by 6.4.3 and 6.6.5.
6.6.6 Definitions and Notations ((1], (154]). Let A be a complex algebra with unit lA. The product and the unit allow us to define linear mappings, mA from the tensor product A 0 A to A and 7JA from C to A, by, for any a, bin A and a inC:
mA(a 0 b)= ab 1JA(a) =alA
A coproduct over A is a multiplicative linear mapping d from A to A 0 A, such that: ( d 0 i)d = ( i 0 d)d The dual space A*, equipped with the transposed d* : A* 0 A* -+ A*, is then a complex algebra. Moreover, let us suppose that there exists on A a co-unit c, i.e. an algebra morphism A-+ C, such that: (c0i)d=(i0c)d=i which implies that d is injective. Then, by transposing, we get an application
g* from C to A*, which gives an unit c*{l) to the algebra A*. Moreover, the
transposed mappings mA. and 7JA are, respectively, a coproduct over A*, and a co-unit on A*. Let us suppose now that there is an antipode j, i.e. a linear application j : A -+ A such that:
Then, by transposing, we get that j* is an antipode on A*. The quadruple (A, d, c, j) is called a complex Hopf algebra; then the quadruple (A*, m A, 7JA, j*) will be called the dual complex H opf algebra. If A is finite dimensional, it is clear that the bidual complex Hopf algebra is equal to the initial one. If A is an involutive algebra, and d,e,j preserve the involution, we shall say that (A,d,c,j) is a *-Hopf algebra.
238
6. Special Cases
6.6.7 Lemma ([1], [154]). Let (A,d,c:,j) be a finite-dimensional complex Hopf algebra. Then, we have:
(i) c;d(j 0 j)d = dj (ii) The application j is antimultiplicative.
=€ j(1) = 1. c:j
(iii) (iv) Proof. As mA®A(i 0 i 0 c;) have:
= (mA 0
i)(i 0 c;)(i 0 mA 0 i), for any x in A, we
mA®A(i 0 i 0 c;(j 0 j)d)(d 0 i)d(x) = (mA 0 i)(i 0 c;)(i 0 mA 0 i)(i 0 i 0 j 0j)(i 0 i 0 d)(i 0 d)d(x) = (mA 0 i)(i 0 c;)(i 0 mA 0 i)(i 0 i 0 j 0j)(i 0 d 0 i)(i 0 d)d(x) by 6.6.6 = (mA 0 i)(i 0 c;)(i 0 mA(i 0 j)d 0 j)(i 0 d)d(x) by 6.6.6 = (mA 0 i)(i 0 c;)(i 01JA€ 0j)d(x) by 6.6.6 = (mA 0 i)(i 0 c;)(i 01JA 0 j)d(x) = mA(i 0 j)d(x) 01A by 6.6.6 = e(x)(1A 01A) Moreover, as mA®A(d 0 d)= do mA, we have, too:
mA®A(dj 0 d)d(x) =do mA(j 0 i)d(x)
= d(c:(x)1A) = c:(x)(1A 01A)
And then, as:
(mA®A 0 i 0 i)(dj 0 d 0 c;(j 0 j)d)(d 0 i)d(x) = 1A®A 0 (c: 0 c;(j 0 j)d)d(x) = 1A®A 0 c;(j 0 j)d(x) and: (i 0 i 0 mA®A)(dj 0 d 0 c;(j 0 j)d)(d 0 i)d(x)
= (dj 0 e)d(x) 01A®A = dj(x) 01A®A we get:
c;(j 0 j)d(x)
= mA®A(mA®A 0 i 0 i)(dj 0 = mA®A(i 0 i 0 mA®A)(dj 0 = dj(x)
which gives (i).
d 0 c;(j 0 j)d)(d 0 i)d(x) d 0 c;(j 0 j)d)(d 0 i)d(x)
6.6 Finite Dimensional Kac Algebras
239
Let us now apply (i) to the dual Hop£ algebra; we get, for all fin A*:
<;(j* ® j*)m"ACJ)
= m'Ai*Cf)
which, for all x, yin A, implies:
J(j(y)j(x))
= {(j* ® j*)m'A(J), y ® x} = J(j(xy))
which implies (ii). We have, too:
e(x)
= e(x)e(lA) = e(e(x)lA) = e(mA(j®i)d(x)) = (e®e)(j®i)d(x) = ej(x)
which is (iii). Applying (iii) to the dual Hop£ algebra, we get, for all fin A*:
Jj(l) = f(l) which implies (iv). 6.6.8 Lemma. Let (A,d,e,j) be a finite-dimensional *-Hopf algebra; then: J·2 =
°
2.
Proof. Let x be in A; we have:
mA((j ® i)<;d(x))
= mA(<;(i ® j)d(x)) = (mA((i ® j)d(x*)))* = (e(x*)lA)* = e(x)lA
So, we have:
mA((j 2 ® j)d(x)) = mA((j ® i)(j ® j)d(x)) = mA(j ® i)<;dj(x)) = ej(x)lA = e(x)lA
by 6.6.7(i) by 6.6. 7 (iii)
Moreover, we have:
(rnA® i)(j 2 ® j ® i)(d ® i)d(x) =(rnA® i)((j 2 ® j)d ® i)d(x) = mA(1JAe ® i)d(x) = lA ® (e ® i)d(x) =lA®X
240
and:
6. Special Cases
(i ® mA)(j 2 ® j ® i)(i ® d)d(x) = (j 2 ® mA(j ® i)d)d(x)
= (j 2 ® 7JAc)d(x) = j 2(x) ® lA and so:
x = mA(mA ® i)(j 2 ® j 0 i)(d 0 i)d(x) = mA(i 0 mA)(j 2 0 j 0 i)(i 0 d)d(x)
= j2(x) 6.6.9 Theorem. Let (A,d,c,j) be a finite-dimensional *-Hopf algebra, such that A is a semi-simple algebra. Then, A is a finite sum of finite-dimensional matrix algebras (see, for instance [49], appendix Ila), and so, it is a finitedimensional von Neumann algebra, and its predual A* is equal to the dual A*; let us put: n
A=
E9 .C(H;) i=l
n
cp = :Ld;Tr; i=l
with d; = dimH; < oo, and where Tr; is the canonical trace on .C(H;). Let us denote by I the linear isomorphism from A to A* defined, for all x in A, by I( x) = Wz (with the notations of 6.1.1 ). We have then: (i) The trace cp is equal to 7JA o I (ii) The quadruple (A, d, j, cp) is a finite-dimensional Kac algebra. (iii) The dual A* is a von Neumann algebra, moreover, the quadruple (A*,mA_,j*, c oi- 1 ) is a finite-dimensional Kac algebra, and the Fourier representation .A is a 18!-isomorphism between this Kac algebra and the dual Kac algebra (A, c;d,], cp). Proof. For all x in A, we have:
7JA oi(x) = 7JA(wz) = Wz(l) = cp(x) which gives (i). By 6.6.7 (i), (ii) and 6.6.8, (A, d,j) is a finite-dimensional co-involutive Hopf-von Neumann algebra, and so, by 6.6.4, we get (ii). As A* is finite-dimensional, we get, by 4.1.3, that .A is an isomorphism from the algebra A* to A. So A* is a finite-dimensional von Neumann algebra, and we can apply (i) and (ii) to the quadruple (A*,m:A_,7J:4,j*), and so we get that (A*, mA_,j*, c oi- 1 ) is a finite-dimensional Kac algebra. Let now 81,82 be in A*, w in A*; we have:
(c;d.A(w),81 ®82} =((.Ax .A)(w),81 ®82}
by 3.2.2(iv) and (ii)
6.6 Finite Dimensional Ka.c Algebras
= (.\.{th).\.(92),w) = mA(.\.(91) ® .\*(82)),w) = (((.\®.\)mA_(w),81 ®82)
241 by 1.4.3
from which we get that: (.\ ® .\)mA_ =<;d.\
which gives the result, thanks to 5.5.6. 6.6.10 Corollaries. (i) Let I be a finite set, and d,e,j such that (Cl,d,e,j) is a *-Hopf algebra. Then I has a group structure, with identity element e, such that, for all f in C1 :
df(s, t) = f(st) e(f) = f(e) j(f)(s) = f(s- 1 ) (ii) Let (A, d, e, j) be a finite dimensional symmetric *-Hopf algebra, such that A is a semi-simple algebra. Then the set { x E A; x =/: 0, dx = x ® x} is a finite group G, the algebra A is the vector space CG, and d,e,j verify, for all s in G: ds=s®s
=1 -1 JS = S es 0
•
Proof. By 6.6.8 and 4.2.5(ii), there is a group r such that (C1 ,d,j) is JH[-isomorphic with JH[0 (G). That allows us to put on I a group structure satisfying (i). By 6.6.8 and 3.6.10, the set {x E A; x =/: 0, dx = x ® x} is the intrinsic group G of the co-involutive Hopf-von Neumann algebra (A, d,j), and, by 6.6.8 and 4.2.5 (i), we get that (A, d, j) is JH[-isomorphic to JH[s( G), which gives (ii).
