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EMS Textbooks in Mathematics EMS Textbooks in Mathematicsis a book series aimed at students or professional mathematicians seeking an introduction into a particular field. The individual volumes are intended to provide not only relevant techniques, results and their applications, but afford insight into the motivations and ideas behind the theory. Suitably designed exercises help to master the subject and prepare the reader for the study of more advanced and specialized literature. Jørn Justesen and Tom Høholdt, A Course In Error-Correcting Codes Peter Kunkel and Volker Mehrmann, Differential-Algebraic Equations. Analysis and Numerical Solution
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Markus Stroppel
Locally Compact Groups
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European Mathematical Society
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Author: Markus Stroppel Institut für Geometrie und Topologie Universität Stuttgart D-70550 Stuttgart Germany
2000 Mathematical Subject Classification (primary; secondary): 22D05, 22-01; 20E18, 22A25, 22B05, 22C05, 22D10, 22D45, 22F05, 12J10, 43A05, 54H15, 22A15
Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.ddb.de.
ISBN 3-03719-016-7 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2006 European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum FLI C4 CH-8092 Zürich Switzerland Phone: +41 (0)1 632 34 36 Email:
[email protected] Homepage: www.ems-ph.org Cover: The stones of the wall represent the fundamental building blocks in the theory of locally compact groups: finite (and thus compact) groups, and connected Lie groups (including the additive group of real numbers, and linear groups over the reals). From these blocks, general locally compact groups are built using constructions like cartesian products and projective limits. The commuting diagram describes a projective limit, thus indicating a concept central to the theory discussed in the book. Typeset using the author’s TEX files: I. Zimmermann, Freiburg Printed on acid-free paper produced from chlorine-free pulp. TCF °° Printed in Germany 987654321
Preface This book introduces the reader to the theory of locally compact groups, leading from the basics about topological groups to more involved topics, including transformation groups, the Haar integral, and Pontryagin duality. I have also included several applications to the structure theory of locally compact Abelian groups, to topological rings and fields. The presentation is rounded off by a chapter on topological semigroups, paying special respect to results that identify topological groups inside this wider class. In order to show the results from Pontryagin theory at work, I have also included the determination of those locally compact Abelian groups that are homogeneous in the sense that their automorphism group acts transitively on the set of non-trivial elements. A crucial but deep tool for any deeper understanding of locally compact groups is the approximation by Lie groups. The chapter on Hilbert’s fifth problem gives an overview. The chart following this preface gives a rough impression of the logical dependence between the sections. During my academic career, I have repeatedly lectured on topics from topological algebra. Apart from a regular seminar including topics from the field, I have given graduate courses on topological groups (1994/95), locally compact groups (1995/96), Pontryagin duality (1996/97), Haar measure (1997), and topological algebra (1999/2000). The present notes reflect the topics treated in these courses. A suitable choice from the material at hand may cover one-semester courses on topological groups (Sections 3, 4, 5, 6, 7, 8, 9, 10, 11), locally compact Abelian groups (Sections 3, 4, 6, 12, 14, 20, 21, 22, 23, 24), topological algebra (groups, rings, fields, semigroups: Sections 3, 4, 5, 6, 9, 10, 11, 26, 28, 29, 30, 31). I have tried to keep these notes essentially self-contained. Of course, as with any advanced topic, there are limits. A reader is supposed to have mastered linear algebra, but only a basic acquaintance with groups is required. Fundamental topological notions (topologies, continuity, neighborhood bases, separation, compactness, connectedness, filterbases) are treated in Section 1 and in Section 2. The more advanced topic of dimension is included only as a reference for the outline in Chapter H. The section about Haar integral draws (naturally) from functional analytic sources. Almost every section (with the systematic exception of those in Chapter H) is accompanied by exercises. These have been tested in class, but of course this is no guarantee that they will work well again with any other group of students. Every reader is advised to use the exercises as a means to check her understanding of the topics treated in the text. Occasionally, the exercises also provide further examples. A remark on the bibliography is in order. The present book is meant as a student text, and historic comments are kept to a minimum. We give references to the literature where we need results or techniques that are beyond the scope of this
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book. Suggestions for further reading would surely include the two volumes by E. Hewitt and K. A. Ross [15], [16]. Quite recent contributions are the impressive monographs by K. H. Hofmann and S. A. Morris [23], [24]. About locally compact abelian groups, we only mention the book by D. L. Armacost [2]. The notes by I. Kaplansky [35] treat abelian groups but go well beyond into the solution of Hilbert’s Fifth Problem. Topological fields are the subject of the monographs by N. Shell [57], S. Warner [65], W. Wiesław [68], and (under the pretext of doing num‘ ber theory) the one by A. Weil [67]. The theory of locally compact groups naturally incorporates deep results from Lie theory. Among the many books about that subject, we mention the ones by N. Bourbaki [4], S. Helgason [14], G. Hochschild [18], A. L. Onishchik and E. B. Vinberg [43], and V. S. Varadarajan [64]. The abstract notion of a topological group seems to appear first in a paper by F. Leja [38]. Historical hallmarks of the theory are the books by L. S. Pontrjagin [48] and A. Weil [66]. I was introduced to the theory of topological groups in a course at the university at Tübingen, given by H. Reiner Salzmann, during the summer term in 1987. That course started with an introduction to the basics (including subgroups, quotients, separation properties, and connectedness), general properties of locally compact groups (existence of open subgroups, extension properties), a discussion of topological transformation groups (leading to Freudenthal’s results about locally compact orbits of locally compact Lindelöf groups). The main part of that course consisted of a discussion of Pontryagin duality, its proof and several consequences, culminating in the classification of compact Abelian groups. Surely, these lectures contributed to my decision to take up research in mathematics. What impressed me deeply was the interplay of subtly interwoven theories (topology, group theory, functional analysis), leading to deep results, with applications in pure as well as applied mathematics. Later on, I had the opportunity to work with Karl Heinrich Hofmann at Darmstadt. During this lasting collaboration, the seed of fascination with the topics was cultivated, ripening into a full-grown addiction to the theory of locally compact groups and its various applications. In my own teaching, I try to pass on the beauty of the subject as well as the fascination that my academic teachers have instilled in me. I do hope that these notes help to advance this fascination. Many students and colleagues have read and criticized scripts accompanying my lecture courses, and parts of the present version. Explicitly, I wish to thank Andrea Blunck, Martin Bulach, Agnes Diller, Helge Glöckner, Jochen Hoheisel, Martin Klausch, Peter Lietz, and Bernhild Stroppel. The errors that remain are mine. Stuttgart, December 2005
Markus Stroppel
Logical Dependence between the Sections 89:;W ?>=< t 1 WWWWWWWW t WWWWW tt WWWWW tt t WWWWW tt WWW+ t t 89:; ?>=< 89:; 3 ?TTTT 2 tt gg?>=< g t g g T g t ? T g
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Contents
Preface
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A Preliminaries 1 Maps and Topologies . . . . . . . . . . . . . . . . . . . . . . . . 2 Connectedness and Topological Dimension . . . . . . . . . . . .
1 1 19
B Topological Groups 3 Basic Definitions and Results . . . . . 4 Subgroups . . . . . . . . . . . . . . . 5 Linear Groups over Topological Rings 6 Quotients . . . . . . . . . . . . . . . 7 Solvable and Nilpotent Groups . . . . 8 Completion . . . . . . . . . . . . . .
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C Topological Transformation Groups 91 9 The Compact-Open Topology . . . . . . . . . . . . . . . . . . . . 91 10 Transformation Groups . . . . . . . . . . . . . . . . . . . . . . . 99 11 Sets, Groups, and Rings of Homomorphisms . . . . . . . . . . . . 105 D The Haar Integral 12 Existence and Uniqueness of Haar Integrals . . . . . . . . . . . . 13 The Module Function . . . . . . . . . . . . . . . . . . . . . . . . 14 Applications to Linear Representations . . . . . . . . . . . . . . .
113 113 123 128
E Categories of Topological Groups 15 Categories . . . . . . . . . . . . . . . . . . . 16 Products in Categories of Topological Groups 17 Direct Limits and Projective Limits . . . . . . 18 Projective Limits of Topological Groups . . . 19 Compact Groups . . . . . . . . . . . . . . .
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143 143 147 156 165 169
F Locally Compact Abelian Groups 20 Characters and Character Groups . . . . . . . . . . . 21 Compactly Generated Abelian Lie Groups . . . . . . 22 Pontryagin’s Duality Theorem . . . . . . . . . . . . 23 Applications of the Duality Theorem . . . . . . . . . 24 Maximal Compact Subgroups and Vector Subgroups
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174 174 181 191 194 203
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Contents
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Automorphism Groups of Locally Compact Abelian Groups . . . 207 Locally Compact Rings and Fields . . . . . . . . . . . . . . . . . 212 Homogeneous Locally Compact Groups . . . . . . . . . . . . . . 230
G Locally Compact Semigroups 28 Topological Semigroups . . . . . . . . . . . . . . . . . . 29 Embedding Cancellative Directed Semigroups into Groups 30 Compact Semigroups . . . . . . . . . . . . . . . . . . . . 31 Groups with Continuous Multiplication . . . . . . . . . .
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242 242 250 254 259
H Hilbert’s Fifth Problem 32 The Approximation Theorem . . . . . . 33 Dimension of Locally Compact Groups 34 The Rough Structure . . . . . . . . . . 35 Notions of Simplicity . . . . . . . . . . 36 Compact Groups . . . . . . . . . . . . 37 Countable Bases, Metrizability . . . . . 38 Non-Lie Groups of Finite Dimension . . 39 Arcwise Connected Subgroups . . . . . 40 Algebraic Groups . . . . . . . . . . . .
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261 261 264 268 272 276 279 280 281 285
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Bibliography
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Index of Symbols
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Subject Index
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Chapter A
Preliminaries 1 Maps and Topologies This chapter introduces the basic notions from topology that are needed later on. An experienced reader will need it only as a reference to the notation used. Among many other books on general topology, the one by Dugundji [11] is a good source for the topological notions that we use here. Most of our maps will be applied from the right, and we will use exponential notation. Thus a map ϕ : A → B maps a ∈ A to a ϕ ∈ B, and composition with ψ : B → C will be written as a ϕψ = (a ϕ )ψ . We write N for the set of nonnegative integers (including 0). 1.1 Basic notions. A topology on a set X is a system T of subsets of X such that ∅ ∈ T and X ∈ T , and T is closed with respect to arbitrary unions and finite intersections. A topological space is a pair (X, T ), where X is a set and T is a topology on X. If the topology T is fixed by the context, we will also denote a topological space (X, T ) merely by the underlying set X. The elements of T are called open; a subset C of X is called closed if X C is open. For x ∈ X, we write Tx := {T ∈ T | x ∈ T }. Then N x := {N ⊆ X | ∃T ∈ Tx : T ⊆ N } is the filter of neighborhoods of x, and N := x∈X Nx is the system of all neighborhoods in X. The interior Y ◦ of a subset Y ⊆ X (in (X, T )) is the union of all open sets that are contained in Y . The closure Y of Y is the intersection of all closed sets that contain Y . A subset D ⊆ X is called dense in X if D = X. If (X, T ) is a topological space and Y is a subset of X, we have that T |Y := {Y ∩ T | T ∈ T } is a topology on Y , called the topology induced on Y (by T ). We also say that (Y, T |Y ) is a subspace of (X, T ). If (Z, Z) is a topological space and ε : Z → X is an injection, we call ε an embedding of (Z, Z) into (X, T ) if T |Z ε = {T ε | T ∈ Z}. Let X be a set, and let (Xα )α∈A be a family of topologies on X. Then it is easy to see that α∈A Xα is a topology on X. If S is any set of subsets of X, we find therefore a smallest topology T on X such that S ⊆ T . We call T the topology generated by S, and write S top := T . Conversely, if T is a topology on X and S is a set of subsets of X such that T = S top we say that S is a subbasis for T . A subbasis B for T is called a basis for T if F = B ∈ B | B ⊆ F for every finite subset F of B. In this case, the topology T is obtained from B by forming arbitrary unions. Note that, in particular, a subbasis of a topology T is a basis if it is closed with respect to finite intersections.
2
A Preliminaries
A set B of neighborhoods of a point x in a topological space (X, X) is called a neighborhood basis at x if every neighborhood of x contains an element of B. 1.2 Examples. (a) On the set R of real numbers, we use the usual metric to define ε-balls Bε (x) := {y ∈ R | |x − y| < ε}. Then B := {Bε (x) | x ∈ R, ε > 0} is a basis for a topology, to which we refer as the usual topology on R. For any natural num-
ber n, the euclidean norm (x1 , . . . , xn ) := x12 + · · · + xn2 defines ε-balls which in turn form a basis for the usual topology on Rn . It is a well known fact that for every vector space norm on Rn the ε-balls generate the same topology on Rn .
(b) More generally, let d be a metric on a set Z. Then the set {Bε (x) | x ∈ Z, ε > 0} of (open) ε-balls Bε (x) := {y ∈ Z | d(x, y) < ε} forms a basis for a topology. This topology is called the topology induced by the metric d. (c) If X is any set, then the set of all subsets of X is a topology on X, called the discrete topology. The set {∅, X} is also a topology on X, and called the indiscrete topology. 1.3 Definitions. (a) A topological space whose topology is induced by some metric as in 1.2 (b) is called a metrizable space. (b) A topological space (X, T ) is said to be first countable if for every point x ∈ X there is a countable neighborhood basis at x. (c) A topological space is called second countable if it has a countable basis. (d) A topological space X is called separable if there exists a countable dense subset, that is, a countable subset Y ⊆ X such that Y = X. We leave it as an exercise to show that every metrizable space is first countable, and that every separable metrizable space is second countable.
Continuity 1.4 Definition. Let (X, X) and (Y, Y) be topological spaces, and let ϕ : X → Y be a map. Then ϕ is called continuous (from (X, X) to (Y, Y)) if the pre← image T ϕ := {x ∈ X | x ϕ ∈ T } of T under ϕ belongs to X for each T ∈ Y. The map ϕ is called open if it maps open sets to open sets. Analogously, we have the notion of closed map. A bijection between topological spaces is called a
1. Maps and Topologies
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homeomorphism if it is continuous and has a continuous inverse. Obviously, this is equivalent to being a continuous open bijection, or a continuous closed bijection. We write Homeo((X, X), (Y, Y)) for the set of all homeomorphisms from (X, X) onto (Y, Y), and abbreviate Homeo((X, X)) := Homeo((X, X), (X, X)). 1.5 Lemma. Let (X, X) and (Y, Y) be topological spaces, and let ϕ : X → Y be a map. Then the following are equivalent: (a) The map ϕ : (X, X) → (Y, Y) is continuous. (b) For every closed subset C of (Y, Y), the pre-image C ϕ
←
is closed in (X, X).
(c) For every x ∈ X and every neighborhood N of x ϕ in (Y, Y), the pre-image ← N ϕ is a neighborhood of x in (X, X). ←
(d) There is a subbasis S for Y such that for each S ∈ S the pre-image S ϕ belongs to X. A map ϕ between topological spaces (X, X) and (Y, Y) is called continuous ← at x ∈ X if there is a neighborhood basis B at x ϕ in (Y, Y) such that N ϕ is a neighborhood of x for each N ∈ B. Using 1.5, one sees that a map ϕ between topological spaces is continuous exactly if it is continuous at every point. We provide a simple tool that will come handy when dealing with piecewise defined functions. 1.6 Lemma. Let X and Y be topological spaces, and let S1 and S2 be subsets of X such that X = S1 ∪ S2 . Consider maps ϕj : Sj → Y , and assume that x ϕ1 = x ϕ2 holds for every x ∈ S1 ∩ S2 . Then a map ϕ : X → Y is well defined by x ϕ = x ϕj ⇐⇒ x ∈ Sj . Moreover, we have: (a) If ϕ1 and ϕ2 are continuous and S1 and S2 are open, then ϕ is continuous. (b) If ϕ1 and ϕ2 are continuous and S1 and S2 are closed, then ϕ is continuous. Proof. Assume first that ϕ1 and ϕ2 are continuous and both S1 and S2 are open. If ← U is an open subset of Y , our definition of ϕ yields that the intersection U ϕ ∩ Sj ← ← ← ← coincides with the pre-image U ϕj . Therefore, the pre-image U ϕ = U ϕ1 ∪ U ϕ2 is open, and we have shown that ϕ is continuous. If ϕ1 and ϕ2 are continuous and both S1 and S2 are closed, we proceed analo← ← ← ← ← gously: we have that C ϕ = (C ϕ ∩ S1 ) ∪ (C ϕ ∩ S2 ) = C ϕ1 ∪ C ϕ2 is closed for any closed subset C of Y . Using 1.5, we see that ϕ is continuous. 2 1.7 Definitions. A topological space X is called locally euclidean if there is a natural number n such that every point has an open neighborhood which is homeomorphic to an open neighborhood in Rn .