The Gat's head began fading away ...
(Alice's Adventures in Wonderland) 0 teachers, are my lessons done I cannot do another one They laughed and laughed and said Well, child, are your lessons done (Leonard Cohen, Teachers)
Postface
The Kac algebras, to the study of which the authors have had a major contribution since the beginning of the subject, have been introduced as a natural framework for duality of type Pontrjagin-van Kampen. A different approach, linking Kac algebras to the classical groups via quantization, has been described in the preface of this book. I would like to mention here the way in which Kac algebras appear as symmetry structures. Like groups, Kac algebras can act automorphically on algebras. Let :K be a Kac algebra, which, for convenience, will be assumed to be finite-dimensional, acting on a factor A; one can then construct the crossed product B of A by IK, which contains A. We shall consider an outer action, which means that the relative commutant A' n B is scalar. The relative position of A in B can then be used to recover the structure of :K. In fact, more generally, we shall consider an inclusion of factors A C B, with A' n B = C, where again for convenience, we assume B to be finitely generated over A (or, in other words, we assume that the Jones index [B: A] is finite). Then the bimodule endomorphisms M = End(ABA) have a bialgebra structure. If we write:
so that any x in M has the form x = Ei=l bi ®A q, with bi and Ci in B, and satisfies ax = xa for all a in A, then we have, for x = Ei bi ®A Ci and y = E; d; ®A e; as above, the multiplication of endomorphisms, given by:
xy = L
biEA(cid;) ®A e;
i,j
where EA is the conditional expectation on A, and a convolution operation defined by:
x
* Y = Lbidj ®A e;ci = LbiYCi i,j
(the convolution is well defined since y commutes with A).
244
Postface
When B is the crossed product of A by JK, then this recovers the bialgebra structure of lK (up to completions) in an invariant way. In the general case, one can show that, under our hypothesis, B is the crossed product of A by a Kac algebra][{ with a cocycle twisted action, if and only if End(AB®ABB) is a factor. This type of result generalizes beyond our finite index assumption, and shows that the Kac algebras arise naturally by means of their actions, as symmetry structures associated to inclusions of von Neumann algebras. I can only hope thus that the authors will continue their beautiful monograph with a study of the Kac algebras in action.
Paris, October 1991
Adrian Ocneanu
Bibliography
[1] E. Abe: Hopf algebras. Cambridge University Press, 1977. [2] C.A. Akemann and M.E. Walter: Non-abelian Pontriagin duality. Duke Math. J. 39(1972), 451-463. [3] S. Baaj et G. Skandalis: c•-algebres de Hopf et theorie de Kasparov equivariante. K-Theory 2(1989), 683-721. [4] S. Baaj et G. Skandalis: Unitaires multiplicatifs et dualite pour les produits croises de c• -algebres. To appear in Ann. Sci. E.N .S. [5] A.A. BenaBHH, B.r. .llpnHtjlenb.a; (A.A. Belavin, V.G. Drinfel'd): 0 pemeHH.IIX xnaccuqecxoro ypaBHeHH.II .HHra-Baxc-repa .a;n.11 npocTbiX anre6p Jlu. ~YHKit. aHanH3 Hero npun. 16-3(1982), 1-29; translated in: On the solutions ofthe classical Yang-Baxter equation for simple Lie algebras. Funct. Anal. appl. 16(1982), 159-180. [6] IO.M. Bepe3aHcxult, A.A. KaniOlKBbllt (Yu.M. Berezansku, A.A. Kalyuzhni1): runepKOMnneKCHble CHCTeMbl C noKanbHO KOMnaKTHbiM 6a3HCOM. Preprint 40(1983) HH. MaT. AH YCCP, KneB; translated in: Hypercomplex systems with a locally compact basis. Sel. Math. Sov. 4(1985), 151-200. [7) IO.M. Bepe3&HCKHit, A.A. KaniOH
246
Bibliography
[16] A. Cannes: Une classification des facteurs de type III. Ann. Sci. E.N.S. 6(1973), 133-252. [17] A. Cannes: Geometrie non commutative. InterEditions, Paris, 1990. [18] J. De Canniere: On the intrinsic group of a Kac algebra. Proc. London Math. Soc. 39(1979), 1-20. [19] J. De Canniere: Produit croise d'une algebre de Kac par un groupe localement compact. Bull. Soc. Math. France 107(1979), 337-372. [20] J. De Canniere: An illustration of the non-commutative duality theory for locally compact groups. Bull. Soc. Math. Belg. 31B(1979), 57-66. [21] J. De Canniere, M. Enock, J.-M. Schwartz: Algebres de Fourier associees a une algebre de Kac. Math. Ann. 245(1979), 1-22. [22] J. De Canniere, M. Enock, J.-M. Schwartz: Sur deux resultats d'analyse harmonique non-commutative: une application de la theorie des algebres de Kac. J. Operator Th. 5(1981), 171-194. [23] J. Dixmier: Algebres quasi-unitaires. Comm. Math. Helv. 26(1952), 275-322. [24] J. Dixmier: Les algebres d'operateurs dans l'espace hilbertien. Gauthier-Villars, Paris, 1969. Translated in: Von Neumann algebras. North-Holland, Amsterdam, 1981. [25] J. Dixmier: Les c•-algebres et leurs representations. Gautier-Villars, Paris, 1969. [26] S. Doplicher and J. Roberts: Endomorphisms of c•-algebras, cross products and duality for compact groups. Ann. of Math. 130(1989), 75-119. [27] S. Doplicher and J. Roberts: A new duality theory for compact groups. lnvent.Math. 98(1989), 157-218. [28] B.I'. llpHBcjlen&.A (V.G. Drinfel'd): I'aMH.IIhTOHOBhl cTpyKTyphl B& rpynnax JIH, 6Hanre6pbl JIH H reOMeTpHQecKHit CMhiC.II K.II&CCHQeCKHX ypaBBeBHii .flHr&B&KCTepa . .lloKn. AKa..zt. HayK CCCP 268(1983), 285-287; translated in: Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of the classical Yang-Baxter equations. Soviet Math. Dokl. 27(1983), 68-71. [29] B.I' . .llpHBcjlenh.A (V.G. Drinfel'd): 0 nocTOJIBBhiX KB&3HK.II&CCHQeCKHX pemeBHJIX KB&BTOBoro ypasBeBHJI .HBra-BaKcTepa . .lloK.II. AKa..zt. HayK CCCP 273(1983), 531-534; translated in: On constant, quasiclassical solutions of the YangBaxter quantum equation. Soviet Math. Dokl. 28(1983), 667-671. [30] B.I'. .llpHBcjlenh.A (V.G. Drinfel'd): Anre6phl Xoncjla H KB&BTosoe ypasBeBHe .HBra-BaKCTepa . .lloKn. AK&,A. HayK CCCP 283(1985), 1060-1064; translated in: Hopfalgebras and the quantum Yang-Baxter equation. Soviet Math. Dokl. 32(1985), 254-258. [31] V.G. Drinfel'd: Quantum groups. Proceedings ICM Berkeley (CA) 1986, vol.l, 798820. [32] B.I'. llpHBcjlen&.A (V.G. Drinfel'd): 0 DOQTH KOKOMMYT&THBBhiX anre6pax Xoncjla. Anre6pa H &H&.IIH3 1-2(1989), 30-47; translated in: On almost cocommutative Hopf algebras. Leningrad Math. J. 1(1990), 321-342. [33] M. Enock: Produit croise d'une algebre de von Neumann par une algebre de Kac. J. Funct. Anal. 26(1977), 16-47. [34] M. Enock et J.-M. Schwartz: Une dualite dans les algebres de von Neumann.Note C.R. Acad. Sc. Paris 277(1973), 683-685. [35] M. Enock et J.-M. Schwartz: Une categorie d'algebres de Kac. Note C.R. Acad. Sc. Paris 279(1974), 643-645. [36] M. Enock et J.-M. Schwartz: Une dualite dans les algebres de von Neumann. Supp. Bull. Soc. Math. France Memoire 44(1975), 1-144. [37] M. Enock et J.-M. Schwartz: Produit croise d'une algebre de von Neumann par une algebre de Kac, II. Publ. R.I.M.S. Kyoto 16(1980), 189-232. [38] M. Enock et J .-M. Schwartz: Deux resultats sur les representations d'algebres de Kac. Yokohama Math. J. 31(1983), 170-184
Bibliography
247
[39] M. Enock et J .-M. Schwartz: Systemes dynamiques generalises et correspondances. J.Operator Th. 11{1984), 273-303. [40] M. Enock et J .-M. Schwartz: Extension de la categorie des algebres de Kac. Ann. lnst. Fourier, 36{1986), 105-131. [41] M. Enock et J.-M. Schwartz: Algebres de Kac moyennables. Pacific J. Math. 125{1986), 363-379. [42] M. Enock et J .-M. Vallin: c• -algebres de Kac et algebres de Kac. To appear in Proc. London Math. Soc. [43] J. Ernest: A new group algebra for locally compact groups 1,11. Amer. J. Math. 86{1964), 467-492; Canad. J. Math. 17{1965), 604-615. [44] J. Ernest: Hopf-von Neumann algebras. Proc. Conf. Functional Anal. {Irvine, CA) Academic press, New York {1967), 195-215. [45] J. Ernest: Math. Rev. 56{1978) # 1091, 156-157. [46] P. Eymard: L'algebre de Fourier d'un groupe localement compact. Bull. Soc. Math. France 92{1964), 181-236. [47] J.M.G. Fell: The dual spaces of c•-algebras. Trans. Amer. Math. Soc. 94(1960), 365-403. [48] J .M.G. Fell: Weak containment and induced representations of groups. Canad. Math. J. 14{1962), 237-268. [49] F.M. Goodman, P. de laHarpe and V.F.R. Jones: Coxeter Graphs and Towers of Algebras. M.S.R.I. Publ. n° 14, Springer-Verlag, New York, 1989. [50] J. Grabowski: Quantum SU{2) group of Woronowicz and Poisson structures. Conf. Diff. Geom. and appl. World Scientific, Singapore, 1990. [51] ll.:H. rypeBH'I {D.I. Gurevich): OnepaTOphl o6o6~eHHOI'O C~HI'a Ha rpynnax JIH. :H3B. A.H. ApM. CCCP MaT. 18(1983), 393-408; translated in: Generalized translation operators on Lie Groups. Soviet J. of Cont. math. Analysis 18-4{1983), 57-70. [52] ll.:H. rypeBH'I {D.I. Gurevich): Ypa.sHeHHe .HHra-BaxcTepa H o6o6~eHHe tjlopManbHOit TeOpHH JIH. lloKn. AK8..l{. HayK CCCP 288 {1986) 797-801; translated in: The Yang-Baxter equation and a generalization of a formal Lie theory. Soviet Math. Dokl. 33{1986), 758-762. [53] ll.:H. rypeBH'I (D.I. Gurevich): Cne.zt H .zteTepMHHaHT B anre6pax, &CCOI{HHpOBaHHhiX C ypa.sHeHHeM .flHra-BaKCTepa. cJ.)yHKI{. &H&nH3 H ei'O DpHn. 213{1987), 79-80; translated in: Trace and determinant in the algebras associated with the Yang-Baxter equation. Funct. Anal. appl. 21{1987), 239-240. [54] ll.:H. rypeBH'I (D.I. Gurevich): 0 CK06Kax IlyaccoHa, &CCOD;HHpOB&HHhiX c KnaCCH'IeCKHM ypaBHeHHeM .flHra-BaKCTepa. cJ.)yHKI{. aHanH3 H ero DpHn., 231{1989), 68-69; translated in: Poisson brackets associated with the classical YangBaxter equation. Funct. Anal. appl. 23{1989), 57-59. [55] D.I. Gourevitch: Equation de Yang-Baxter et quantification des cocycles. C.R. Acad. Sc. Paris 310{1990), 845-848. [56] U. Haagerup: Normal weights on w•-algebras. J. Funct. Anal. 19(1975), 302-317. [57] U. Haagerup: The standard form of von Neumann algebras. Math. Scand. 37{1975), 271-283. [58] U. Haagerup: On the dual weights of crossed products of von Neumann algebras 1,11. Math. Scand. 43{1978), 99-118; 119-140. [59] U. Haagerup: Operator-valued weights in von Neumann algebras 1,11. J. Funct. Anal. 32{1979), 175-206; 33{1979), 339-361. [60] R.H. Herman and A. Ocneanu: Index theory and Galois theory for infinite index inclusions of factors. C.R. Acad. Sc. Paris 309{1989), 923-927. [61] R.H. Herman and A. Ocneanu: in preparation [62] E. Hewitt and K.A. Ross: Abstract harmonic analysis 1,11. Springer-Verlag, Berlin, 1963.