4
A Preliminaries
We say that X is locally homogeneous if for every pair (x, y) of points in X there exist open neighborhoods U and V of x and y, respectively, and a homeomorphism from U to V mapping x to y. If one can take U = V = X for each pair (x, y), we say that X is homogeneous.
Products 1.8 Definition. Let (Xα )α∈A be an arbitrary family of sets (that is, a map from A to the class of all sets, mapping α ∈ A to Xα ). Then the set Xα∈A Xα := ϕ : A → α∈A Xα | ∀α ∈ A : α ϕ ∈ Xα is called the (cartesian) product of the family (Xα )α∈A . Mostly, we will use the special notation ϕα := α ϕ and ϕ = (ϕα )α∈A for ϕ ∈ Xα∈A Xα . If Xα is the same set X for each α ∈ A, we write X A := Xα∈A Xα . (If one regards a natural number n as the set of its predecessors n = {0, . . . , n − 1}, the usual notation Xn = {(x0 , . . . , xn−1 ) | xα ∈ X} fits neatly in here.) For each β ∈ A, the canonical projection πβ : Xα∈A Xα → Xβ is just evaluation at β; that is, ϕ πβ = ϕβ . 1.9 Lemma. The cartesian product of a family (Xα )α∈A of sets has the following universal property: (P)
For every set W and every family (ψα )α∈A of maps ψα : W → Xα there is : W → Xα∈A Xα such that ψπ α = ψα for each α ∈ A. a unique map ψ
Xα∈A Z XHHαRWRWWW
HH RRRRWWWWW πi H πj RR Wπk WWWW HH RRR RRR WWWWWWW+ H$ ( X XO i ψ 6 Xk · · · > j | | llll | l l ψ ψi ψ | j lll k |l|llll W
Proof. In fact, for w ∈ W we obtain that wψ has to map α to w ψα , and this 2 determines the map ψ. 1.10 Definition. Now let ((Xα , Xα ))α∈A be an arbitrary family of topological spaces. The product topology P on the cartesian product Xα∈A Xα is generated by the following subbasis S: for every α ∈ A we let πα denote the projection to the factor Xα , and let Sα be thesystem of all πα -pre images of elements of Xα . Then S := α∈A Sα . We write α∈A Xα := P and α∈A (Xα , Xα ) :=
1. Maps and Topologies
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( Xα∈A Xα , P ). The product of two spaces (X1 , X1 ) and (X2 , X2 ) is written (X1 , X1 ) × (X2 , X2 ) := α∈{1,2} (Xα , Xα ). 1.11 Lemma. Let (Y, Y) be a topological space, and let ((Xα , Xα ))α∈A be an arbitrary familyof topological spaces. Then a map ϕ : Y → Xα∈A Xα is continuous from (Y, Y) to α∈A (Xα , Xα ) exactly if ϕπβ : (Y, Y) → (Xβ , Xβ ) is continuous for each β ∈ A. ϕ / (Y, Y) S α∈A (Xα , Xα ) SSS SSS SSS πβ ϕπβ SSSSS ) (Xβ , Xβ ) Proof. Using 1.5 (d), this follows from the definition of the product topology.
2
1.12 Lemma. The product topology has the following universal property: For every topological space (W, W ) and every family (ψα )α∈A of contin(W, W ) → (Xα , Xα ) there is a unique continuous map (P) uous maps ψα : : (W, W ) → ψ α∈A (Xα , Xα ) such that ψπα = ψα for each α ∈ A. In fact, the product topology is characterized by this universal property: whenever T is a topology on Xα∈A Xα such that for each α ∈ A the map πα is continuous (from ( Xα∈A Xα , T ) to (Xα , Xα )), and that (P) holds with ( Xα∈A Xα , T ) instead of α∈A (Xα , Xα ) then T is the product topology.
) α∈A (Xα_ , X O α VYYYY
OOO VVVVVYYYYYYY YYYYY VVV O π V πi OO π Y OOO j VVVVVVVk YYYYYYYYYYYY VV+ YYY, ' (Xj , Xj ) (X , X ) (Xk , Xk ) · · · i i O 8 ψ hhhh3 h h rrr h r h h ψi rψj hhhhhψk rrhrhhhhh (W, W )
has been established in 1.9, and Proof. Existence and uniqueness of the map ψ continuity is a consequence of 1.11. If Tis a topology on Xα∈A Xα such that (P) holds with ( Xα∈A Xα , T ) instead of α∈A (Xα , Xα ) then the identity from α∈A (Xα , Xα ) to ( Xα∈A Xα , T ) is continuous; in fact, it coincides with the map π . Conversely, the identity from ( Xα∈A Xα , T ) to α∈A (Xα , Xα ) is continuous as well, for the same reason. 2 1.13 Lemma. Let ((Xα , Xα ))α∈A be a family of topological spaces. For every α ∈ A, let (Yα , Yα ) be a topological space, and let ηα : (Yα , Yα ) → (Xα , Xα ) be
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A Preliminaries
a continuousmap. Define the map ϕ : α∈A (Yα , Yα ) → (Xα , Xα ) by ϕ = πα ηα , where πα : α∈A (Yα , Yα ) → (Yα , Yα ) is the canonical projection. Then the following hold. (a) If ηα is injective for each α ∈ A then ϕ is a continuous injection. (b) If ηα is an embedding for each α ∈ A then ϕ is an embedding, as well. Proof. We already know that ϕ is continuous. Let ψα : α∈A (Xα , Xα ) → (Xα , Xα ) be the canonical projection. For y = (yα )α∈A and z = (zα )α∈A we have that ϕ = z ϕ implies y ηα = y ϕ ψα = z ϕ ψα = zηα for each α ∈ A. If η is injective, this y α α α means yα = zβ , and assertion (a) follows. ←ϕ Now fix β ∈ A and pick U ∈ Yβ . Then U πβ coincides with the intersection ← of U ηβ ψβ with the image of ϕ . Therefore, the map ϕ induces an open map onto its image, and assertion (b) is established. 2
Separation Properties 1.14 Definition. Let (X, X) be a topological space. We say that (X, X) or X belongs to the class T0
if for each pair (x, y) of different points of X there exists an open set containing only one of them.
T1
if for each pair (x, y) of different points of X there exists a pair of open sets (Ux , Uy ) ∈ Xx × Xy such that x ∈ / Uy and y ∈ / Ux .
T2
if for each pair (x, y) of different points of X there exists a pair of disjoint open sets (Ux , Uy ) ∈ Xx × Xy .
T3
if for each closed subset A ⊂ X and each point x ∈ X A there exists a pair of disjoint open sets (U, V ) such that A ⊆ U and x ∈ V .
T4
if for each pair of disjoint closed sets (A, B) there exists a pair of disjoint open sets (U, V ) such that A ⊆ U and B ⊆ V .
The spaces in class T2 are also called Hausdorff spaces, those in T2 ∩ T3 are called regular, and those in T2 ∩ T4 are called normal. The class T1 consists exactly of the spaces with the property that each oneelement subset is closed. A space X belongs to the class T2 exactly if the diagonal {(x, x) | x ∈ X} is closed in X2 , compare 1.16 below. The defining property of T3 may be rephrased as follows: each point has a neighborhood basis consisting of closed neighborhoods. The following is an easy consequence of the definitions.
1. Maps and Topologies
1.15 Lemma. We have inclusions T0 ⊃ T1 ⊃ T2 ⊃ T2 ∩ T3 ⊃ T2 ∩ T4 .
7 2
It is easy to construct examples of topological spaces that show that all these inclusions are proper, and that T3 T0 and T4 T0 ∪ T3 . The reader should be aware of the fact that some authors (including J. Dugundji [11]) include condition T2 in the definition of T3 , and of T4 . 1.16 Lemma. Let X and Y be topological spaces. (a) The space Y is Hausdorff exactly if the diagonal Y := {(y, y) | y ∈ Y } is closed in Y 2 . (b) Let ϕ and ψ be continuous maps from X to Y . If Y is Hausdorff then the set {x ∈ X | x ϕ = x ψ } is closed in X. (c) Let be a set of continuous maps from X to X. If X is Hausdorff then Fix(ϕ) := {x ∈ X | x ϕ = x} is closed in X for each ϕ ∈ , and Fix( ) := ϕ∈ Fix(ϕ) is closed, as well. Proof. If Y is Hausdorff then for each pair (y, z) ∈ Y 2 Y we find neighborhoods U of y and V of z such that U × V is contained in Y Y . Thus Y Y is open, and Y is closed. Conversely, if Y is closed, we find for each (y, z) ∈ Y 2 Y a neighborhood W in Y 2 such that W ∩ Y = ∅. According to the definition of the product topology, there are neighborhoods U of y and V of z such that U × V is contained in W ⊆ Y Y . This proves assertion (a). Let ϕ and ψ be continuous maps from X to Y . We obtain a continuous map π : X → Y 2 by putting x π = (x ϕ , x ψ ). Now apply assertion (a) to infer that ← the pre-image {x ∈ X | x ϕ = x ψ } = ( Y )π is closed in X if Y is Hausdorff. Therefore, assertion (b) holds. Finally, assertion (c) is an application of assertion (b) to the case ψ = idX . 2 There is an equivalent characterization of normal spaces, due to P. Urysohn. A proof of the following assertion may be found, for instance, in [11], Chap. VII, Th. 4.1. 1.17 Lemma. Let Y be a Hausdorff space. Then the following two properties are equivalent: (a) The space Y is normal. (b) For each pair of disjoint closed sets A, B ⊆ Y , there exists a continuous function ϕ : Y → [0, 1] such that Aϕ ⊆ {0} and B ϕ ⊆ {1}.
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A Preliminaries
Compactness 1.18 Definition. A topological space is called compact if every covering by a family of open sets contains a finite sub-covering. A topological space is called locally compact if every point x has a neighborhood basis consisting of compact neighborhoods. A topological space is called σ -compact if it is the union of a countable family of compact subspaces. Passing to complements, one easily sees that the defining property of compactness is equivalent to the following property: every family of closed subsets of a compact space with empty intersection contains a finite subfamily with empty intersection. Note that we do not include the Hausdorff property in our definition of compactness. 1.19 Lemma. Let X and Y be topological spaces, and let ϕ : X → Y be continuous. If C ⊆ X is compact then C ϕ is compact, as well. Proof. Without loss, we assume C = X and C ϕ = Y . If (Uα )α∈A is a covering ϕ← of C ϕ by open sets then (Uα )α∈A is a covering of C by open sets, and compactness ϕ← of C allows to pick a finite subset F of A with C = α∈F Uα . We conclude that ϕ←ϕ 2 (Uα )α∈F = (Uα )α∈F is a finite sub-covering of C ϕ , as required. 1.20 Lemma. (a) Every closed subspace of a compact space is compact. Conversely, every compact subspace of a Hausdorff space is closed. (b) Every closed subspace of a locally compact space is locally compact. There are locally compact subspaces of Hausdorff spaces that are not closed. (c) Every compact Hausdorff space is normal. (d) Every locally compact Hausdorff space is regular. (e) A Hausdorff space is locally compact exactly if every point has a compact neighborhood. In particular, compact Hausdorff spaces are locally compact. Proof. Let (C, C) be a compact space, and assume that B ⊆ C isclosed. If B is B) ∪ α∈A (Uα ), and covered by a family (Uα )α∈A of open sets, then C = (C we find a finite subset F ⊆ A such that C = (C B) ∪ α∈F (Uα ). This implies that B is covered by the finite family (Uα )α∈F . Now assume that (X, X) is a Hausdorff space and C is a compact subspace. If x ∈ X C, we find for each c ∈ C a pair of disjoint open sets (Uc , Vc ) ∈ Xc × Xx .
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The compact space C is covered (Uc )c∈C . Thus there exists a finite by the family subset F of C such that C ⊆ c∈F Uc . Now c∈F Vc is an open neighborhood of x that is disjoint to C. This completes the proof of assertion (a). Let (Y, Y) be a locally compact space, and consider a closed subspace D. Every point of D has a neighborhood basis consisting of compact sets in (Y, Y). The intersection of such a compact neighborhood with D is compact again by assertion (a). Thus we obtain a neighborhood basis of compact neighborhoods in D. An example of a non-closed locally compact subspace of a Hausdorff space isobtained by taking a convergent sequence without its limit; for instance, the subset n1 | n ∈ N {0} . in R. Now consider a compact Hausdorff space (C, C), and let A and B be disjoint closed subspaces. Then both A and B are compact. Fix an element a ∈ A. For every b ∈ B we find a pair of disjoint open Cb . There is a sets (Ub , Vb ) ∈ Ca × finite subset Fa of B such that B ⊆ Ta := b∈Fa Vb . Then Wa := b∈Fa Ub is an open neighborhood we find a finite subset E of A such of a. Since A is compact, that A ⊆ W := a∈E Wa . Now W and T := a∈E Ta are disjoint open sets that contain A and B, respectively. This proves assertion (c). In a locally compact Hausdorff space, every point has a neighborhood basis consisting of compact sets, which are closed by assertion (a). Thus assertion (d) follows. Finally, let (X, X) be a Hausdorff space, and assume that x ∈ X has a compact neighborhood C. Then C is regular by assertion (d). Since C is a neighborhood in (X, X), this gives a neighborhood basis at x of closed sets in (X, X), and asser2 tion (e) is established. 1.21 Closed Graph Theorem. Let X and Y be topological spaces, let ϕ : X → Y be a map, and let ϕ := {(x, x ϕ ) | x ∈ X} ⊆ X × Y be the graph of ϕ. (a) If Y ∈ T2 and ϕ is continuous then ϕ is closed in X × Y . (b) If X ∈ T2 and Y is compact the closedness of ϕ implies that ϕ is continuous. Proof. In order to prove assertion (a), let ψ : X × Y → Y × Y be defined by (x, y)ψ := (x ϕ , y). Then ψ is continuous, and our assumption Y ∈ T2 implies that ψ←
y and ϕ = y are closed in Y × Y and in X × Y , respectively. Now assume that X ∈ T2 , that Y is compact, and that ϕ is closed in X × Y . ← We are going to show that every closed subset C of Y has closed pre-image C ϕ ← in X. To this end, consider a point a ∈ X C ϕ . For each c ∈ C, we then have (a, c) ∈ / ϕ . As ϕ is closed, we find open neighborhoods of Uc of a in X and Vc of c in Y such that Uc × Vc has empty intersection with the graph ϕ. As C is compact, there is a finite subset F ⊆ C such that C ⊆ c∈F Vc . The intersection U := c∈F Uc is an open neighborhood of a in X. In order to see that ← U has empty intersection with C ϕ , we assume to the contrary that there is some
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A Preliminaries
u ∈ U such that uϕ ∈ C. Then there is some c ∈ F with uϕ ∈ Vc , and the graph ϕ contains (u, uϕ ) ∈ U × Vc ⊆ Uc × Vc . This contradicts our choice of Uc and Vc . ← We have shown that the complement of C ϕ in X is open. Thus assertion (b) is established. 2 1.22 Definition. A set F of nonempty subsets of a set X is called a filterbasis in X if for each pair (F, G) ∈ F 2 there exists H ∈ F such that H ⊆ F ∩ G. The filter generated by F is F fil := {Y ⊆ X | ∃F ∈ F : F ⊆ Y } If X is a topology on X, we say that the filterbasis F converges to x ∈ X if every neighborhood of x contains a member of F . 1.23 Lemma. Let (C, C) be a compact space,and assume that F is a filterbasis in C consisting of closed subsets. Then S := F ∈F F is not empty. If S consists of a single element c ∈ C then F converges to c. Proof. Assume that S is empty. Then C = F ∈F C F and we find a finite subset E of F such that C = E∈E C E because (C, C) is compact. As F is a filterbasis, there exists H ∈ F with H ⊆ E∈E E = ∅, a contradiction. Now assume S = {c}, and let U be a neighborhood of c. Then the set G = {F U ◦ | F ∈ F } consists of closed subsets of (C, C). The intersection over all members of G satisfies G∈G G ⊆ S U ◦ = ∅. Passing to complements, we obtain a covering of the compact space C U ◦ by open subsets, and conclude ◦ that there is a finite subset E of F such that E∈E E U F is a is empty. As ◦ filterbasis, this means that we find H ∈ F such that H ⊆ E∈E E ⊆ U . 2 1.24 Example. Let X be a set, and let S := {T ⊆ X | X T is finite}. Then S top = S ∪ {∅} is the so-called co-finite topology on X. If X is infinite then the co-finite topology on X belongs to T1 T2 . It is compact, and every subspace is compact. However, every proper closed subset is finite. Thus not every compact subspace is closed. 1.25 Lemma. Every locally compact Hausdorff space X has the following property (known as “complete regularity” ): for every closed subset A ⊆ X and every point p in X A, there is a continuous function ϕ : X → [0, 1] such that pϕ = 1 and Aϕ ⊆ {0}. Proof. We indicate the main idea, and leave the details as an exercise. Pick compact neighborhoods V and W of p such that V ⊆ W ◦ and A∩W = ∅. As the space W is compact Hausdorff, it is normal, and we find a continuous function ϕ : W → [0, 1] such that {p}ϕ = {1} and (W V ◦ )ϕ ⊆ {0}. Now it remains to note that the extension ϕ : X → [0, 1] mapping each element of X W to 0 is continuous. 2 1.26 Corollary. Every locally compact Hausdorff space belongs to the class T3 .
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Proof. Let X be a locally compact Hausdorff space, and consider a closed subset A ⊂ X and a point p ∈ X A. According to 1.25, we pick a continuous function ϕ that Aπ ⊆ {0} and p ϕ = 1. The pre-images of the intervals
:X → [0,
11] such 1 0, 2 and 2 , 1 under ϕ are disjoint open sets with the properties required here. 2 1.27 Corollary. Let X be a locally compact Hausdorff space. For every compact C ⊆ X and every open U ⊆ X such that C ⊆ U , there is a continuous function ϕ : X → [0, 1] such that C ϕ ⊆ {1} and (X U )ϕ ⊆ {0}. Proof. Put A := X U . For every c ∈ C, we find a continuous function ϕc : X → [0, 1] such that Aϕ ⊆ {0} and cϕc = 1. As C is compact, we find elements
ϕ ← c1 , . . . , cn such that C is contained in the union nj=1 21 , 1 cj . Now x ϕ := ϕ 2 inf 1, n2 nj=1 x cj defines a function ϕ as required. 1.28 Definition. A subset A of a topological space is called nowhere dense if the closure of A has empty interior in X. A topological space X is called meager if it is the union of a countable family of nowhere dense subspaces. 1.29 Lemma. Let X be a nonempty locally compact Hausdorff space. Then X is not meager. Proof. Let (An )n∈N be a sequence of closed subspaces of a locally compact space X, and assume that each An has empty interior. Then Un := X An is open in X, and Un = X. Since X is locally compact, we find a compact neighborhood C0 . The intersection C0◦ ∩ U0 is not empty, and we find a compact neighborhood C1 ⊆ compact C0◦ ∩ U0 . Proceeding inductively, we obtain a decreasing sequence of neighborhoods Cn+1 ⊆ Cn◦ ∩ Un . Since C0 is compact, the intersection n∈N Cn is not empty; compare 1.23. Thus X n∈N An = n∈N Un ⊇ n∈N Cn is not empty. We have shown that X is not meager. 2 We need a characterization of compact metric spaces. The following definitions will be convenient (see Section 8 for a more general notion of completeness). 1.30 Definition. Let (X, d) be a metric space. (a) The space (X, d) is called pre-compact if it has the following property: for every ε > 0, there is a finite subset F of X such that X ⊆ f ∈F Bε (f ). (b) A sequence (xn )n∈N in X is called a Cauchy sequence if for ε > 0 there is a natural number Nε such that n, m > Nε implies d(xn , xm ) < ε. The metric space (X, d) is called complete if every Cauchy sequence in X converges. Note that pre-compact spaces are often called “totally bounded”. The name “pre-compact” is justified by the following two observations.
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A Preliminaries
1.31 Lemma. Let (X, d) be a metric space, and consider a subspace Y . If Y is pre-compact then its closure Y is pre-compact, as well. Proof. Fix ε > 0, and pick a finite subset F of Y such that Y ⊆ f ∈F B 2ε (f ). For x ∈ Y , we find y ∈ Y such that d(x, y) < 2ε . Then there exists z ∈ F such that 2 d(y, z) < 2ε , and we obtain Y ⊆ f ∈F Bε (f ), as required. 1.32 Proposition. Let (X, d) be a metric space. If (X, d) is complete and precompact then (X, d) is compact. Proof. Assume, to the contrary, that there is a family U = (Uα )α∈A of open subsets Uα ⊆ Xsuch that X = α∈A Uα but X = α∈B Uα for any finite subset B ⊂ A. As X is pre-compact, we find finite subsets Fn ⊆ X such that X = f ∈Fn B2−n (f ); for every natural number n. Then there exists x1 ∈ F1 such that B1 := B2−1 (x1 ) is not covered by any finite subfamily of U. Now we proceed inductively: if xn ∈ Fn and Bn are chosen, we pick xn+1 ∈ Fn+1 such that Bn+1 := B2n+1 (xn+1 ) has nonempty intersection with Bn , and cannot be covered by a finite subfamily of U. Then d(xn , xn+1 ) < 2−n + 2−n−1 < 21−n holds because Bn and Bn+1 have a point in common. For n, m > N we may assume m > n, and obtain d(xn , xm ) < kj =n 21−j < 21−n j ≥0 2−j = 22−n < 22−N . This shows that the sequence (xn )n∈N is a Cauchy sequence. Since X is complete, this sequence converges to some z ∈ X. We pick α ∈ A such that z ∈ Uα , and ε > 0 such that Bε (z) ⊆ Uα . Then we find a natural number n such that d(xn , z) < 2ε and 21−n < ε. But this entails Bn ⊆ Uα , contradicting our choice of Bn . 2 The following deep result is of importance for the construction of compact spaces (and groups!). It depends on the axiom of choice; in fact, it is equivalent to it. A proof can be found in [36], Chapter 5, Theorem 13, p. 143. Proofs for the special case of Hausdorff spaces (which is the only case that is really needed in this book) can be found in most books on general topology, for instance in [11], Chapter XI, 1.4. 1.33 Tychonoff’s Theorem: Products of compact spaces are compact. Let ((Xα , Xα ))α∈A be an arbitrary family of compact spaces. Then the product space
Xα∈A Xα , α∈A Xα is compact.
Quotient Topologies and Quotient Maps 1.34 Definition. Let (X, X) be a topological space, and let ϕ : X → Y be a ← surjection. We put X/ϕ := {T ⊆ Y | T ϕ ∈ X}, and call X/ϕ the quotient topology (with respect to ϕ).
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A surjection ψ : (X, X) → (Y, Y) between topological spaces is called a quotient map if Y = X/ψ. 1.35 Lemma. (a) Every quotient map is continuous. (b) Every open (or closed) continuous surjection between topological spaces is a quotient map. (c) Let ϕ : (X, X) → (Y, Y) be an open (or closed) continuous map. If A ⊆ X is ← ϕ-saturated (that is, if A = Aϕϕ ), then ϕ induces a quotient map ψ : A → Aϕ . (d) Let ϕ : (X, X) → (Y, Y) be a quotient map. If A ⊆ X is ϕ-saturated and open or closed then ϕ induces a quotient map ψ : A → Aϕ . (e) Let f : X → Y be a surjection, and let X and Y be topologies on X and Y , respectively. Then f is a quotient map from (X, X) onto (Y, Y) exactly if the following condition is satisfied: (Q)
For every topological space (Z, Z) and for each map g : Y → Z, continuity of f g implies continuity of g, and vice versa. XO A=
ϕ
?
← Aϕϕ
ϕ|A
2, YO ,2 ? ϕ A
Regarding assertion (c).
X@ @@ @@ f @@
fg
Y
/Z ? g
Regarding condition (Q).
Proof. Assertions (a) and (b) are obvious. In order to prove assertions (c) and ← ← (d), observe first that B ψ = B ϕ for each B ⊆ Aϕ . We have to show that Y|Aϕ = (X|A )/ψ. The inclusion Y|Aϕ ⊆ (X|A )/ψ is obvious because ψ is continuous. In order to prove the reverse inclusion, consider U ∈ (X|A )/ψ. Then ← there is an open set V in X such that V ∩ A = U ψ . If ϕ is an open map, we obtain V ϕ ∈ Y and conclude U = V ϕ ∩ Aϕ ∈ Y|Aϕ . If ϕ is a closed map, we observe ← A U ψ = A V and apply similar reasoning to the closed set X V . If A is closed or open in X, we can replace V and X V by the ϕ-saturated open set V ∪ (X A) and the ϕ-saturated closed set X (A ∪ V ), respectively, and argue as above without using that ϕ is open or closed. It remains to prove assertion (e). Let f : (X, X) → (Y, Y) be a quotient map, and let g : Y → Z be a map. If g is continuous with respect to some topology Z on Z, the map f g is continuous as well. Conversely, assume that f g is continuous. For every U ∈ Z we then have ← ← ← ← that (U g )f = U (f g) belongs to X. Thus U g ∈ Y, and g is continuous.
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A Preliminaries
Finally, assume that f : (X, X) → (Y, Y) has the property (Q), consider (Z, Z) = (Y, X/f ) and let g be the identity on Y . Then f g : (X, X) → (Y, X/f ) is a quotient map, and continuous by assertion (a). Moreover, we obtain that g −1 : (Y, X/f ) → (Y, Y) is continuous, because quotient maps have property (Q). Our assumption that f has the property (Q) yields that g : (Y, Y) → (Y, X/f ) is continuous; that is, X/f = Y. 2 1.36 Corollary. Let ϕ : (C, C) → (H, H) be a continuous map from a compact space (C, C) to a Hausdorff space (H, H ) such that C ϕ = H . Then ϕ is a quotient map. Proof. Every closed subset of C is compact. Thus ϕ maps closed subsets of C to compact subsets of H . Since H is Hausdorff, this means that ϕ is a closed map. In particular, we have C ϕ = C ϕ = H . Now the assertion follows from 1.35 (b). 2 1.37 Example. The map ψ : [0, 3] → {0, 1, 2} defined by ⎧ ⎪ ⎨0 if x ∈ [0, 1], ψ x = 1 if x ∈ ]1, 2[, ⎪ ⎩ 2 if x ∈ [2, 3], shows that quotient maps need not be open, and need not preserve any of the properties T1 – T4 . A similar construction yields a quotient of [0, 3] which does not belong to T0 .
Continuity, Revisited There are situations where plain open or closed sets are too clumsy to deal with. If the topologies are induced by metrics, a familiar criterion for continuity of maps uses convergent sequences. In the general case, sequences do not suffice to describe continuity: more precisely, this happens for spaces that are not first countable. We proceed to give a generalization of sequences. 1.38 Definition. A directed set (J, ) is a nonempty set J with a pre-order (that is, a reflexive and transitive binary relation on J ) such that for any two elements i, j ∈ J there exists k ∈ J satisfying i k and j k. A net (with domain (J, ), or just J if is assumed to be known) in a topological space (X, T ) is a map a : J → X. Usually, one writes this map as a = (aj )j ∈J , interpreting it as an element of XJ as in 1.8. Each net a ∈ XJ defines a filterbasis T (a) := {Tj (a) | j ∈ J } in X with terminal sets Tj (a) := {an | j n ∈ J }: in fact, for i, j ∈ J we pick k ∈ J satisfying i k and j k and obtain Tk (a) ⊆ Tj (a) ∩ Tj (a).
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We say that a net a ∈ XJ accumulates at x ∈ X if for every neighborhood U of x in (X, T ) and every j ∈ J we have U ∩ Tj (a) = ∅. The net a converges to x if every neighborhood U of x contains some terminal set of a. The difference (and the similarity) of these definitions becomes clear if one rewrites them using logical quantifiers: a accumulates at x a converges to x
⇐⇒ ⇐⇒
∀U ∈ Nx ∀i ∈ J ∃j i : aj ∈ U, ∀U ∈ Nx ∃i ∈ J ∀j i : aj ∈ U.
If a converges to x then x is called the limit of a, and we write x = lim a = limj ∈J aj . 1.39 Examples. Perhaps the simplest nontrivial example of a directed set is given by (N, ≤), where ≤ denotes the usual order relation. Nets with domain (N, ≤) are called sequences. We leave it as an exercise to verify that convergence of sequences in the well known sense is the same as convergence of these sequences, considered as nets. Every filterbasis F defines a directed set (F , ⊆). In particular, this applies to the filterbasis Nx of all neighborhoods of a point x in a topological space (X, T ). Simple comparison of the definitions yields the following. 1.40 Lemma. Let (X, T ) be a topological space, let x ∈ X, and let a : (J, ) → X be a net. Then a converges to x if, and only if, the filterbasis T (a) converges to x. 1.41 Lemma. Let (X, X) and (Y, Y) be topological spaces, let ϕ : X → Y be a map, and let a : J → X be a net in X. Then a ϕ : J → Y is a net in Y (mapping ϕ j ∈ J to aj ). If ϕ : (X, X) → (Y, Y) is continuous then the following hold. (a) If a accumulates at x ∈ X then a ϕ accumulates at x ϕ . (b) If a converges to x ∈ X then a ϕ converges to x ϕ . Conversely, one can show that ϕ is continuous if every convergent net is mapped to a net converging to the image of the limit under ϕ. However, we will not use that fact. Proof of 1.41. Let U be a neighborhood of x ϕ in (Y, Y). As ϕ is continuous, the ← pre-image U ϕ is a neighborhood of x in (X, X). ← Let i ∈ J . If a accumulates at x, we find j ∈ J such that i j and aj ∈ U ϕ . ϕ This means aj ∈ U , and we see that a ϕ accumulates at x ϕ . ← If a converges to x then there exists some j ∈ J with Tj (a) ⊆ U ϕ . This ϕ ϕ ϕ 2 implies Tj (a ) ⊆ U , and we obtain that a converges to x . 1.42 Lemma. Let (X, T ) be a topological space, and let A be a subset of X.