248
Bibliography
[63] M. Jimbo: A q-difference analogue of U(g) and the Yang-Baxter equation. Lett. Math. Phys. 10(1985), 63-69. [64] M. Jimbo: A q-analogue of U(gl(N + 1)), Heeke algebra and the Yang-Baxter equation. Lett. Math. Phys. 11(1986), 247-252. [65] B.E. Johnson: Isometric isomorphisms of measure algebras. Proc. Amer. Math. Soc. 15(1964), 186-188. [66] r.l1. Ka~ (G.I. Kac): 06o6meHHe rpynnoaoro npHHI{Hila )l;BOi!:CTBeHHOCTH. lloKJI. AKa.n;. HayK CCCP 138(1961), 275-278; translated in: Generalization of the group principle of duality. Soviet Math. Dokl. 2(1961), 581-584. [67] r.l1. Ka~ (G.I. Kac): llpe.n;cTaBJieHHR KOMilaKTHhiX KOJihl{eBh!X rpynn. lloKJI. AKa.n;. HayK CCCP 145(1962), 989-992; translated in: Representations of compact ring groups. Soviet Math. Dokl. 3(1962), 1139-1142. [68] r.l1. Ka~ (G.I. Kac): KoHe"!Hhle KOJihl{eBh!e rpynnhl. lloKJI. AKa.n;. HayK CCCP 147(1962), 21-24; translated in: Finite ring groups. Soviet Math. Dokl. 3(1962), 1532-1535. [69] r.l1. Kal{ (G.I. Kac): KoMilaKTHhle H )l;HCKpeTHhle KOJih~eBh!e rpynnhl. YKp. MaTeM. lf<. 14(1962), 260-269. [70] r.l1. Kal{ (G.I. Kac): KoJib~eBh!e rpyllllhl H npHH~Hil )l;BOi!:CTBeHHOCTH, I,II. Tpy.n;hi MocK. MaTeM. o6-Ba 12(1963), 259-301; 13(1965) 84-113; translated in: Ring-groups and the principle of duality, I,II. Trans. Moscow Math. Soc. (1963), 291-339; (1965), 94-126. [71] r.l1. Kal{ (G.I. Kac): PacmHpeHHR rpynn, RBJIRIOillHeCR KOJib~eBh!MH rpynnaMH. MaTeM. c6. 76(1968), 473-496; translated in: Extensions of groups to ring groups. Math. USSR Sbornik 5(1968), 451-474. [72] r.l1. Kal{ (G.I. Kac): HeKOTOphle apHcpMeTH"!eCKHe CBOi!:CTBa KOJihl{eBh!X rpynn. ct>yHK~. aHaJIH3 H ero npHJI. 6-2(1972), 88-90; translated in: Certain arithmetic properties of ring groups. Funct. Anal. appl. 6(1972), 158-160. [73] r.l1. Kal{, B.r. llaJIIOTKHH (G.I. Kac, V.G. Paljutkin): llpHMep KOJihl{eBoi!: rpynnhl, nopom.n;eHHoi!: rpynnaMH JIH. YKp. MaTeM. m. 16(1964), 99-105. [74] r.l1. Kal{, B.r. llaJIIOTKHH (G.I. Kac, V.G. Paljutkin): llpHMep KOJihl{eBoi!: rpynnhl BOChMoro nopn.n;Ka. YcnexH MaT. HayK, 20-5(1965), 268-269. [75] r.l1. Kal{, B.r. llaJIIOTKHH (G.I. Kac, V.G. Paljutkin): KoHe"!Hhle KOJihl{eBh!e rpynnhi. Tpy.n;hl MocK. MaTeM. o6-aa 15(1966), 224-261; translated in: Finite ring-groups. Trans. Moscow Math. Soc. 1966, 251-294. [76] R.V. Kadison: Isometries of operator algebras. Ann. of Math. 34(1951) 325-338. [77] Y. Katayama: Takesaki's duality for a non-degenerate coaction. Math. Scand. 55(1985), 141-151. [78] E. Kirchberg: Representations of a coinvolutive Hopf-W*-algebra and non-abelian duality. Bull. Acad. Pol. Sc. 25(1977), 117-122. [79] E. Kirchberg: Darstellungen coinvolutiver Hopf-W*-Algebren und ihre Anwendung in der nicht-Abelschen Dualitiits-Theorie lokalcompakter Gruppen. Berlin, Akademie der Wissenschaften der DDR, 1977. [80] H.T. Koelink and T.H. Koornwinder: The Clebsch-Gordan coefficients for the quantum group SJ.IU(2) and q-Hahn polynmnials. Proc. Kon. Neder. Akad. Weten. A92(1989), 443-456. [81] T.H. Koornwinder: Representations of the twisted SU(2) quantum group and some qhypergeometric orthogonal polynomials. Proc. Kon. Neder. Akad. Weten A92(1989), 97-117. [82] M.r. Kpei!:H (M.G. Krein): 0 noJIOH
Bibliography
249
[84) M.r. Kpel!:u (M.G. Krein): 3pMHTOBo-noJIOlKHTeJibHhle .R,ll;pa ua o,Z~;Hopo,ll;HhiX npocTpaHcTBax 1,11. YKp. MaTeM. lK. 1-4(1949), 64-98; 2-1(1950), 10-59; translated in: Hermitian-positive kernels in homogeneous spaces 1,11. Amer. Math. Soc. Transl. 34(1963), 69-108; 109-164. [85) II.II. KynHm, E.K. CKJI.RHHH (P.P. Kulish, E.K. Sklyanin): 0 pemeHH.RX ypasHeHH.R ..flura-BaKcTepa. B c6.: .llHijlljlepeHI{HaJILHa.R reoMeTpH.R, rpynnhl JIH H MexaHHKa. 3an. uayt~H. ceMHH. JleHHHrpa,ll;. OT,!J;eJI. MaTeM. HucT. HM. CTeKJIOBa (JIOMH) 95(1980), 129-160; translated in: Solutions of the Yang-Baxter equation. J. Soviet Maths 19(1982), 1596-1620. [86) II.II. KynHm, H.IO. PemeTHXHH (P.P. Kulish, N.Yu. Reshetikhin): KBaHTOBa.R JIHHeitHa.R 3a,ll;at~a ,ll;JI.R ypaBHeHH.R CHHyc-rop,ll;OH H BhiCWHe npe,ll;CTaBJieHH.R. B c6.: Bonpochl KB. TeopHH non.R H cTaT. lj!H3HKH. 3an. uayt~H. ceMHH. JleHHHrpa,ll;. OT,!J;eJI. MaTeM. 11HcT. HM. CTeKJIOBa (JIOMH) 101(1981), 101110; translated in: Quantum linear problem for the sine-Gordon equation and higher representations. J. Soviet Maths 23(1983), 2435-2441. [87) P.P. Kulish, N.Yu. Reshetikhin and E.K. Sklyanin: Yang-Baxter equation and representation theory I. Lett. Math. Physics 5(1981), 393-403. [88) M.B. Landstad: Duality theory for covariant systems. Trans. A.M.S. 248(1979), 223267. [89) M.B. Landstad: Duality for dual covariance algebras. Comm. Math. Phys. 52(1977), 191-202. [90) M.B. Landstad, J. Phillips, I. Raeburn and C.E. Sutherland: Representations of crossed products by coactions and principal bundles. Trans. A.M.S. 299(1987), 747784. [91) J.