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A Preliminaries
(a) For each x ∈ A there exists a net in A that converges to x in (X, T ). (b) If there exists a net in A that accumulates at x in (X, T ) then x belongs to A. Proof. Assume first x ∈ A. Then every neighborhood U of x contains some element aU of A. This defines a net a := (aU )U ∈Nx with domain (Nx , ⊆), compare 1.39. Quite obviously, this net converges to x since U contains the terminal set TU (a). Conversely, assume that there exists a net b : J → A such that b accumulates at x. Pick any j ∈ J . Then every neighborhood of x contains some member of Tj (b) ⊆ A, and x ∈ A. 2 1.43 Lemma. Let C be a compact space, and let a : J → C be a net. Then there is some point x ∈ C such that a accumulates at x. Proof. The filterbasis T (a) associated with the net a gives another filterbasis F := {T | T ∈ T (a)}. Since F consists of closed subsets of a compact space, we know from 1.23 that F ∈F F contains some point x. In order to show that a accumulates at x, we consider a neighborhood U of x in C and an element j ∈ J . Then x ∈ Tj (a) 2 implies U ∩ Tj (a) = ∅. 1.44 Lemma. Let (Xα , Xα )α∈A be a family of topological spaces, and let P := α∈A (Xα , Xα ) be the product space, with projections πβ : P → (Xβ , Xβ ). Let x = (xα )α∈A be a point in P , and let a : J → P be a net. (a) The net a converges to x in P if, and only if, each of the nets a πβ converges to xβ in (Xβ , Xβ ), where β runs over A. (b) If a accumulates at x then each of the nets a πβ accumulates at xβ in (Xβ , Xβ ). (c) If there is an index γ ∈ A such that for each α ∈ A {γ } the net a πα converges to xα in (Xα , Xα ) and a πγ accumulates at xγ in (Xγ , Xγ ) then a accumulates at x in P . Proof. If a accumulates at x in P then a πβ accumulates at xβ = x πβ because πβ is continuous, see 1.41. If a converges to x then a πβ converges to xβ , again by 1.41. Thus we have proved assertion (b) and one half of assertion (a). Conversely, consider a neighborhood U of x in P . Then U contains a neighborhood of the form Xα∈A Uα , where Uα is a neighborhood of xα in (Xα , Xα ), and there is a finite set F ⊆ A such that Uα = Xα holds for all α ∈ A F . If a πβ converges to xβ then we find jβ ∈ J such that Tjβ (a πβ ) ⊆ Uβ . As the domain (J, ) of a is directed and F is finite, we find k ∈ J such that jβ k holds for each β ∈ F . This implies Tk (a) ⊆ Tjβ (a), and we obtain Tk (a) ⊆ Xα∈A Uα ⊆ U , as required for convergence of a to x. This completes the proof of assertion (a). Now assume that a πβ converges to xβ , for each β ∈ A {γ }, and that a πγ accumulates at xγ . Moreover, let i ∈ J be given. For each β ∈ F , we pick jβ ∈ J
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17
such that Tjβ (a πβ ) ⊆ Uβ . Again, directedness of (J, ) and finiteness of F ∪ {i} yield the existence of k ∈ J such that i k and Tk (a πβ ) ⊆ Uβ holds for each β ∈ F . As a πγ accumulates at xγ , we have Uγ ∩Tk (a πγ ) = ∅, and U ∩Tk (a) = ∅ 2 follows. 1.45 Example. In fact, the convergence assumption in 1.44 (c) is needed: for instance, consider X0 := {−1, 1} and X1 := {−1, 1} with the discrete topologies. The net a : N → X0 × X1 given by an := ((−1)n , (−1)n ) accumulates at each of the points (1, 1) and (−1, −1), but neither at (1, −1) nor at (−1, 1). However, the net a π0 accumulates at 1 (and at −1), and a π1 accumulates at −1 (and at 1).
Extension of Continuous Maps 1.46 Lemma. Let (X, X) and (Y, Y) be topological spaces such that (Y, Y) ∈ T2 ∩ T3 , let D be a dense subset of X, and let ϕ : (D, X|D ) → (Y, Y) be a continuous map. Then the following are equivalent: (a) There is a continuous map : (X, X) → (Y, Y) such that |D = ϕ. (b) For each x ∈ X, there exists xˆ such that for each filterbasis F in D converging to x, the filterbasis D ϕ := {F ϕ | F ∈ F } converges to x. ˆ (c) For each x ∈ X, there exists xˆ such that for each net a : J → D converging to x, the net a ϕ : J → Y converges to x. ˆ Proof. Clearly, assertion (a) implies assertions (b) and (c): the image of the filterbasis and the net converge to xˆ := x , cf. 1.41. Conversely, we have to show that : X → Y : x → xˆ is well defined, and continuous. Uniqueness of xˆ follows from our assumption that Y is Hausdorff. Thus the extension exists (and is in fact unique, cf. 1.16 (b)). In order to show continuity of at x under assumption (b), it suffices to consider closed neighborhoods of x, ˆ because Y ∈ T3 . Applying (b) to the filterbasis {D ∩ U | U ∈ Tx }, we find for any given closed neighborhood V of xˆ an open neighborhood U of x such that (U ∩ D)ϕ ⊆ V . For each u ∈ U , the filterbasis {D ∩ T | T ∈ Xu , T ⊆ U } converges to u, and assumption (b) yields u ∈ (U ∩ D)ϕ and thus U ⊆ V , as required. 2 The proof that assertion (c) implies continuity of is similar. An important application of 1.46 will be given in 8.38 below, in the context of completions of uniform spaces.
18
A Preliminaries
Exercises for Section 1 Exercise 1.1. Let T be a topology on X, and let Y be a subset of X. Show that T |Y is a topology. Exercise 1.2. Let d be a metric on a set Z. Show that {Bε (x) | x ∈ Z, ε > 0} forms a basis for a topology. Exercise 1.3. Show that every metrizable space is first countable, and that every separable metrizable space is second countable. Exercise 1.4. Let ϕ : X → Y be a surjection, and let T be a topology on X. Show that T /ϕ is a topology on Y . Give examples where {T ϕ | T ∈ T } is not a topology. Exercise 1.5. Prove Lemma 1.15. Exercise 1.6. Find examples of topologies that show that each of the classes T3 T4 , T2 T3 , T1 T2 , T0 T1 , and T3 T2 is nonempty. Exercise 1.7. Consider the interval [0, 2π ] and the circle C := {(x, y) ∈ R2 | x 2 + y 2 = 1} with the topologies induced by the usual topologies on R and on R2 , respectively. Show that the map ϕ : [0, 2π] → C defined by x ϕ = (cos x, sin x) is a quotient map, but not open. Show that the restriction of ϕ to ]0, 2π ] is surjective, but not a quotient map. Exercise 1.8. Show that the product of two compact spaces is compact, without using the axiom of choice. Exercise 1.9. Show that the usual topology on Rn is locally compact and Hausdorff, but not compact for n ∈ N {0}. Exercise 1.10. Show that the usual topology on Rn coincides with the product topology, if we identify Rn with a product of n factors, all homeomorphic to R with the usual topology. Exercise 1.11. Let X be a topological space. For any A ⊆ X show that A◦ = X (X A), and A = X (X A)◦ . Exercise 1.12. Prove that the closure of A in a topological space X consists exactly of those points x ∈ X that have the following property: every neighborhood of x meets A. Exercise 1.13. Let X and Y be topological space, and endow X × Y with the product topology. Show A × B = A × B for any choice of A ⊆ X and B ⊆ Y . Exercise 1.14. Fill in the details in the proof of 1.25. Exercise 1.15. Prove the “net version” of 1.46.
2. Connectedness and Topological Dimension
19
2 Connectedness and Topological Dimension Connectedness 2.1 Definition. A topological space (X, X) is called connected if ∅ and X are the only closed open subsets of X. Equivalently, there is no pair of disjoint proper closed subsets (Y, Z) such that X = Y ∪ Z. 2.2 Remark. We leave as an exercise the proof of the following suggestive characterization of connectedness: a topological space (X, X) is connected exactly if it has the property that every continuous map from (X, X) to a discrete space is constant. 2.3 Lemma. Let X be a topological space, and assume that (Cα )α∈Ais a family of connected subspaces with nonempty intersection. Then the union α∈A Cα is connected. Proof.Pick an element c ∈ α∈A Cα , and a closed open subset Y of the union U = α∈A Cα . Replacing Y by U Y , if necessary, we achieve c ∈ Y . Now for each α ∈ A, the intersection Y ∩ Cα is a nonempty closed open subset of Cα , and coincides with Cα . Thus Y contains each of the Cα , and coincides with U . 2 2.4 Definition. For every a ∈ X, we have by 2.3 a maximal connected subset Xa containing a. We refer to Xa as the connected component of a in X. A topological space is called totally disconnected if every connected component consists of a single point. 2.5 Lemma. Let X and Y be topological spaces, and assume that ϕ : X → Y is continuous and surjective. Then connectedness of X implies that Y is connected. ←
Proof. Let A be a nonempty closed open subset of Y . Then Aϕ shares these ← properties, since ϕ is a continuous map. Now connectedness of X implies Aϕ = X ϕ and A = X = Y . 2 2.6 Lemma. Let ((Xα , Xα ))α∈A be a family of nonempty topological spaces. Then α∈A (Xα , Xα ) is connected exactly if each (Xα , Xα ) is connected. Proof. If α∈A (Xα , Xα ) is connected then (Xα , Xα ) is connected by 2.5, since the natural projection is continuous and surjective. Conversely, assume that (Xα , Xα ) is connected for each α ∈ A. For each π← y y = (yα )α∈A ∈ P := Xα∈A Xα and each β ∈ A put Sβ := α∈A{β} yα α . Then y Sβ is homeomorphic to Xβ , and therefore connected. Now let Y be a nonempty y closed open subset of P . For y ∈ Y we then have that Y ∩ Sβ is a nonempty closed y y open subset of Sβ , and infer Sβ ⊆ Y . This means that we can change the value of
20
A Preliminaries
the function y ∈ Y at finitely many arguments without leaving Y . As Y is open in the product topology, one finds a finite subset F ⊆ A such that each function differing from y only at members of F belongs to Y . This implies Y = P . 2 2.7 Lemma. Let ((Xα , Xα ))α∈A be a family of topological spaces, and let x = (xα )α∈A be an element of Xα∈A Xα . For each α ∈ A, let Cα be the connected component of xα in (Xα , Xα ). Then Xα∈A Cα is the connected component of x in α∈A (Xα , Xα ). Proof. We denote the connected component of x in α∈A (Xα , Xα ) by C, and the topology induced on Cα by Cα . For the sake of readability, we abbreviate D = Xα∈A Cα . As the natural projection πα is continuous, we infer using 2.5 π← that C πα ⊆ Cα . Therefore,we have C ⊆ α∈A Cα α = D. Now endow D with the product topology C = α∈A Cα , then (D, C) is connected by 2.6. According to 1.13, the inclusion map ι : (D, C) → α∈A (Xα , Xα ) is an embedding. In particular, we have that D = D ι is connected. This means D ⊆ C, and we have shown D = C. 2 2.8 Lemma. Let (X, T ) be a topological space, and assume that Y ⊆ X is closed and open. Then Y = y∈Y Xy is the union of connected components. Proof. For each y ∈ Y the intersection Xy ∩ Y is a nonempty closed open subset of Xy . Thus Xy = Xy ∩ Y ⊆ Y . 2 2.9 Lemma. Let (X, X) be a topological space. (a) For every connected subspace C of X the closure C is connected. (b) Every connected component is closed. (c) Every totally disconnected space belongs to T1 . (d) Let ϕ : X → Y := {Xa | a ∈ X} be the map that maps every point to its connected component. Then (Y, X/ϕ) is totally disconnected. Proof. Let C be connected, and consider y ∈ C. Then every neighborhood of y meets C. If C ∪ {y} were disconnected, we could find disjoint nonempty closed subsets Z and Y of C ∪ {y} such that y ∈ Y and C ∪ {y} = Z ∪ Y . Then Y is a neighborhood of y in C ∪ {y}, and Y ∩ C = ∅. Now Z = Z ∩ C and Y ∩ C are nonempty disjoint closed subsets of C, and C = Z ∪ (Y ∩ C), contradicting our assumption that C is connected. This yields assertion (a). Assertions (b) and (c) are immediate consequences. In order to prove assertion (d), consider a connected subset C of Y . By asser← tion (a), we may assume that C is closed. Then ϕ induces a quotient map from C ϕ ← onto C, see 1.35 (e). If A is a nonempty closed open subset of C ϕ , we note that
2. Connectedness and Topological Dimension
21
every connected set that meets A is already contained in A. Thus A is ϕ-saturated, ← and Aϕ is closed and open in C. This implies that Aϕ = C and A = C ϕ . We ← obtain that C ϕ is connected, and C has at most one element. 2 2.10 Lemma. Let (C, C) be a Hausdorff space. (a) If (C, C) is compact then, for each c ∈ C, the connected component Cc coincides with the intersection of all closed open sets containing c. (b) If (C, C) is locally compact and totally disconnected then the compact open sets form a basis for C. Proof. Assume first that (C, C) iscompact. Let S denote the set of all closed open sets containing c, and put D := S. Then D is closed in C. For each S ∈ S, we have Cc ⊆ S by 2.8. Thus Cc ⊆ D. We show that D is connected; this implies Cc = D and completes the proof of assertion (a). Assume that A is closed and open in the space (D, C|D ), and that c belongs to A. Then both A and D A are closed in C. As (C, C) ∈ T4 by 1.20 (c), wefind an open subset Uof C such that A ⊆ U and U ∩ (D A) = ∅. Thus S ∩ (U U ) | S ∈ S = D ∩ (U U ) = ∅. Since D is compact, we find finitely many S1 , . . . , Sn ∈ S such that S1 ∩ · · · ∩ Sn ∩ (U U ) = ∅. Then S := S1 ∩ · · · ∩ Sn belongs to S, and S ∩ U = S ∩ U is closed and open in C. From c ∈ A ⊆ U we infer S ∩ U ∈ S, and D ⊆ S ∩ U ⊆ U . We obtain D A ⊆ U and thus D = A. In order to prove assertion (b), assume now that (C, C) is locally compact and totally disconnected, and consider a neighborhood U of c. We have to find a compact open set V such that c ∈ V ⊆ U . Without loss, we may assume that U is open, and that the closure U is compact. According to assertion (a), we find / Vx c. Now for each x ∈ U {c} a closed open set Vx ⊆ U such that x ∈ V U ⊆ U U = ∅, and V U is closed in the compact space U U . x x∈U U x Therefore, we find a finite setF ⊂ C such that x∈F Vx U = ∅. This means that the compact open set V := x∈F Vx satisfies our requirement c ∈ V ⊆ U . 2 2.11 Remark. Assertion (c) of 2.10 does not extend to locally compact Hausdorff spaces, as the following
example shows: For each positive integer i, consider the subset Si := 1i , y | −1 ≤ y ≤ 1 of R2 . Let C be the union of all these Si plus the set S0 := {(0, y) | −1 ≤ y ≤ 1, y = 0}. Then C is closed in R2 {(0, 0)}. Being open in R2 , the latter set is locally compact, so C is locally compact by 1.20 (b). However, the connected component of (0, 1) in C clearly is Cc = {(0, y) | 0 < y ≤ 1} while the intersection of all closed open subsets of C containing c is S0 = Cc . We will see in 6.8 how 2.10 (c) does extend to locally compact Hausdorff groups.
22
A Preliminaries
Pathwise Connectedness 2.12 Definition. Let X be a topological space. A path from x to y in X is a continuous map ϕ : [a, b] → X such that a ϕ = x and bϕ = y, where a and b are real numbers, the interval [a, b] is endowed with the topology induced from the usual topology on R. A topological space Y is called pathwise connected if there exists a path from each point of Y to each other point of Y . The path component of x in X is the maximal pathwise connected subspace of X containing x. 2.13 Lemma. Let X and Y be topological spaces, and assume that ϕ : X → Y is continuous and surjective. Then pathwise connectedness of X implies that Y is pathwise connected. Proof. Let y0 and y1 be points in Y . As ϕ is surjective, we find x0 , x1 ∈ X such ϕ that xi = yi . If α : [0, 1] → X is a path joining x0 and x1 then αϕ : [0, 1] → Y is a path joining y0 and y1 . 2 2.14 Lemma. Let (Xα )α∈A be a family of nonempty topological spaces. Then the space α∈A Xα is pathwise connected exactly if each Xα is pathwise connected. Proof. If α∈A Xα is pathwise connected then Xα is pathwise connected by 2.13, since the natural projection is continuous and surjective. Conversely, assume that Xα is pathwise connected for each α ∈ A. For elements (xα )α∈A and (yα )α∈A of Xα∈A Xα , pick paths ϕα : [0, 1] → Xα from xα to yα in Xα . Then the map ϕ : [0, 1] →Xα∈A Xα defined by t ϕ = (t ϕα )α∈A is a path joining (xα )α∈A and (yα )α∈A in α∈A Xα . 2 2.15 Remark. Some authors use the terms “arcwise connected” and “arc component” instead of “pathwise connected” or “path component”, respectively. As an arc is a subspace homeomorphic to the unit interval; that is the image of an embedding rather than a merely continuous map, the notion of arcwise connectedness is stronger than that of pathwise connectedness: for instance, the topology {∅, {0, 1}} on {0, 1} is pathwise connected but not arcwise connected. However, the two notions coincide for Hausdorff spaces, see [20], Theorem 3-24, Theorem 3-17.