-H. LuandA. Weinstein: Poisson-Lie groups, dressing transformations and Bruhat decompositions. J. Diff. Geom. 31(1990), 501-526. [92) G. Lusztig: Quantum deformations of certain simple modules over enveloping algebras. Adv. Math. 70(1988), 237-249. [93) G. Lusztig: Modular representations and quantum group. Contemp. Math. 82(1989), 59-77. [94) G.W. Mackey: Borel structures in groups and their duals. Trans. A.M.S. 85(1957), 134-165. [95) S. Mac Lane: Categories for the Working Mathematician. Springer, New York, 1971. [96) V.B. de Magalhaes IOrio: Hopf-IC*-algebras and locally compacts groups. Pacific J. of Maths 87 (1980), 75-96. [97) S. Majid: Hopf-von Neumann algebra bicrossproducts, Kac algebra bicrossproducts and the classical Yang-Baxter equation. J. Funct. Anal. 95(1991), 291-319. [98) S. Majid and Ya.S. Soibelman: Rank of quantized universal enveloping algebras and modular functions. Com. Math. Phys. 137(1991), 249-262. [99) Yu.I. Manin: Quantum groups and non-commutative geometry. CRM. Univ. de Montreal (1988). [100) T. Masuda, K. Mimachi, Y. Nakagami, M. Noumi and K. Ueno: Representations of quantum groups and a q-analogue of orthogonal polynomials. C.R. Acad. Sc. Paris 307(1988), 559-564. [101) T. Masuda, K. Mima.chi, Y. Nakagami, M. Noumi and K. Ueno: Representations of the quantum SU9 (2) and the little q-Jacobi polynomials. J. Funct. Anal. 99(1991), 357-386. [102) Y. Nakagami: Dual action on a von Neumann algebra and Takesaki's duality for a locally compact group. Publ. RIMS Kyoto Univ. 12(1976) [103) Y. Nakagami: Some remarks on crossed products of a von Neumann algebra by Kac algebras. Yokohama Math. J. 27(1979), 141-162. [104) Y. Nakagami and M. Takesaki: Duality for crossed products of von Neumann algebras. Lect. Notes in Math. n° 731 (1979), Springer-Verlag.
250
Bibliography
[105] M.A. HaliMapK (M.A. Na.lmark): HopMHpOBaHHhle KOJibl{a, rocTeXH3.l{&T, MocKBa, 1956. Translated in: Normed rings. Noordhoff, Groningen, Nederlands, 1964. (106] R. Nest and A. Ocneanu: in preparation [107] A. Ocneanu: A Galois theory for operator algebras. Preprint 1986 [108] A. Ocneanu: Quantized groups, string algebras and Galois theory for algebras. in Operators algebras and applications, vol. 2, Londres, Math. Soc. Lect. Note Series 136(1988), 119-172. (109] A. Ocneanu: Manuscript notes on extended groups and discrete Kac algebras. Univ. of Warwick, England, 1982. [110] B.r. IIaniOTKHH (V.G. Paljutkin): 06 3KBHBaJieHTHOCTH .nsyx onpe.neneHHii KOHeqaoft KOJibl{eBOft rpynnhl. YKp. MaTeM. lK. 16(1964), 402--406. [111] B.r. IIaniOTKHH (V.G. Paljutkin): HaBapHaaTHII.ll Mepa aa KOMnaKTaoli KOJiblleBolt rpynne. YKp. M&TeM. lK. 18(1966), 49-59; translated in: Invariant measure of a compact ring group. Am. Math. Soc. Transl. 84(1969), 89-102. (112] B.r. IIa.niOTKHH (V.G. Paljutkin): 06 annpoKCHM&THBBoli e.nHHHlle B rpynnoBolt anre6pe KOJiblleBoli rpynnhl. YKp. MaTeM. lK. 20(1968), 176-182; translated in: An approximate unit in the group algebra of a ring group. Ukr. Math. J. 20(1969), 162-177. (113] G.K. Pedersen: c•-Algebras and their Automorphism Groups. Academic Press. London, 1979. [114] G.K. Pedersen and M. Takesaki: The Radon-Nikodym theorem for von Neumann algebras. Acta Math. 130(1973), 53-87. [115] S. Petrescu and M. Joita: Amenable actions of Kac algebras on von Neumann algebras. Revue Roumaine de Math. pures et Appl 35(1990), 151-160. (116] S. Petrescu and M. Joita: Property (T) for Kac algebras. INCREST Bucure§ti (Romania), preprint series n° 31 (1989). [117] J.-P. Pier: L'analyse harmonique. Son developpement historique. Masson, Paris, 1990. [118] P. Podle8: Quantum spheres. Lett. Math. Phys. 14(1987), 193-202. [119] P. Podle8: Complex quantum groups and their real representations. RIMS Kyoto preprint 1991. [120] P. Podle8 and S.L. Woronowicz: Quantum deformation of Lorentz group. Comm. Math. Physics 130(1990), 381--431. [121] L.S. Pontrjagin: The theory of topological commutative groups. Ann. of Math. 35(1934), 361-368. [122] H.IO. PemeTHXHH, JI.A. TaxT&.n:IKIIH, JI ..ll. 4>a.n.neeB (N.Yu. Reshetikhin, L.A. Takhtadzhan, L. D. Faddeev): KB&HTOB&HHe rpynn H anre6p JIH. Anre6pa H aaaJIH3 1-1(1989), 178-207; translated in: Quantization of Lie groups and Lie algebras. Leningrad Math. J. 1(1990), 193-226. [123] N.Yu. Reshetikin, V.G. Turaev: Ribbon graphs and their invariants derived from quantum groups. Com. Math. Phys. 127(1990), 1-26. [124] C. Rickart: Banach algebras, Krieger, Huntington, NY, 1974. [125] M.A. Rieffel: Some solvable quantum groups. Proceeding of the Conf. on Operator Algebras and their connections with Topology and Ergodic Theory II, Craiova, Romania., sept. 89 [126] M.A. Rieffel: Deformation quantization and operator algebras. Operator theory: operator algebras and applications I, A.M.S., Providence, RI, (1990), 411--423. [127] M. Rosso: Comparaison des groupes SU(2) quantiques de Drinfel'd et Woronowicz. Note C.R.A.S. Paris 304(1987), 323-326. [128] M. Rosso: Finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple Lie algebra. Com. Math. Phys. 117(1988), 581-593. [129] M. Rosso: Groupes quantiques et modeles a vertex de V. Jones en theorie des noeuds. Note C.R.A.S. Paris 307(1988), 207-210.