2. Connectedness and Topological Dimension
23
Topological Dimension For deeper investigations into the structure of locally compact groups, we need a notion of (topological) dimension. Mainly, we will use the so-called small inductive dimension, denoted by ind. 2.16 Definition. Let X be a topological space. We write ind X = −1 if, and only if, X is empty. If X is nonempty, and n is a natural number, then we write ind X ≤ n if, and only if, for every point x ∈ X and every neighborhood U of x in X there exists a neighborhood V of x such that V ⊆ U and the boundary ∂V = V V ◦ satisfies ind ∂V ≤ n − 1. Finally, let ind X denote the minimum of all n such that ind X ≤ n; if no such n exists, we say that X has infinite dimension. Obviously, ind X is a topological invariant. If ind X is finite then X belongs to T3 because every neighborhood of any point x ∈ X contains a closed neighborhood. A nonempty space X satisfies ind X = 0 if, and only if, there exists a neighborhood basis consisting of closed open sets. Consequently, ind X = 0 implies that X is totally disconnected. See [30] for a thorough study of the properties of ind for separable metrizable spaces. Although it is quite intuitive, our dimension function does not work well for arbitrary spaces. Other dimension functions, notably covering dimension ([47], 3.1.1), have turned out to be better suited for general spaces, while they coincide with ind for separable metrizable spaces. See [47] for a comprehensive treatment. Note, however, that small inductive dimension coincides with large inductive dimension and covering dimension, if applied to locally compact groups [1], [45]. The duality theory for compact Abelian groups uses covering dimension rather than inductive dimension. We will prove the equality in 36.7. We collect some important properties of small inductive dimension. The following assertion is an easy exercise, if one uses the fact that a totally disconnected locally compact Hausdorff space has a basis consisting of compact open sets, see 2.10. 2.17 Proposition. Let X be a locally compact Hausdorff space. Then X is totally 2 disconnected if, and only if ind X = 0. 2.18 Sum Theorem for small inductive dimension. Let X be a separable metrizable space. If (An )n∈N is a countable family of subsets of X then ind n∈N An = supn∈N ind An . Proof. See, for instance, [42], p. 14.
2
2.19 Theorem. For every natural number n, we have ind Rn = n. Proof. See [30], Th. IV 1, or [47], 3.2.7 in combination with [47], 4.5.10.
2
24
A Preliminaries
2.20 Lemma. Let X be a nonempty regular topological space. (a) For every subspace Y of X, we have ind Y ≤ ind X. (b) ind X ≤ n holds exactly if every point x ∈ X has some neighborhood Ux such that ind Ux ≤ n. (c) ind X ≥ n holds exactly if there exists a point x ∈ X with arbitrarily small neighborhoods of dimension at least n. (d) If X is locally homogeneous, then ind X = ind U for every neighborhood U in X. (e) If X is locally compact, then ind X = 0 exactly if X is totally disconnected. (f) If X is a product of a family of nonempty finite discrete spaces, then ind X = 0. (g) If ind X = 0, then ind(Rn × X) = n for every natural number n. Proof. We prove assertion (a) by induction on ind X. In fact, there is nothing to do for ind X = −1 and in the case where X has infinite dimension. Let x be a point in Y ⊆ X, and let U be a neighborhood of x in Y . Then there is a neighborhood U of x in X such that U = Y ∩ U . We find a neighborhood W of x in X such that W ⊆ U and ind ∂W < ind X. Now V := Y ∩ W ◦ is an open neighborhood of x in Y whose closure C in Y satisfies C ⊆ Y ∩ W ⊆ U and C V ⊆ Y ∩ (W W ) ⊆ ∂W . Our induction hypothesis yields ind (C V ) ≤ ind ∂W , and we obtain ind Y ≤ ind X. Assertions (b) and (c) are immediate consequences of the definition and our assumption X ∈ T3 , and they imply assertion (d). By assertion (b), it suffices to prove assertion (e) for compact spaces. A compact Hausdorff space has at every point a neighborhood basis of closed open sets exactly if it is totally disconnected, see 2.10. Assertion (f) follows immediately from assertion (e) since the product of a family of finite discrete spaces is compact Hausdorff and totally disconnected. Finally, assume ind X = 0. We proceed by induction on n. If n = 0, then ind(R0 × X) = ind X = 0. So assume n > 0, and ind Rn−1 × X = n − 1. Let U be a neighborhood of (r, x) in Rn × X. Since ind X = 0, there exists a closed open set V in X and a ball B around r in Rn such that (r, x) ∈ B × V ⊆ U . Since the boundary ∂(B × V ) is contained in ∂B × V , we infer from our induction hypothesis that ind U ≤ n. The subspace Rn × {x} of Rn × X has dimension n. This completes 2 the proof of assertion (g). 2.21 Remark. Assertion 2.20 (d) applies, in particular, to locally euclidean spaces, and to homogeneous spaces of topological groups; see 6.3 below. Note that, if a space X is not locally homogeneous, it may happen that there exists a point x ∈ X with the property that ind U < ind X for every sufficiently small neighborhood U of x. For instance, consider the disjoint union of R and a singleton, where both R and the singleton are open, and R carries the usual topology.
2. Connectedness and Topological Dimension
25
2.22 Halder’s Lemma. Assume that the topological space X ∈ T3 is the countable union of neighborhoods Un such that for every natural number n we have that U n is compact, and ind Un = d. Let Y be a separable metrizable space. If ϕ : X → Y is a continuous injection, then d = ind X = ind Xϕ ≤ ind Y . Proof. Small inductive dimension is defined locally, whence ind X = ind Un for ϕ every n. Since Un is compact, we obtain that Un and Un are homeomorphic, and ϕ ϕ ind Un = ind Un . Now we have ind Xϕ = ind Un by the Sum Theorem 2.18. Finally, monotony of ind yields ind Xϕ ≤ ind Y . 2
Exercises for Section 2 Exercise 2.1. Prove the assertion of 2.2. Exercise 2.2. Show that the connected subsets of R (with its usual topology) are exactly the intervals. Explain how this yields the Intermediate Value Theorem. Exercise 2.3. Determine the connected components in the spaces R, Q, RQ (each with the topology induced from the usual topology on R) and in the spaces R2 {(0, 0)}, R2 (R×{0}) (with the topology induced from the usual topology on R2 ). Exercise 2.4. Show that R and any open interval in R are homeomorphic (with the topologies induced by the usual one). Exercise 2.5. Show that R and R2 are not homeomorphic (with the usual topologies). Hint. Study connected components of complements of points. Exercise 2.6. Verify 2.11 in detail. Exercise 2.7. Show ind R = 1. Exercise 2.8. Let Sn be the unit sphere in Rn+1 (with respect to the usual euclidean metric). Show ind Sn = ind Rn . Exercise 2.9. Use induction on n to show ind Rn ≤ n for each natural number n. (Equality holds, but is hard to prove. Where is the problem?) Exercise 2.10. Prove that a locally compact Hausdorff space X is totally disconnected exactly if ind X = 0.
Chapter B
Topological Groups 3 Basic Definitions and Results Before we introduce the notion of topological group (and other structures) by a bulk of formal definitions, let us have a look at one of the fundamental examples: the additive group of real numbers. We consider the usual topology T on R (which is generated by the familiar basis B = {]x − ε, x + ε[ | x, ε ∈ R, ε > 0}, consisting of open intervals), and notice that addition is continuous “in both arguments simultaneously”. Formalizing the latter assertion, we introduce a map μ : R2 → R by putting (x, y)μ = x+y, and observe that μ : (R, T )2 → (R, T ) is continuous. This is considerably stronger than the assumption that the maps λx : (R, T ) → (R, T ) and ρy : (R, T ) → (R, T ) defined by y λx = (x, y)μ = x ρy are continuous for all x, y ∈ R. While μ is far from being injective, the maps λx and ρy are. This means that the equations (a, y)μ = b and (x, a)μ = b have unique solutions x, y ∈ R for all a, b ∈ R. We now observe that these solutions depend continuously on a and b. Generalizing this fundamental example, we obtain the notion of a topological group: 3.1 Definitions. For our purposes, it will be convenient to consider a group as an algebra (in the sense of universal algebra) (G, μ, ι, ν), where G is a set endowed with a binary operation μ : G×G → G, a unary operation ι : G → G and a nullary operation (that is, a constant) ν : {0} → G. The group axioms then say that the following equations hold (for all a, b, c ∈ G): (a, 0ν )μ = a = (0ν , a)μ (a ι , a)μ = 0ν = (a, a ι )μ
(two-sided neutral element) (two-sided inverses)
and
(a, b)μ , c
μ
μ = a, (b, c)μ
(associativity).
G × G ×NG NNN p Nid NN×μ p NNN p p p N& p xp G × GN G×G NNN p p p NNN pp pppμ μ NNNN p p N& xpp G μ×idpppp
3. Basic Definitions and Results
27
A subgroup of a group (G, μ, ι, ν) is a substructure (H, μ , ι , ν), where H is a subset of G, and μ , ι are the restrictions of μ to H × H and ι to H , respectively. In other words: a subgroup of G is a subset which is closed under the group operations; in particular, every subgroup contains the neutral element. Every subgroup is a group. Mostly, we use multiplicative notation for groups: the operation μ is called multiplication and written as xy := (x, y)μ ; and x −1 := x ι is called the inverse of x. In this case, we write 1 := 0ν . If the group G is commutative; that is, if (x, y)μ = (y, x)μ for all pairs (x, y) ∈ G × G, we prefer additive notation: the operation μ is called addition and written as x + y := (x, y)μ ; and we write −x := x ι and 0 := 0ν . Commutative groups are called Abelian as well. Fearing cumbersome notation more than confusion by sparse notation, we will often denote a group just by the underlying set G rather than by (G, μ, ι, ν). Sometimes we will indicate just the set and the binary operation; in fact, the group axioms imply that the remaining operations are uniquely determined. A topological group is given as (G, μ, ι, ν, T ), where (G, μ, ι, ν) is a group, and T is a topology on G such that μ : (G, T ) × (G, T ) → (G, T ) and ι : (G, T ) → (G, T ) are continuous. Subgroups of topological groups are usually endowed with the induced topology; this makes the subgroup a topological group, again. We introduce some related concepts. 3.2 Definitions. A semigroup is a nonempty set S with an associative binary operation μ : S × S → S. If S is a topological space and μ is continuous, we speak of a topological semigroup. For the construction of important examples we also need the notions of topological ring and topological field. A ring is an algebra (R, α, σ, ζ, μ, ν), where ζ and ν are nullary operations (the zero and the neutral element), and α and μ are binary operations (addition and multiplication) such that (R, α, σ, ζ ) is a group and μ is associative. Moreover, one requires that the following equations hold for all a, x, y ∈ R:
μ
μ a, 0ν = a = 0ν , a (multiplicative unit)
α μ μ μ α = (a, x) , (a, y) a, (x, y)
μ
α α (x, y) , a = (x, a)μ , (y, a)μ (distributivity). Usually, the operations α and μ are written additively, resp. multiplicatively. Accordingly, one writes 0 = 0ζ and 1 = 0ν . Note that no commutativity assumptions are made here. Using distributivity and the multiplicative unit, it is easy to show that α is commutative. If μ is commutative as well, we say that the ring is commutative. A field is an algebra (F, α, σ, ζ, μ, ι, ν), where (F, α, σ, ζ, μ, ν) is a ring such that 0ζ = 0ν , and the restrictions of μ, ι and ν induce a group on F {0ζ }.
28
B Topological Groups
Let (R, α, σ, ζ, μ, ν) be a ring. Then a (right) R-module (or a module over R) is a group (M, α, σ, ¯ ζ¯ ) with an operation μ¯ : M × R → M called multiplication by scalars (from the right) satisfying the following equations for all m, n ∈ M and all r, s ∈ R: ((m, n)α , r)μ¯ = ((m, r)μ¯ , (n, r)μ¯ )α , (m, (r, s)α )μ¯ = ((m, r)μ¯ , (m, s)μ¯ )α , (m, (r, s)μ )μ¯ = ((m, r)μ¯ , s)μ¯ , (m, 0ν )μ¯ = m. Similarly, one defines left modules over R, where the scalars operate from the left: the crucial difference occurs in the equation ((r, s)μ , m)μ¯ = (r, (s, m)μ¯ )μ¯ . ¯ ζ¯ ) is written additively, and μ¯ is denoted by juxtaUsually, the group (M, α, σ, position like multiplication. With the analogous conventions for rings, our equations take the suggestive form (m + n)r = mr + nr, m(r + s) = mr + ms, m(rs) = (mr)s, m1 = m. Again, it is a consequence of the definition that the addition α in a module is commutative. If F is a field, then (right/left) modules over F are also called (right/left) vector spaces. A topological ring is a ring (R, α, σ, ζ, μ, ν) together with a topology T on R such that the operations α, σ and μ are continuous. A topological field is a field (F, α, σ, ζ, μ, ι, ν) with a topology T such that (F, α, σ, ζ, μ, ν, T ) is a topological ring and the restriction of ι to F {0ζ } is continuous. Finally, a topological module is a topological group which is a module over a topological ring such that multiplication by scalars is a continuous map, and a topological vector space is a topological module over a topological field. 3.3 Definition. A (locally) compact group (G, μ, ι, ν, T ) is a topological group where T is (locally) compact. Similarly, we define (locally) compact semigroups, rings, fields, modules, and vector spaces.
Basic Examples 3.4 Examples. Rather trivial examples of topological groups are the discrete ones; they are of course locally compact. In particular, we have the infinite cyclic group
3. Basic Definitions and Results
29
(Z, +) and the finite cyclic groups Z(n) := Z/nZ (obtained by addition “modulo n” on the set {0, . . . , n − 1}) for n ∈ N {0}. Via infinite products (compare 3.35), the latter provide examples of infinite compact (and thus non-discrete) groups. 3.5 Examples. With the usual topology, the additive group (R, +) and the multiplicative group (R× , ·) are locally compact Hausdorff groups; here R× := R {0} carries the induced topology. The set C of complex numbers can be identified with R2 ; thus we obtain the natural topology on C. Again, the additive group (C, +) and the multiplicative group (C× , ·) are locally compact Hausdorff groups. If R is a topological ring, then the ring R n×n of all n × n matrices with entries in R is a topological ring, if it is endowed with the product topology (here 2 we identify R n×n with R (n ) ). If R is a commutative topological ring such that inversion is continuous on the set of invertible elements, then Cramer’s rule shows that the inverse of a matrix with nonzero determinant may be expressed by addition, multiplication and inversion in R. Thus the group GL(n, R) of all invertible n × n matrices with entries in R is a topological group. The additional assumption on R is satisfied, for instance, if R is a topological commutative field. The subgroup SL(n, R) of all matrices with determinant 1 is another interesting topological group. We will prove in Section 5: for each topological ring K such that the set K × of invertible elements is open and inversion is continuous in K × , the group GL(n, K) is a topological group. For special results in the locally compact case, compare Section 31, as well. 3.6 Lemma. (a) If R is a (locally) compact ring and n is a natural number then R n×n is a (locally) compact ring. (b) If R is a commutative Hausdorff ring then SL(n, R) is closed in R n×n and therefore (locally) compact if R is (locally) compact. (c) If F is a commutative Hausdorff field then GL(n, F ) is open in F n×n . As a consequence, this group is locally compact exactly if F is. Proof. If R is (locally) compact then the product topology turns R n×n into a (locally) compact space. Over a commutative Hausdorff ring, the inverse image SL(n, R) of the set {1} under the continuous determinant function is closed in R n×n . Over a commutative Hausdorff field the same reasoning shows that the complement of GL(n, F ) in F n×n is closed. 2 3.7 Examples. Let F be a field. The center of F is C := {c ∈ F | ∀f ∈ F : f c = cf }. It is easy to see that C is a commutative subfield of F . We can consider F as a vector space over C, with scalars from C operating from the left. If the dimension d := dimC F of F over C is finite, we can interpret each F -linear map
30
B Topological Groups
ϕ from F k to F l as a C-linear map ϕC : C dk → C dl . In particular, we have an embedding of GL(n, F ) in GL(dn, C). If C is a topological field, this means that F n×n becomes a topological ring, and that GL(n, F ) becomes a topological group. Specializing n = 1, we also obtain that F is a topological field. The group GL(n, F ) coincides with the centralizer of the scalar multiplication of elements of F n by elements of F : GL(n, F ) = {ϕ ∈ GL(dn, C) | ∀f ∈ F ∀v ∈ F n : (f v)ϕ = f (v ϕ )}. Thus GL(n, F ) is closed in GL(dn, C) if F is Hausdorff. If, in particular, the field C is locally compact Hausdorff, we have that F is a locally compact Hausdorff field and that GL(n, F ) is a locally compact Hausdorff group. The deep fact that every non-discrete locally compact Hausdorff field has finite dimension over its center (cf. 26.43) justifies our restriction to the case where dimC F is finite. A detailed analysis of an algorithm used to obtain the inverse of a matrix in a suitable neighborhood of 1 leads to a proof of the fact that GL(n, K) is a topological group for each Hausdorff field K, irrespective of commutativity, see Section 5 below. 3.8 Examples. Another series of important examples of locally compact groups is obtained as follows. For each natural number n, we define the (real) orthogonal group as O(n, R) := {M ∈ Rn×n | MM = 1}, where M denotes the transpose of M. The special orthogonal group SO(n, R) is defined as the intersection of O(n, R) with SL(n, R). These are closed subgroups of GL(n, R), and therefore locally compact Hausdorff groups. In fact, they are even compact. Passing from R to C, we write M ∗ for the matrix obtained from M ∈ Cn×n by transposition and complex conjugation of all entries, and define the unitary group as U(n, C) := {M ∈ Cn×n | MM ∗ = 1} and the special unitary group SU(n, C) := U(n, C) ∩ SL(n, C). Again, these groups are even compact. As an important special case, we also mention the group U(1, C), which can easily be identified with {c ∈ C | cc¯ = 1}. For rather obvious reasons, this group is called the circle group.
Easy Topological Properties of Topological Groups 3.9 Lemma. Let (G, μ, ι, ν) be a group, and let T be a topology on G. Then the following are equivalent: (a) (G, μ, ι, ν, T ) is a topological group. (b) The map α : (G, T ) × (G, T ) → (G, T ) defined by (x, y)α := (x, y ι )μ is continuous. (c) The map β : (G, T ) × (G, T ) → (G, T ) defined by (x, y)β := (x ι , y)μ is continuous.
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It may be instructive to use multiplicative notation for the maps α and β: one has (x, y)α = xy −1 and (x, y)β = x −1 y. Proof of 3.9. Obviously, continuity of α or β implies continuity of ι. If ι is continuous then the maps γ and δ defined by (x, y)γ = (x, y ι ) and (x, y)δ = (x ι , y) are homeomorphisms of (G, T ) × (G, T ). The equations α = γ μ, β = δμ and μ = γ α = δβ yield that continuity of one of the maps μ, α, β implies continuity of the other two. 2 3.10 Lemma. Let (G, T ) be a topological group. For each g ∈ G the maps ρg , λg , and ig , defined by x ρg := (x, g)μ , x λg := (g, x)μ , and x ig := ((g ι , x)μ , g)μ , respectively, are homeomorphisms from (G, T ) onto (G, T ). Proof. The maps ρg and λg may be considered as restrictions of μ, and are therefore continuous. The map ig = λg ι ρg = ρg λg ι is then continuous as well. 2 3.11 Corollary. Assume that B is a neighborhood basis at g ∈ G and that x is an arbitrary element of G. Put r = (g ι , x)μ and l = (x, g ι )μ . Then {U ρr | U ∈ B} and {U λl | U ∈ B} are neighborhood bases at x. 3.12 Corollary. If there exists a compact neighborhood in (G, T ) then T is locally compact. Proof. We will see later in 6.6 that every group topology T belongs to the class T3 . Thus an element of G has a neighborhood basis of compact sets if it has one compact neighborhood. The homeomorphisms ρg then show that every point in G shares 2 this property. 3.13 Corollary. If G is a locally compact group such that {1} is closed then either G is discrete or G is uncountable. Proof. Assume that G is a countable locally compact group, and that {1} is closed in G. As G = {{g} | g ∈ G} is not meager, one of the (countably many) sets {g} has nonempty interior. Thus every one-element subset of G is open, and G is discrete. 2
Closure of Subsets in Topological Groups 3.14 Definition. Let (G, μ, ι, ν) be a group. A subset S ⊆ G is called normal if S ig = S for each g ∈ G. As ig is a homeomorphism, we obtain:
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B Topological Groups
3.15 Lemma. The closure of a normal subset of a topological group is a normal subset. From now on, we use multiplicative notation for our groups, unless stated otherwise. Let G be a group. For elements x, y ∈ G and subsets X, Y ⊆ G we write Xy x ∈ X}, xY := {xy | y ∈ Y } and XY := {xy | x ∈ X, y ∈ Y } = := {xy | xY = x∈X y∈Y Xy. If G is a topological group and X is open in G then 3.10 implies that Xy, yX, XY and Y X are open for each y ∈ G and each Y ⊆ G. Similarly, if Y is closed in G, then Y x and xY are closed. However, XY need not be closed, even if both X and Y are closed. 3.16 Example. We consider the group of real numbers, written additively, with its usual topology. Let r be an irrational real number. Then the subgroups Z and rZ are both closed in R. However, the subgroup Z + rZ is not closed. 3.17 Lemma. Let G be a topological group, and let B be a neighborhood basis at 1 in G. For each subset X of G we have X = B∈B XB = B∈B BX and also X = B∈B XB = B∈B BX. Proof. An element g ∈ G belongs to X exactly if gB meets X for each B ∈ B. This means that g ∈ B∈B XB, and we have shown that X = B∈B XB. Analogously, one sees that X = B∈B BX. Continuity of multiplication implies that the set C := {BC | B, C ∈ B} is a neighborhood basis at 1. Thus B∈B XB = 2 B∈B C∈B (XB)C = D∈C XD = X. 3.18 Lemma. Let G be a topological group. Assume that U ⊆ G is open and that C ⊆ U is compact. Then there exists a neighborhood V of 1 in G such that V C ∪ CV ⊆ U . Proof. For each c ∈ C, we find an open neighborhood Wc of 1 such that Wc c and cWc are contained in U . By continuity of multiplication, we find open neighborhoods Vc of 1 such that Vc Vc ⊆ Wc . The compact set C is covered by the )c∈C . Therefore we find a finite set F ⊆ C such that families (Vc c)c∈C and (cVc C ⊆ f ∈F Vf f and C ⊆ f ∈F f Vf . We put V := f ∈F Vf . For each c ∈ C, we find f ∈ F such that c ∈ Vf f . Then V c ⊆ V Vf f ⊆ Vf Vf f ⊆ Wf f ⊆ U . 2 Analogously, we see that cV ⊆ U . 3.19 Lemma. Let G be a topological Hausdorff group, and assume that B ⊆ G is closed and C ⊆ G is compact. Then BC and CB are closed. Proof. For x ∈ G BC we have that B ι x is closed and disjoint to C. According to 3.18, we find a neighborhood V of 1 such that B ι x ∩ CV = ∅. But then xV ι is a neighborhood of x disjoint to BC. Closedness of CB may be proved analogously, or using CB = (B ι C ι )ι . 2
3. Basic Definitions and Results
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3.20 Lemma. Let (G, μ, ι, ν, T ) be a topological group. Then μ is an open map. Proof. Let U be open in G × G, and fix (x, y) ∈ U . Then there is a pair of open subsets (X, Y ) ∈ Tx × Ty such that X × Y ⊆ U . The image (X × Y )μ = XY is open in G. 2
Local Characterization of Group Topologies 3.21 Lemma. Let (G, T ) be a topological group. If F is a neighborhood basis at 1 then the following hold. (FB) For all U, V ∈ F there exists W ∈ F such that W ⊆ U ∩ V . (M) For each U ∈ F there exist V , W ∈ F such that V W ⊆ U . (I) For each U ∈ F there exists V ∈ F such that V ι ⊆ U . (C) For each g ∈ G and for each U ∈ F there exists V ∈ F such that V ig ⊆ U . If the neighborhood basis F consists of open sets, it also satisfies (O) For each U ∈ F and for each g ∈ U there exists V ∈ F such that V g ⊆ U . Proof. Property (FB) holds for every neighborhood basis. Properties (M), (I) and (C) hold because multiplication and inversion are continuous. If U ∈ F is open, we find for each g ∈ U a neighborhood X of g contained in U . Then Xg −1 is a neighborhood of 1, and we find V ∈ F such that V ⊆ Xg −1 . Now V g ⊆ X ⊆ U , and property (O) is established for all open sets in F . 2 3.22 Theorem. Let G be a group, and let F be a nonempty set of subsets of G such that 1 ∈ F . If F satisfies (M), (I), (FB) and (C) of 3.21 then T := {T ⊆ G | ∀t ∈ T ∃U ∈ F : U t ⊆ T } is a topology on G, and for each g ∈ G the set Bg (F ) := {Ug | U ∈ F } forms a neighborhood basis at g for T . Moreover, (G, T ) is a topological group. If F also satisfies (O) then Bg (F ) ⊆ T ; that is, B(F ) = g∈G Bg (F ) is a basis for T .
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B Topological Groups
Proof. We check first that T is a topology. Obviously, the empty set and G belong to T , and T is closed with respect to arbitrary unions. For S, T ∈ T and t ∈ S ∩ T we find U, V ∈ F such that U t ⊆ S and V t ⊆ T . According to (FB), there exists W ∈ F such that W ⊆ U ∩ V . Now W t ⊆ S ∩ T , and we have shown that S ∩ T belongs to T . Thus T is a topology. For U ∈ F and g ∈ G, consider the set T := {x ∈ Ug | ∃S ∈ F : Sx ⊆ Ug}. We claim that T belongs to T . In fact, for x ∈ T and S ∈ F with Sx ⊆ Ug we find V , W ∈ F such that V W ⊆ S. Then V wx ⊆ Ug for each w ∈ W , and W x ⊆ T . This shows that the set Bg (F ) is a neighborhood basis at g for T . If F satisfies (O) then Bg (F ) is contained in T . It remains to show that (G, T ) is a topological group. Applying (I) and (C), we obtain continuity of ι from the observation (Ug)ι = g ι U ι = U ιig g ι . Finally, an application of (M) and (C) to (U x)(V y) = U V ix ι xy yields that multiplication is continuous. 2 Note that the topology T in 3.22 belongs to T1 exactly if F = {1}. 3.23 Examples. (a) Let G be a group, and assume that for a normal subgroup N of G there exists a topology on N that renders N a topological group. Moreover, assume that for each g ∈ G the restriction of ig to N is continuous. Applying 3.22 to any neighborhood basis F at 1 in N, we can then topologize the group G. The normal subgroup will be open in G; in particular, local compactness of N yields local compactness of G. (b) Let G be a group, and let F be the set of all subgroups of finite index in G. The intersection of a subgroup A of index a and a subgroup B of index b has index at most ab. Thus (FB) holds for F . Properties (M), (I) and (O) are trivially satisfied (just put V = W = U ), and property (C) follows from the fact that the maps ig are automorphisms, and do not change the index of subgroups. Note that, in general, the topology thus obtained does not belong to T1 . (c) Similarly, one can apply 3.22 to the set of all normal subgroups of finite index. 3.24 Lemma. Let (R, +, −, 0) be a group, let · : R ×R → R be a binary operation (called multiplication) on R such that the distributive laws hold, and assume that there exists an element 1 ∈ R {0} such that 1 · x = x = x · 1 holds for each x. Let T be a topology on R. Then multiplication and additive inversion are continuous if (and only if ) the following hold: (a) (R, +, T ) is a topological semigroup (i.e., addition is continuous). (b) Multiplication is continuous at (0, 0).
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(c) For each a ∈ R, both the maps ρa : R → R : x → x · a, λa : R → R : x → a · x are continuous. Proof. Consider (a, b) ∈ R × R. Our continuity assumptions yield that the map ϕ : (x, y) → (x + a) · (y + b) = xy + ay + xb + ab is continuous at (0, 0), and that (u, v) → (u − a, v − b) is continuous, for each (a, b) ∈ R × R. Therefore, multiplication (u, v) → u · v = (u − a, v − b)ϕ is continuous at (a, b). It remains to note that additive inversion λ−1 : x → −x in (R, +, −, 0, T ) is continuous. 2 We remark that all assumptions on the multiplication in 3.24 are satisfied for a ring: we have just left out associativity of multiplication, which is not needed in the proof. The result may (and will, see 8.53) be used to show continuity of multiplication before discussing associativity. Afterwards, continuity may help to prove associativity.
Homomorphisms 3.25 Definitions. (a) Let (G, μ, ι, ν) and (G , μ , ι , ν ) be groups. A (group) homomorphism from (G, μ, ι, ν) to (G , μ , ι , ν ) is a map ϕ : G → G such that ← (g, h)μϕ = (g ϕ , hϕ )μ for all g, h ∈ G. The kernel of ϕ is ker ϕ := {0ν }ϕ . A (group) anti-homomorphism from (G, μ, ι, ν) to (G , μ , ι , ν ) is a map that reverses the order of multiplication, that is, a map ψ : G → G such that (g, h)μψ = (hψ , g ψ )μ for all g, h ∈ G. (b) If G and H are groups then the set of all homomorphisms from G to H is denoted by Hom(G, H ). (c) Let (R, α, σ, ζ, μ, ν) and (R , α , σ , ζ , μ , ν ) be rings. A (ring) homomorphism from (R, α, σ, ζ, μ, ν) to (R , α , σ , ζ , μ , ν ) is a map ϕ : R → R such that (r, s)αϕ = (r ϕ , s ϕ )α and (r, s)μϕ = (r ϕ , s ϕ )μ hold for all r, s ∈ R, νϕ ν and 0 = 0 . Note that this implies that ϕ is a group homomorphism from (R, α, σ, ζ ) to (R , α , σ , ζ ). The kernel of ϕ is just the kernel of this group ← homomorphism; that is, ker ϕ = {0ζ }ϕ . A (ring) anti-homomorphism from (R, α, σ, ζ, μ, ν) to (R , α , σ , ζ , μ , ν ) is a map ψ : R → R such that (r, s)αψ = (r ψ , s ψ )α and (r, s)μψ = (s ψ , r ψ )μ hold for all r, s ∈ R, and 0νψ = 0ν . Note that only the order of multiplication is reversed.
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B Topological Groups
3.26 Examples. Let G be a group. Then the map i from G to the group of all bijective homomorphisms defined by mapping g ∈ G to ig is a homomorphism: in fact, we have igh = ig ih . The inversion map is an anti-homomorphism of the group G onto itself. Therefore, group anti-homomorphisms yield homomorphisms simply by composing them with inversion in one of the groups, and one obtains a bijection from Hom(G, H ) onto the set of all anti-homomorphisms from G to H in this way. If G is any group then the set Hom(G, G) is a semigroup with respect to composition. The subset of all bijective homomorphisms in Hom(G, G) forms a group; in fact one verifies easily that the inverse of a homomorphism is a homomorphism again. 3.27 Definition. It is sometimes convenient to use the following notion: a sequence · · · An−1
ϕn−1
/ An
ϕn
/ An+1
ϕn+1
/ An+2 · · ·
of homomorphisms between groups is called exact at n (or, less accurate, at An ) if ϕn−1 the image An−1 equals the kernel ker ϕn . The sequence is called exact if it is exact at each n. A short exact sequence is a sequence A0
ϕ0
/ A1
ϕ1
/ A2
ϕ2
/ A3
ϕ3
/ A4
where the groups A0 and A4 are trivial. This means exactly that ϕ1 is injective (having trivial kernel) and ϕ2 is surjective (since Aϕ2 is the kernel of the constant ϕ morphism ϕ3 ), and that the kernel of ϕ2 equals A1 1 . 3.28 Definition. Let R be a ring. A subgroup I of the additive group of R is called a right (resp. left) ideal of R if I r ⊆ I (resp. rI ⊆ I ) for each r ∈ R. We call I an ideal of R if it is both a right and a left ideal. The following are easy consequences of the axioms. 3.29 Lemma. Let (G, μ, ι, ν) and (G , μ , ι , ν ) be groups. (a) Each group homomorphism ϕ from (G, μ, ι, ν) to (G , μ , ι , ν ) also satisfies 0νϕ = 0ν and g ιϕ = g ϕι for each g ∈ G. (b) The kernel of a group homomorphism is a normal subgroup. (c) Conversely, let N be a normal subgroup of G. Then we have (N x)(Ny) = N xy for every pair (x, y) ∈ G2 . Thus the setting (N x, Ny)μ¯ = N xy defines a binary operation on the set G/N = {Ng | g ∈ G}. With (Ng)ι¯ = Ng ι and 0ν¯ = N (= N0ν ) we obtain that (G/N, μ, ¯ ι¯, ν¯ ) is a group and the map πN defined by g πN = Ng is a group homomorphism.