Bibliography
251
[130] M. Rosso: Algebres enveloppantes quantifiees, groupes quantiques compacts de matrices et calcul differentiel non commutatif. Duke Mat. J. 61(1990), 11-40. [131] M. Rosso: Representations des groupes quantiques: Sem. Bourbaki n° 744 (juin 1991). [132] R. Rousseau and A. Van Daele: Crossed products of commutation systems. Math. Ann. 239(1979), 7-20. [133] S. Sakai: C*-algebras and W*-algebras, Springer-Verlag, Berlin, 1971. [134] J.-L. Sauvageot: Implementation canonique pour les co-actions d'un groupe localement compact et les systemes dynamiques generalises. Math. Ann. 270(1985), 325337. [135] J .-M. Schwartz: Relations entre "ring groups" et algebres de Kac. Bull. Sc. Math. 100(1976), 289-300. [136] J.-M. Schwartz: Sur la structure des algebres de Kac 1,11. J. Funct. Anal. 34(1979), 370-406; Proc. London Math. Soc. 41(1980), 465-480. [137] M.A. CeMeHOB-T.RH-illa.HCKHA (M.A. Semenov-Tyan-Shanski'l'): qTO TaKoe KnaccuqecKa.R r-MaTpn~a? ~YHK~. aaanu3 u ero npun. 17-4(1983), 17-33; translated in: What is a classical R-matrix? Funct. Anal. appl. 17{1983), 259-272. [138] A.J.-L. Sheu, J.-H. LuandA. Weinstein: Quantization of the Poisson SU{2) and its Poisson homogeneous space. The 2-sphere. Com. Math. Phys. 135(1991), 217-232. [139] A.H. illTepB {A.I. Shtern): .llsoltcTBeHHOCTh ~n.R cKpell{eHHhiX npoH3Be~eauA anre6p 4jloa HeAMaaa. HToru HayKH H Tex. Cosp. npo6n. MaT. 27{1985), 3365; translated in: Duality for crossed products of von Neumann algebras. J. Soviet Math. 37{1987), 1396-1421. [140] E.K. CKn.RHHH (E.K. Sklyanin): MeTO~ o6paTaoA 3a~aqu pacce.RHH.R H KBaHTOBoe aenuaeltaoe ypasaeaue illpe~arepa . .lloKn. AKa..zt. HayK CCCP 244{1979), 1337-1341; translated in: Method of the inverse scattering problem and the nonlinear quantum SchrOdinger equation. Soviet Phys. Dokl. 24{1979), 107-109. [141] E.K. CKn.RBHH {E.K. Sklyanin): KsaHTOBhiA sapHaHT MeTO~a o6paTaoA 3a~aqu pacce.RHH.R. B c6.: .llu4j14jlepeH~Han&Ha.R reoMeTpH.R, rpynnhl Jlu H MexaHHKa. 3an. Hayqa. ceMHH. Jleuuarpa..zt. OT~en. MaTeM. HacT. HM. CTeKnosa (JIOMH) 95(1980), 55-128; translated in: Quantum version of the inverse scattering problem. J. Soviet Maths 19{1982), 1546-1596. [142] E.K. CKn.RHHH (E.K. Sklyanin): 06 o~aoA anre6pe, nopom~aeMoA KBa..ztpaTHQHhiMH COOTBOmeHH.RMH. YcneXH MaTeM. BayK, 40-2(1985), 214. [143] E.K. CKn.RBHH, JI.A. TaxTa~:IK.RH, Jl ..ll. ~a~ees (E.K. Sklyanin, L.A. Takhtadzhan, L. D. Faddeev): KsaBTOBhiA MeTO~ o6paTaoA 3a~aqu. I. TeopeT. H MaTeM. 4jiH3HKa 40{1979), 194-220; translated in: Quantum inverse problem method I. Theoret. and Math. Physics 40(1980), 688-706. [144] E.K. CKn.RHHH, Jl ..ll. ~~ees {E.K. Sklyanin, L.D. Faddeev): KsaHTOBOMexaauqecKHA no~o~ K BHonae HHTerpupyeMhiM Mo~en.RM Teopuu non.s . .lloKn. AKa~. HayK CCCP 243{1978), 1430-1433; translated in: Quantuum-mechanical approach to completely integrable models of field theory. Sov. Phys. Dokl. 23(1978), 902-904. [145] .H.C. Colt6en&Ma.JI (Ya.S. So'l'bel'man): HenpHBO~HMhle npe~cTasneHH.R anre6phl 4jlyaK~HA aa KB&HTOBoA rpynne SU(n) u KneTKH illy6epTa . .lloKn. AKa~. HayK CCCP 307(1989), 41-45; translated in: Irreducible representations ofthe function algebra on the quantum group SU(n), and Schubert cells. Soviet Math. Dokl. 40(1990), 34-38. [146] .H.C. CoA6en&MaH (Ya.S. So'l'bel'man): Anre6pa 4jlyaK~HA aa KOMn&KTHoA rpynne H ee npe~cTasneau.s. Anre6pa u aaanH3 2-1(1990), 190-212; translated in: The algebra of functions on a compact quantum group, and its representations. Leningrad Math. J. 2(1991), 161-178. [147] .H.C. CoA6en&MaH (Ya.S. So'l'bel'man): CocTO.RHH.R ren&4jlaH~a-HaAMapKa Cura.na. H rpynna. Beltn.s ~n.R KBa.JITOsoA rpynnhl SU(n). ~yu~. a.aa.nu3 u
252
[148] [149] [150] [151] [152] (153] [154] [155] (156] [157] [158] [159] [160] [161] [162] [163] [164]
[165]
[166] [167] [168] [169] [170]
[171]
Bibliography ero npHn. 24-3(1990), 92-93; translated in: Gel'fand-Naimark-Segal states and Weil group for the quantum group SU(n). Funct. Anal. appl. 24(1990), 253-255. W.F. Stinespring: Integration theorems for gauges and duality for unimodular locally compact groups. Trans. Amer. Math. Soc. 90 (1959), 15-56. E. St!yHKn. auanH3 1 23-2(1989), 75-76; translated in: Noncommutative homology of quantum tori. Funct. Anal. appl. 23(1989), 147-149. JI.A. TaxTa.AlKIIH, JI ..Ll. Cl>a.A.AeeB (L.A. Takhtadzhan, L.D. Faddeev): IyHKI{. aHaHH3 Hero UpHn. 8-1(1974), 7576; translated in: Characterization of objects dual to locally compact groups. Funct. Anal. appl. 8(1974), 66-67. JI.H. BaituepMaH (L.I. Valnerman):)) KoHe'IHble rHnepKoMnneKCHble cHcTeMbl Kana. Cl>yHKn. auanH3 H ero npHn. 17-2(1983), 68-69; translated in: The finite hypercomplex systems of Kac. Funct. Anal. appl. 17(1983), 135-136.