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Now let (R, α, σ, ζ, μ, ν) and (R , α , σ , ζ , μ , ν ) be rings. (d) Each ring homomorphism from (R, α, σ, ζ, μ, ν) to (R , α , σ , ζ , μ , ν ) also satisfies 0ζ ϕ = 0ζ and r νϕ = r ϕν for each r ∈ R. (e) The kernel of a ring homomorphism is an ideal. (f) Conversely, let I be an ideal of R. We obey the usual convention and write α as addition and μ as multiplication. The set I is a normal subgroup of the additive group of R because the latter is commutative, and we obtain a group (R/I, α, ν¯ , ζ¯ ) as in assertion (c). Moreover, we have (I + r)(I + s) ⊆ I I + rI + I s + rs ⊆ I + rs for every pair (r, s) ∈ R 2 . Thus (I + r, I + s)μ¯ = I + rs defines a binary operation μ¯ on R/I . With 0ν¯ = I + 0ν we obtain a ring ¯ μ, (R/I, α, σ, ¯ ζ, ¯ ν¯ ), and πI is a ring homomorphism. The group (G/N, μ, ¯ ι¯, ν¯ ) is called the factor group of G by N, and πN is referred to as the natural map from G onto G/N . If no confusion is possible, we will denote the factor group simply by G/N. Analogously, we call R/I the factor ring of R by I . Combining assertions (b) and (c) or assertions (d) and (f), respectively, we see that the normal subgroups of a given group are exactly the kernels of group homomorphisms from this group to arbitrary groups, and that the ideals of a given ring are exactly the kernels of ring homomorphisms starting from this ring. 3.30 Example. We have now a formal interpretation for the quotient Z(n) = Z/nZ, both as a group and as a ring. 3.31 Example. Let G be a commutative group, written additively. n
(a) For each positive integer n define nG by = g + · · · + g. Let 0G be the constant mapping every g to 0, and put −nG = nG ι. Then zG is a homomorphism from G to G for each z ∈ Z. If no confusion is possible we will also write zg := g zG . g nG
(b) We obtain a binary operation on Hom(G, G) as follows: for elements ϕ, ψ ∈ Hom(G, G) we define ϕ + ψ by putting g ϕ+ψ = g ϕ + g ψ . It is easy to verify that this operation turns Hom(G, G) into a group. The map G from Z to Hom(G, G) mapping z to zG is a homomorphism. 3.32. We will often deal with homomorphisms between topological groups. In general, such a homomorphism need not be continuous. Even if it is continuous and bijective, it need not be a homeomorphism; for instance, consider the identity between the additive group of real numbers with the discrete topology, and the same group with its usual topology. Two topological groups may only be considered as
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“isomorphic as topological groups” if there is a bijective homomorphism between them which is at the same time a homeomorphism; such a map will be called a topological isomorphism. (Simple examples show that it is not sufficient to require the existence of a bijective homomorphism and separately the existence of a homeomorphism, compare Exercise 3.9 and Exercise 3.10.) A bijective homomorphism of a topological group onto itself which is a homeomorphism will be called an automorphism of the group. Again, the set of all automorphisms of a topological group G form a group, which is denoted by Aut(G). We will see in 9.15 that Aut(G) can be turned into a topological group in a quite natural way if G is locally compact. If G is a commutative topological group then it is easy to see that for each integer z the homomorphism zG defined in 3.31 is continuous. If zG is bijective, however, the inverse zG −1 need not be continuous. Checking continuity of homomorphisms is greatly facilitated by the following observation. 3.33 Lemma. Let (G, μ, ι, ν, T ) and (G , μ , ι , ν , T ) be topological groups. Then a homomorphism ϕ : G → G is continuous exactly if it is continuous at a single point (for instance, at the neutral element). Proof. Assume that ϕ is continuous at x ∈ G, let y be another element of G, and put z := (y ι , x)μ . Then ρz ϕ is continuous at y, and therefore ψ := ρz ϕρzιϕ is continuous at y. But ψ = ϕ since ϕ is a homomorphism. 2 3.34 Definition. If G is a group and N is a normal subgroup of G, we say that G is an extension of N by G/N . More precisely, the extension is given by the short exact sequence {1}
/N
ε
/G
πN
/ G/N
/ {1}
where ε is just inclusion. In this manner, it is also possible to speak of an extension of N by Q, where N and Q are arbitrary groups: this means a short exact sequence /N /G /Q / {1} . {1} Of course, an exact sequence of topological groups is one where all homomorphisms involved are continuous. We say that a topological group (G, G) is an extension of (N, N ) by (Q, Q) if {1}
/ (N, N )
ε
/ (G, G)
π
/ (Q, Q)
/ {1}
is a short exact sequence of topological groups, ε is a topological embedding (that is, a homeomorphism from (N, N ) onto (N ε , G|N ε ) and π is a quotient map. Occasionally, we will encode this information in arrows of special form, as follows: π ,2 / (N, N ) ε / (G, G) / {1} . {1} (Q, Q) However, we promise to give all necessary information in the text as well.
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Let an extension (G, G) of (N, N ) by (Q, Q) be given by / (N, N )
{1}
ε
/ (G, G)
2, (Q, Q)
π
/ {1} .
We say that the extension splits (or is a split extension) if there is a continuous homomorphism σ : (Q, Q) → (G, G) such that σ π = idQ . While it is rather difficult (if not impossible) to determine (G, G) from a given extension up to isomorphism, this is quite easy in the case of a split extension. We will do that in 10.14. The subgroup lattice of a group is the set of all its subgroups, partially ordered by inclusion. Occasionally, it is helpful to visualize a part of the subgroup lattice by a so-called Hasse diagram: the subgroups are represented by dots, and inclusion relations by (sequences of) lines. It is customary to indicate normal subgroups by double lines. For instance, a split extension {1}
/N
/Go
πN σ
/
Q
/ {1}
then looks like G P · PPPP | | | PPP ||| | | PPP | PP ||||| σ N · RRRR {· Q RRR { { RRR {{ RRR R {{ · {1} The reader should be aware that a Hasse diagram may sometimes disguise topological peculiarities.
Products The following construction of a topological group from a collection of topological groups is of great importance. 3.35 Theorem: Cartesian products. Let ((Gα , Tα ))α∈A be a family of topological groups. (a) The set Xα∈A Gα , endowed with the product topology and the multiplication defined by (xα )α∈A (yα )α∈A = (xα yα )α∈A is a topological group.
(b) The topological group Xα∈A Gα , α∈A Tα has the following universal property (which in fact characterizes it up to isomorphism):
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B Topological Groups
For every topological group (H, H) and every family ψ = (ψα )α∈A of continuous homomorphisms ψα : (H, H) → (Gα , Tα ) there is a unique contin- (P) from (H, H ) to the product Xα∈A Gα , uous homomorphism ψ α∈A Tα such that ψπα = ψα for each α ∈ A.
Tα Xα∈A Gα , a α∈A QQQ WWWWYWYYYYYY YY W Q Qπ
i
ψ
YY WW QQQ Wπj WWWWYWYYπk YYYYYYY YYYYYY WWWWW QQQ YYYY, WW+ ( (Gj , Tj ) (Gi ,O Ti ) (Gk , Tk ) · · · r9 hhh3 h r h r h h r hh ψi rψj hhhhψk rrhrhhhhh (H, H)
Proof. Let μα be the multiplication in Gα , and let μ denote the multiplication in the product Xα∈A Gα . In order to see that μ is continuous, we consider the canonical projection πβ : Xα∈A Gα → Gβ for β ∈ A, and the map ϕβ : ( Xα∈A Gα ) × ( Xα∈A Gα ) → Gβ × Gβ defined by ((xα )α∈A , (yα )α∈A )ϕβ = (xβ , yβ ). Then ϕβ and therefore μπβ = ϕβ μβ are continuous. Thus μ is continuous by 1.11. Inversion ι in Xα∈A Gα is continuous since ιπβ = πβ ιβ is. This completes the proof of assertion (a). has been established in 1.12. Existence, uniqueness and continuity of the map ψ Computing
x ψ y ψ = (x ψ )α (y ψ )α α∈A = x ψα y ψα α∈A = (xy)ψα α∈A = (xy)ψ is a homomorphism. one verifies that ψ
2
The group Xα∈A Gα is called the cartesian product of the family (Gα )α∈A . As the product topology is very natural, we will mostly endow a cartesian product with it. However, sometimes the restriction to a certain class of topological groups (for instance, discrete ones, or locally compact ones) forces one to use different topologies. If A consists of two elements, say A = {0, 1}, we also write (G0 , T0 ) × (G1 , T1 ) :=
X Gα ,
α∈{0,1}
α∈{0,1}
Tα .
3. Basic Definitions and Results
41
The Hasse diagram of the cartesian product G0 × G1 looks like G0 × G1 · MM MMM MMM MMM MMM G0 × {1} · NNN · {1} × G1 NNN NNN NNN NNN · {1} 3.36 Lemma. For α ∈ {0, 1}, let (Gα , Tα ) be a topological group. Then the cartesian product (G0 , T0 ) × (G1 , T1 ) is a split extension of (G0 , T0 ) by (G1 , T1 ). Proof. Define ε : (G0 , T0 ) → (G0 , T0 ) × (G1 , T1 ) by x ε = (x, 1). The sequence {1}
/ (G1 , T1 )
ε
/ (G0 , T0 ) × (G1 , T1 )
π1
/ (G1 , T1 )
/ {1}
is exact because 1 = g1 = (g0 , g1 )π1 is equivalent to (g0 , g1 ) ∈ Gε0 . We define a continuous homomorphism σ : (G1 , T1 ) → (G0 , T0 ) × (G1 , T1 ) by y σ = (1, y), obtain σ π1 = idG1 , and see that the extension splits. 2 Note the symmetry in 3.36: the group (G0 , T0 ) × (G1 , T1 ) is of course also a split extension of (G1 , T1 ) by (G0 , T0 ).
Centralizers and Normalizers 3.37 Definition. Let G be a group, and let X be a subset of G. (a) The set CG (X) := {g ∈ G | ∀x ∈ X : xg = gx} is called the centralizer of X in G. We abbreviate CG (x) := CG ({x}). (b) The set NG (X) := {g ∈ G | g −1 Xg = X} is called the normalizer of X in G. 3.38 Lemma. Centralizers and normalizers are subgroups. If H is a subgroup of G then H is normal in NG (H ). Proof. Straightforward computations show that CG (X) and NG (X) are closed under multiplication, and under inversion. The assumption that H be a subgroup combined with the defining property for NG (H ) forms the definition of the term “normal subgroup”. 2 Note that, in general, the set compr G (X) := {g ∈ G | ∀x ∈ X : g −1 xg ∈ X} is a subsemigroup of G (called a compression semigroup, cf. 28.1). The normalizer is obtained as NG (X) = compr G (X) ∩ (compr G (X))−1 .
42
B Topological Groups
3.39 Proposition. Let G be a topological group, and let X ⊆ G. (a) If X is closed in G then NG (X) is also closed in G. (b) If G is Hausdorff then the centralizer CG (X) is closed in G. The Proof. For each x ∈ G, the map εx : G → G : g → g −1 xg is continuous. ← compression semigroup compr G (X) is obtained as the intersection x∈X X εx of preimages of the set X. If X is closed in G then each of these preimages is closed, and so is compr G (X). But then NG (X) = compr G (X) ∩ (compr G (X))−1 is closed, as well. ← If G is Hausdorff, each preimage CG (x) = {x}εx is closed, and the intersection 2 CG (X) = x∈X CG (x) is closed, too. 3.40 Example. Consider the group a b G := 0 1
a, b ∈ R
endowed with the topology induced from R2×2 . Then G is a Hausdorff group. The (non-closed) subgroup 1 b H := b∈Q 0 1 has the normalizer
NG (H ) =
a b 0 1
a ∈ Q, b ∈ R
which is not closed. 3.41 Remark. One may describe CG (x) and NG (X) as stabilizers, with respect to suitable actions of G (via conjugation on G itself, and on the set of subsets of G, respectively), see 10.9 below. The group CG (X) can be interpreted as the kernel of the action of NG (X) on X.
Exercises for Section 3 Exercise 3.1. Show that the complex numbers with their usual topology form a topological field. a b a, b ∈ C is a field (with noncommutative Exercise 3.2. Show that the set H := −b¯ a¯ multiplication). Determine the center of H.
4. Subgroups
43
Exercise 3.3. Show that H, endowed with the topology induced from the ring C2×2 of 2 × 2 matrices over C, is a locally compact field (known as the field of Hamilton’s quaternions). Exercise 3.4. Show that addition is commutative in any ring, and in any module over a ring. Hint. Apply the distributive laws to (x + y)(1 + 1), where 1 = 0ν is the multiplicative unit. Exercise 3.5. Verify the assertions of 3.16 in detail. Exercise 3.6. Show that a compact neighborhood of Rn does not contain any subgroup apart from {0}. Conclude that {0} is the only compact subgroup of the additive group of Rn . Exercise 3.7. Let G be a group, and let N be the set of all normal subgroups of finite index in G. Assume that N = {1}. Show that there is an injective homomorphism from G into XN∈N G/N . Topologize the group G as in 3.23 (c), and show that the homomorphism from G into XN∈N G/N can be chosen in such a way that it is continuous. Exercise 3.8. Show that there is a unique topology T on the group QN with the following properties: (QN , T ) is a topological group, the subgroup ZN is open, and T |ZN is the product topology. Prove that for each z ∈ Z {0} the homomorphism zG : (QN , T ) → (QN , T ) is continuous and bijective, but not open for z ∈ / {−1, 1}. Hint. Apply 3.22. Exercise 3.9. Let G be the subgroup of GL(2, R) consisting of all matrices of the form a b , where a, b ∈ R and a > 0. Let T1 be the topology defined by the metric d1 given 0 1
by d1 a0 b1 , 0c d1 = (a − c)2 + (b − d)2 , and let T2 be the topology defined by the ! a b c d
|b − d| if a = c, metric d2 given by d2 0 1 , 0 1 = |b − d| + 1 otherwise. Show that (G, T1 ) and (G, T2 ) are topological groups. Exercise 3.10. Let U be the usual topology on R, and let D be the discrete topology on R. Show that (G, T1 ) × (R, D) and (G, T2 ) × (R, U) are homeomorphic, but not isomorphic as topological groups. Exercise 3.11. Let (A, +) be a commutative group, and let F be a subring of End(A, +). Verify that A is a right module over F . If F is a field, then A is a vector space over F . Exercise 3.12. Show that every commutative topological group is a topological module over the discrete ring Z.
4 Subgroups Throughout this section, let (G, T ) be a topological group, written multiplicatively. 4.1 Definition. If G is a group and X is some subset of G we denote by X the smallest subgroup of G that contains X. Note that one can describe X as the union (X ∪ Xι )
, where Y denotes the set of all products of n elements of Y . n∈N
44
B Topological Groups
A group G is called cyclic if it is generated by a single element (that is, G = g for some suitable element g ∈ G). A topological group G is called compactly generated if there exists a compact subset C of G such that G = C . 4.2 Lemma. For every subgroup H of G we have that (H, T |H ) is a topological group. Proof. The group operations μ|H ×H and ι|H of H are restrictions of continuous 2 maps, and thus continuous.