Bibliography
253
[172) JI.H. BaitHepMaH {1.1. Va1nennan): rHuepKOMUJieKCHhle CHCTeMhl c KOMU&KTHhiM H .AHCKpeTHhiM 6a.3HCOM . .lloKn. AK&.A. HayK CCCP 278(1984), 16-19; translated in: Hypercomplex systems with a compact and discrete basis. Soviet Math. Dokl. 30{1984), 305-308. [173) JI.H. BaitHepMa.H (1.1. Vamennan): rapMOHH'IeCKHA a.HaJIH3 Ha. rHnepKOMUJieKCHhiX CHCTeMaX C KOMUaKTHhiM H .I{HCKpeTHhiM 68.3HCOM. CneKTp&JibHII.II TeOpHJI onepaTOpos H 6ecKoHe'IHOMepHhlit aHanH3. HH. MaT. AH YCCP KHeB {1984), 19-32; translated in: Harmonic Analysis on Hypercomplex systems with a compact and discrete basis. Sel. Math. Sov. 10{1991), 181-193. [174) JI.H. Ba.itHepMaH (1.1. V&nennan): IlpHHuHn .ABoitcTBeHHOCTH .AJIJI KOHe'IHhiX rHnepKoMnneKCHhiX CHCTeM. H3B. sy3os.MaT. 29-2{1985), 12-21; translated in: A duality principle for finite hypercomplex systems. Soviet Math. {lz. VUZ) 292{1985), 13-24. [175) JI.H. BaituepMB.H (1.1. V&nennan): .llsoitcTBeHHOCTb anre6p c HHBonrouHeit H onepaTOphl o6o6IQ;eHHoro C.ABHra. HTOrH HayKH H Tex. MaTeM. AHan. 24(1986), 165-205; translated in: Duality of algebras with an involution and generalized shift operators. J. Soviet Math. 42{1988), 2113-2138. [176) JI.H. BaitHepMaH {1.1. V&nennan): 06 a6cTpaKTHoit ljlopMyne IlnaHmepenJI H ljlopMyne o6paueHHJI. YKp. MaT. )1(. 41{1989), 1041-1047; translated in: Abstract Plancherel equation and inversion formula. Ukr. Mat. J. 41{1989), 891-896. [177) 1.1. V&nennan: Correlation between compact quantum group and Kac algebras. to appear in Adv. Soviet Math. [178) JI.H. BaitHepMaH (1. I. Valnerman): HeyHHMO.l(yJIJipHhle KBaHTOB&HHhle rHnepKOMnneKCHhle CHCTeMhl . .lloKn. AKa.l{. HayK YCCP Cep. A. 1989, 7-10. [179) JI.H. BaitHepMa.H, r.H. Kau (1.1. Va.lnerman, G.I. Kac): HeyHHMO.l(yJIJipHhle Kon~>neshle rpynnhl H anre6phl Xonljla-ljloH HeitMaua. .lloKn. AK&.A. HayK CCCP 211(1973), 1031-1034; translated in: Nonunimodular ring-groups and Hopfvon Neumann algebras. Soviet Math. Dokl. 14(1974), 1144-1148. [180) JI.H. Ba.AHepMaH, r.H. Ka.n (1.1. Va1nerman, G.I. Kac): HeyHHMO.I{yJIJIPHhle Kon~>neshle rpynnhl H a.nre6phl Xonljla-ljloH HeAMa.Ha. MaTeM. c6. 94{1974), 194225; translated in: Nonunimodular ring-groups and Hopf-von Neumann algebras. Math. USSR Sbomik 23{1974), 185-214. [181] JI.H. BaAHepMaH, A.A. KaniO)f(Hhlit (1.1. V&nerman, A.A. Kalyuzhnil): KsaHToBaHHhle rHnepKOMJieKCHhle CHCTeMhl. rpaHH'IHhle 3a.l{a'IH .I{JIJI .I{HijlljlepeHnHaJibHhiX ypasHeHHA. HH. MaT. AH YCCP KHeB (1988), 13-29; to be translated in Sel. Math. Sov. [182) JI.H. BaAHepMa.H, r.JI. JIHTBHHOB (1.1. Va.lnerman, G.1. 1itvinov): 4>opMyna IlnaHmepenJI H ljlopMyna o6paiQ;eHHJI .I{JIJI onepaTOpos o6o6ceHHoro IU.ABHra . .lloKn. AK&.A. HayK CCCP 257(1981), 792-795; translated in: The Plancherel formula and the inversion formula for generalized shift operators. Soviet Math. Dokl. 23{1981), 333-337. [183) JI.JI. BaKcMaH (1.1. Vaksman): q-AHanorH K031jlljJHnHeHTOB Kne6ma-rop.AaHa H anre6pa ljlyHKnHA Ha KBaHTosoJt rpynne SU(2) . .lloKn. AKa.A. HayK CCCP 306(1989), 269-272; translated in: q-analogues of Clebsch-Gordan coefficients and the algebra offunctions on the quantum group SU(2). Soviet Math. Dokl. 39{1989), 467-470. [184) JI.JI. BaKcMaH H JI.H. Koporo.ACKHA (1.1. Vaksman and 1.1. Korogodskil): Anre6pa orpB.HH'IeHHhiX ljlyHKnHJt Ha KB&HTOBOJt rpynue .I{BH)f(eHHJt IIJIOCKOCTH H q-aHanorH ljlyHKnHA BeccenJI . .lloKn. AKa.A. HayK CCCP 304{1989); translated in: An algebra of bounded functions on the quantum group of the motion of the plane, and q-analogues of Bessel functions. Soviet Math. Dokl. 39{1989}, 173-177. [185) JI.JI. BaKCMaH H H.C. CoJt6enbM&H {1.1. Vaksman and Ya.S. SoThel'man): Anre6pa ljlyHKnHA Ha KBaHTosoJt rpyuue SU{2). 4>yHKn. aHaJIH3 Hero upHn. 22-
254
[186]
[187] [188] [189] [190] [191] [192]
[193] [194] [195] [196] [197] [198] [199] [200] [201] [202] [203] [204] [205] [206]
Bibliography 3(1988), 1-14; translated in: Algebra of functions on the quantum group SU(2). Funct. Anal. appl. 22{1989), 170-181. JI.JI. BaKcMaH H .H.C. Coi:t6eJI&MaH (L.L. Vaksman and Ya.S. Solbel'man): Anre6pa cjlyHK~Hi:t HR KBRHTOBOi:t rpynne SU(n+1) H He'leTHOMepHbie KB&HTOBble ccJlepbi. Anre6paa aHaJIH3 2-5(1990), 101-120. Translated in: The Algebra of .Functions on the Quantum Group SU(n + 1), and odd-dimensional Quantum Spheres. Leningrad Math. J. 2(1991), 1023-1042. J.-M. Vallin: c•-algebres de Hopf et c•-algebres de Kac. Proc. London Math. Soc. 50(1985), 131-174. A. Van Daele: Continuous crossed product and type III von Neumann algebras. London Math. Soc. Lecture Note 31, Cambridge Univ. Press, 1978. A. Van Daele: A quantum deformation of the Heisenberg group. Preprint L. Van Heeswijck: Duality in the theory of crossed products. Math. Scand. 44(1979), 313-329. J.-L. Verdier: Groupes quantiques (d'apres V.G. Drinfel'd). Sem. Bourbaki n° 685. Asterisque 152-153{1987), 305-319. D. Voiculescu: Amenability and Kac algebras. Proceedings of the Colloque International "Algebres d'operateurs et leurs applications ala physique theorique" Marseille {1977) M.E. Walter: Group duality and isomorphisms of Fourier and Fourier-Stieltjes algebras from a w•-algebra point of view. Bull. Amer. Math. Soc. 76(1970), 1321-1325. M.E. Walter: w•-algebras and non-abelian harmonic analysis. J. of Funct. Anal. 11(1972), 17-38. M.E. Walter: A duality between locally compact groups and certain Banach algebras. J. Funct. Anal. 17{1974), 131-159. M.E. Walter: Dual algebras. Math. Scand. 58(1986), 77-104. A. Weil: L'integration dans les groupes topologiques et ses applications. Act. Sc. Ind. n° 1145. Hermann. Paris. 1953. J.G. Wendel: On isometric isomorphism of group algebras. Pacific J. of Math. 1(1951), 305-311. J.G. Wendel: Left centralizers and isomorphisms of group algebras. Pacific J. of Math. 2(1952), 251-261. S.L. Woronowicz: Twisted SU(2) group. An example of non-commutative differential calculus. Publ. RIMS 23{1987), 117-181. S.L. Woronowicz: Compact matrix pseudogroups. Comm. Math. Phys. 111(1987), 613-665. S.L. Woronowicz: Tannaka-Kreln duality for compact matrix pseudogroups. Twisted SU(N) group. Invent. Math. 93{1988), 35-76. S.L. Woronowicz: Differential calculus on compact matrix pseudogroups (quantum groups). Comm. Math. Phys. 122{1989), 125-170. S.L. Woronowicz: Unbounded elements affiliated with c•-algebras and non-compact quantum groups. Com. Math. Phys. 136{1991), 399-432. S.L. Woronowicz: A remark on compact matrix quantum groups. RIMS Kyoto, preprint 1990. S.L. Woronowicz and S. Zakrzewski: Quantum Lorentz group induced by the Gauss decomposition. Preprint.
Index
Abelian co-involutive Hopf-von Neumann algebra Achieved left Hilbert algebra Affiliated . . . Analytic element Antipode . . .
. . 1.2.5 2.1.1 (iii) 1.1.1 (ii) 2.1.1 (iv) . . 6.6.3
Bochner's theorem
4.4.4 (ii)
C*-algebra Coextension of a lHI-morphism Co-involutive Hopf-von Neumann algebra Commutant Kac algebra . Compact type Kac algebra Complex Hopf algebra . Conditional expectation Connection relations
1.1.1 (i) 5.2.3 1.2.5 2.2.5 6.2.1 6.6.6 2.1.8 (ii) 3.6.1
Discrete type Kac algebra Dual co-involutive Hopf-von Neumann algebra Dual Kac algebra Dual morphism Dual weight . Ernest algebra Ernest's theorem Extended positive part of M Extension of a lHI-morphism . Extension of a representation Eymard algebra . . . . Eymard's duality theorem Eymard's theorem
6.3.1 3.3.2 3.7.4 5.1.2 3.5.3 1.6.8 4.7.3 2.1.8 (i) 5.2.3 4.6.8 3.4.4 4.3.3 3.4.6
256
Index
Fourier algebra . . . . . Fourier-Plancherel mapping . Fourier-Plancherel transform Fourier representation . . . Fourier-Stieltjes algebra Fourier-Stieltjes representation Fourier transform Fundamental operator Generator
3.3.4 3.5.3 4.3.7 2.5.3 1.6.9 1.6.9
4.3.7 2.4.2 1.5.2
llll-morphism Haar weight Heisenberg bicharacter Heisenberg's commutation relation Heisenberg pairing operator Heisenberg's theorem Hopf algebra . . . . . *-Hopf algebra . . . . . Hopf-von Neumann algebra
1.2.6 2.2.1 4.6.6 (ii)
Induced von Neumann algebra Intrinsic group . .
1.1.1 (ii) 1.2.2
Johnson's theorem
4.6.2 4.6.1 6.6.6 6.6.6 1.2.1
5.6.9
Kac algebra . . . Kac-Paljutkin's theorem :K-morphism Kre'ln algebra Kre'ln's theorem Kronecker product Kubo-Martin-Schwinger condition Left Hilbert algebra . . . Left-invariant weight Left regular representation Modular automorphism group Operator-valued weight Opposite Kac algebra Pentagonal relation
4.6.7
.
2.2.5
6.6.5 5.1.1 6.4.2 6.4.6 1.4.1 2.1.1 (iv) 2.1.1 (iii) 2.2.1 . . 1.1.5 2.1.1 (iv) 2.1.8 (i) 2.2.5 2.4.4
Index
Peter-Weyl's theorem Plancherel weight Pontrjagin's duality theorem Positive definite elements Radon-Nykodim derivative of weights Reduced Kac algebra . . . . Reduced von Neumann algebra Representable . . . . . . .
257
6.2.7 (iv) . 3.6.11 4.3.8 . 1.3.1 . 2.1.1 (v) . . 2.2.6 1.1.1 {ii) 1.3.6
Square-integrable element of M. Standard von Neumann algebra Stone's theorem Strict l!ll-morphism . . . . . Sub-Kac algebra . . . . . Symmetric co-involutive Hopf-von Neumann algebra
2.1.6 1.1.1 (iii) 4.4.4 {iii) 5.3.1 2.2.7 1.2.5
'Thnnaka's theorem Takesaki's theorem Tatsuuma's theorem Tensor product of von Neumann algebras Tensor product of weights . . . . . Tensor product of operator-valued weights
6.2.8 4.2.4 4.7.4 1.1.1 {ii) 2.1.1 (iv) 2.1.8 (iii)
Unimodular Kac algebra
. 6.1.3
Von Neumann algebra
1.1.1 {i)
W* -algebra . . . . Walter's theorem on Fourier algebras Walter's theorem on Fourier-Stieltjes algebras ..... . Weight Weil's theorem Wendel's duality theorem Wendel's theorem
1.1.1 {i) . 5.5.12 . 5.6.10 2.1.1 (i) . 4.2.6 . 4.5.9 . 5.5.11