Closure of Subgroups 4.3 Lemma. For every subgroup H of G, the closure H is a subgroup. If H is a normal subgroup then H is a normal subgroup, as well. Proof. We have to show that H ι ⊆ H and that (H ×H )μ ⊆ H . The first assertion is clear since ι is a homeomorphism that leaves H invariant. For the second inclusion, μ observe that H × H = H × H . Since μ is continuous, we infer that H × H ⊆ (H × H )μ = H . If H is normal then each of the homeomorphisms ig also leaves the closure of H invariant. 2 4.4 Lemma. If H is a commutative subgroup of a Hausdorff group G then the closure H of H is a commutative subgroup, as well. Proof. The map ϕ : G×G → G defined by (a, b)ϕ = ab(ba)−1 is continuous, and H × H is contained in the closed pre-image of {1} under ϕ. Thus (H × H )ϕ = {1}, and H is commutative. 2 Endowing a noncommutative group G with the topology {∅, G} and considering H = {1}, one sees that the assumption G ∈ T2 cannot be dispensed with in 4.4. 4.5 Lemma. A subgroup H of G is closed in G exactly if there is a neighborhood U of 1 in G such that U ∩ H is closed in G. Proof. Choose a neighborhood V of 1 in G such that V = V ι and V V ⊆ U ; this is possible since ι and μ are continuous. Let x be an element of H . For every neighborhood W of 1 we find an element hW ∈ W x ∩ H . Since x −1 ∈ H , we also find y ∈ x −1 V ∩ H . For W ⊆ V we obtain that hW y ∈ (W x)(x −1 V ) = W V ⊆ V V ⊆ U and that hW y ∈ H . Therefore the intersection W xy ∩ (U ∩ H ) is nonempty for each neighborhood W of 1. This means that xy ∈ U ∩ H = U ∩ H , and we conclude that x ∈ H . 2
4. Subgroups
45
4.6 Corollary. Let H be a subgroup of G. If T ∈ T1 and T |H is discrete, then H is closed in G. 4.7 Corollary. Let H be a subgroup of G. If T ∈ T2 and T |H is locally compact, then H is closed in G. Proof. Choose a neighborhood U ∈ N1 such that U ∩ H is compact. Since G is Hausdorff, we have that U ∩ H is closed in G. 2 Since closed subspaces of locally compact spaces are locally compact (see 1.20) we have the following corollary. 4.8 Theorem. Let (G, T ) be a locally compact Hausdorff group. Then a subgroup of G is closed in G exactly if it is locally compact.
Connectedness 4.9 Lemma. The connected component G1 of the neutral element 1 is a closed normal subgroup of G. (In fact, it is even invariant under continuous homomorphisms: if H is a topological group and γ : G → H is a continuous homomorphism then (G1 )γ ⊆ H1 .) Proof. Connected components are always closed, see 2.9. The image of G1 under ι is a connected subset of G and contains 1. Therefore (G1 )ι ⊆ G1 . The set G1 × G1 is connected, thus (G1 × G1 )μ is a connected subset of G, and contained in G1 . We have shown that G1 is a closed subgroup of G. If γ is a continuous homomorphism from G to H , we have that (G1 )η is a connected subset of H containing 1, whence (G1 )η ⊆ H1 . Applying this to H = G 2 and η = ig , we obtain that G1 is a normal subgroup of G. 4.10 Lemma. Let H be a subgroup of G. If H contains a neighborhood then H is both open and closed in G, and H contains the connected component G1 of 1 in G. Proof. Let U be a nonempty open set in G. If U ⊆ H we obtain that H ⊆ H U ⊆ H H = H . Thus H is open. For g ∈ G H , we have that Hg is open and disjoint to H . Thus H is closed. Every closed open set is the union of connected components. 2 4.11 Lemma. Let C be a compact subset of G. Then for each neighborhood V of 1 there exists a neighborhood W of 1 such that for each c ∈ C we have W ic ⊆ V .
46
B Topological Groups
Proof. Let ϕ : G × G → G be the map defined by (x, y)ϕ := x −1 yx = y ix . Then (C × {1})ϕ ⊆ {1} ⊆ V , and continuity of ϕ implies that for each c ∈ C we ϕ find neighborhoods Vc of c and Wc of 1 such that (V c × Wc ) ⊆ V . Since C is compact, there is a finite subset F of C such that C ⊆ f ∈F Vf . The neighborhood W := f ∈F Wf satisfies our requirements. 2 4.12 Lemma. Assume that U ⊆ G is compact and open, and that 1 ∈ U . Then there exists an open compact subgroup H of G such that H ⊆ U . If G is compact, there even exists an open compact normal subgroup N of G such that N ⊆ U . Proof. Pick a neighborhood V of 1 in G such that U V ⊆ U ; this is possible by 3.18. Then V ⊆ U , V V ⊆ U , and inductively one sees V := V V ⊆ U for each n ∈ N. This means that the open subgroup H generated by V ∩V ι is contained in U . As open subgroups are closed and U is compact, the subgroup H is compact. If G is compact, we pick a neighborhood W of 1 such that W ig ⊆ V for all g ∈ G; this is possible by 4.11. Then the subgroup N generated by g∈G W ig is an open compact normal subgroup, and N ⊆ H ⊆ U . 2 4.13 Proposition. Let G be a locally compact group, and assume that G is totally disconnected. Then there exists a neighborhood basis at 1 consisting of open compact subgroups. If G is compact, then there is a neighborhood basis at 1 consisting of open compact normal subgroups. Proof. As G is totally disconnected, it belongs to the class T1 . We will see in 6.6 below that this implies G ∈ T2 . Let W be a neighborhood of 1 in G. According to 2.10, there is a compact open neighborhood U of 1 such that U ⊆ W . Applying 4.12, we obtain the assertion. 2
Divisible Subgroups 4.14 Definition. A group G is called divisible if for each positive integer n ∈ N{0} the map nG : G → G defined by g nG = ng is surjective (here we use additive notation for G although we do not require G to be commutative). 4.15 Example. For n ∈ N, the additive groups Qn and Rn are divisible. The group Z is not divisible. Also, the (noncommutative) groups GL(2, R) and SL(2, R) are 2 not divisible: for instance, there is no element A ∈ GL(2, R) such that A = −2 0 . 0 −2−1 The proof of the next assertion is easy, and left as an exercise. 4.16 Lemma. Let G and H be groups. If G is divisible and ϕ : G → H is a surjective homomorphism then H is divisible.
4. Subgroups
47
4.17 Lemma. Let (A, +) be a divisible commutative group. If A does not contain elements of finite order (except 0, of course), then A is a vector space over Q. −1
Proof. Multiplication by n1 is defined by n1 x := x nA : surjectivity of nA follows from divisibility, while ker nA = {x ∈ A | ord x divides n} = {0} yields injectivity. 2 The treatment of divisible groups containing nontrivial elements of finite order needs some more preparation. 4.18 Definitions. Let p be a prime. (a) A group G is called a p-group if {ord x | x ∈ G} ⊆ {pn | n ∈ N}. n (b) For each commutative group A, the nsubgroups ker p A form an ascending chain. Thus the union Ap := n∈N ker p A is a subgroup, called the p-part of A, or a primary subgroup of A.
4.19 Example. Using the isomorphism mapping Z+r ∈ R/Z to e2π ir ∈ U(1, C) = {c ∈ C | cc¯ = 1}, we see that the p-part of the circle group U(1, C) is isomorphic to the group Z(p∞ ) := Z + pzn | z ∈ Z, n ∈ N – we ignore the topology here. In a commutative Hausdorff group A, the kernels nA are always closed, but the primary parts will, in general, not be closed. Consider, for example, the group U(1, C). In every commutative group A, the subset Tors(A) := {x ∈ A | ord x ∈ N} = n∈N{0} ker nA is a subgroup which clearly contains every primary part of A. Each element of Tors(A) generates a finite cyclic group. Using the additive representation of the least common divisor of two integers, we easily see that finite cyclic groups are direct sums of their primary components. Thus we have proved: " 4.20 Proposition. In every commutative group A, we have Tors(A) = p∈P Ap . 2 We will show in 4.23 below that every divisible commutative group is the direct sum of its primary parts, and some vector space over Q. Moreover, we will explicitly determine the divisible p-groups. The following result will also play its part in these characterizations: we may, of course, use the discrete topologies. 4.21 Lemma: Extending homomorphisms to divisible groups. Let A and D be commutative topological groups, and assume that D is divisible. For every open subgroup B of A and each continuous homomorphism ϕ : B → D there is an extension of ϕ to A; that is, a continuous homomorphism ψ : A → D such that ψ|B = ϕ.
/A B @◦ @@ @@ ψ ϕ @@ D
48
B Topological Groups
Proof. As B is open, it suffices to find a homomorphism ψ : A → D such that ψ|B = ϕ; then ψ is continuous by 3.33. We consider the set E of all pairs (C, γ ), where C is a subgroup of A such that B ≤ C ≤ A and γ : C → D is a homomorphism extending ϕ. On E we have a partial ordering defined by (C1 , γ1 ) (C2 , γ2 ) ⇐⇒ C1 ≤ C2 , γ2 |C1 = γ1 . Zorn’s Lemma yields the existence of a maximal element (M, μ) in (E , ). We claim that M = A. Aiming for a contradiction, assume that x ∈ A M. Let X be the subgroup generated by x. Then there is a natural number k such that kx generates X ∩ M. As D is divisible, we find d ∈ D with the property kd = (kx)μ . Since A is commutative, the set X + M := {u + v | u ∈ X, v ∈ M} is a subgroup of A. Defining γ : X + M → D by (lx + m)γ = ld + mμ we obtain (X + M, γ ) ∈ E 2 and (M, μ) ≺ (X + M, γ ), a contradiction. 4.22 Corollary. Let A be a commutative topological group. If B is an open divisible subgroup of A then there exists a discrete subgroup C of A such that B ∩ C = {0} and B + C = A. The group A is isomorphic as a topological group to the cartesian product B × C, with the product topology. Proof. Applying 4.21 to the identity idB : B → B we obtain a continuous homomorphism j : A → B such that j |B = idB . The kernel of j is the subgroup C we want.
/A B @◦ @@ @@ j idB @@ B
Mapping (b, c) to b + c we obtain a surjective continuous homomorphism i : B × C → A. Observing ker i = B ∩ C = {0} we see that i is a bijection. As C is discrete, the set B × {0} is open in B × C, and continuity of i −1 follows from 3.33. 2 4.23 Lemma. Let A be a commutative group, put T := Tors(A), and Q := A/T . (a) If A is divisible then T is divisible. (b) The group T is divisible if, and only if, each of its primary parts is divisible. (c) If A is divisible, then A is isomorphic to Q × T . (d) A commutative p-group P is divisible if, and only if, there exists a cardinal number d such that P ∼ = Z(p∞ )(d) . Proof. Let x ∈ Ap = Tp , and consider n ∈ N {0}. We decompose n = p e · r, where r ∈ N is not divisible by p. Now r A induces a bijection of x , and any element y ∈ A with pe y = x belongs to Ap . This yields that every solution y of ny = x lies in Ap , and the first two assertions are proved. Assertion (c) follows from 4.22.
4. Subgroups
49
Now assume that P is a commutative divisible p-group, consider x ∈ P , and let e ∈ N satisfy pe x = 0 = pe−1 x. We find x1 ∈ P such that px 1 = x, and continue recursively, picking xn+1 such that pxn+1 = xn . Then S := n∈N xn is a 1 extends to an isomorphism from P onto Z(p ∞ ). subgroup of P , and xn → Z+ pn+e Let M be a subgroup of P , maximal with respect to the property of being isomorphic to Z(p∞ )(d) , for some cardinal d. (The existence of such a subgroup follows from Zorn’s Lemma.) According to 4.22, there exists C ≤ P such that P = M ⊕ C, because M is divisible. Aiming at a contradiction, we assume M = P , then C contains a copy D of Z(p∞ ), and M < M ⊕ D ∼ = Z(p∞ )(d+1) contradicts our choice of M. 2 4.24 Theorem. A commutative group A is divisible if, and only a family "if, there is (d ) ∞ ∼ 0 d = (dj )j ∈P∪{0} of cardinal numbers such that A = Q × p∈P Z(p )(dp ) . " Proof. Clearly, every group of the form Q(d0 ) × p∈P Z(p∞ )(dp ) is divisible. Conversely, assume that A is divisible. Then A ∼ = A/ Tors(A)×Tors(A) by 4.23. As A/ Tors(A) is a divisible commutative group, and does not contain any nontrivial elements of finite order, there is a cardinal d0 such that A/ Tors(A) ∼ = Q(d0 ) , cf. 4.17. The rest follows from 4.23 (d) and 4.20. 2
Exercises for Section 4 Exercise 4.1. Find an example of a topological group (G, T ) where T ∈ / T2 . Exhibit a discrete subgroup of G which is not closed. Exercise 4.2. Determine all open subgroups of (F, +) and (F× , ·), where F ∈ {R, C, H} carries the usual topology. Exercise 4.3. Exhibit a neighborhood basis at 1 consisting of open normal subgroups in the group Xα∈A Gα , where (Gα )α∈A is a family of discrete groups, and Xα∈A Gα carries the product topology. Exercise 4.4. Endow (Q, +) with the topology T induced from the usual topology on R. Show that T is totally disconnected, but there is no neighborhood basis at 0 consisting of open subgroups. (This shows that the condition ‘locally compact’ cannot be dispensed with in 4.13.) Exercise 4.5. Show that zψ := eiz defines a continuous homomorphism from (Z, +) to (C× , ·), and that Zψ = {u ∈ C | uu¯ = 1}. Determine the kernel of ψ. Exercise 4.6. Show that r ϕ := (eir , eπ ir ) defines a continuous homomorphism from (R, +) to (C× , ·) × (C× , ·). Determine ker ϕ and Rϕ .
50
B Topological Groups
Exercise 4.7. Let ϕ : G → H a continuous homomorphism between topological groups G and H . Show that ϕ := {(g, g ϕ ) | g ∈ G} is a subgroup of the product G × H , and that
ϕ is closed if H is Hausdorff. Exercise 4.8. Prove 4.16. a b Exercise 4.9. Show that the subgroup D := 0 1 | a, b ∈ R, a > 0 of GL(2, R) is divisible. Hint. Show that D = T ∪ d∈D dH d −1 , where T and H are the subgroups given by 1 b T := 0 1 | b ∈ R and H := a0 01 | a ∈ R, a > 0 . Exercise 4.10. Show that, for each integer n > 1, the groups GL(n, R) and SL(n, R) are not divisible. Hint. Use the fact that cyclic groups are commutative, and consider centralizers of suitable elements in GL(n, R). Exercise 4.11. Let F be a topological commutative field. Show that SL(n, F ) is (pathwise connected) if F is (pathwise) connected. Hint. Choose a basis b1 , . . . , bn , and consider the transvections τjt k mapping v = ni=1 fi bi to v + tfk bj . Show that the set Tj k := {τjt k | t ∈ F } forms a subgroup of SL(n, F ) whenever j and k are different elements of {1, . . . , n}. Now observe that Tj k is isomorphic as a topological group to the additive group of F , and use the Gauss algorithm to show that SL(n, F ) is generated by the union of the subgroups Tj k . Exercise 4.12. Let n be a natural number. Prove that the groups SL(n, R), SL(n, C), GL(n, C) and GL+ (n, R) := {M ∈ GL(n, R) | det M > n} are pathwise connected. Show also that GL+ (n, R) is the connected component of GL(n, R).
5 Linear Groups over Topological Rings Let K be a topological ring such that the set K × of invertible elements is open in K, and inversion is continuous in K × . For instance, every Hausdorff topological field satisfies these hypotheses; see also Section 31. Fix a natural number n. Then the matrix ring K n×n is a topological ring, cf. 3.5. In order to show that GL(n, K) is a topological group, it remains to show (cf. 1.5 and 3.22) that inversion is continuous near the identity matrix E ∈ K n×n . 5.1 Definitions. In the sequel, let bj denote the 1 × n matrix whose j -th entry is 1, while all the other entries are 0. Dually, let bj denote the n × 1 matrix whose j -th entry is 1, while all the other entries are 0. Then we have ! 1 if j = k, bj bk = 0 otherwise,
5. Linear Groups over Topological Rings
51
while the set {bj bk | j, k < n} forms a nice basis for the vector space K n×n . Moreover, the (i, j )-coefficient of A ∈ K n×n is obtained as ai,j = bi Abj . These observations may be helpful when checking the computations in the sequel. 5.2 An algorithm. Starting with an n × n matrix A, we will try to construct a sequence of matrices kA = ( ka i,j )i,j 0 bj bj Ab0 b0 , that is, the matrix defined by ⎧ ⎪ if j = k, ⎨1 0 bj TA bk = 0 for k ∈ / {0, j }, ⎪ ⎩ −aj,0 for j > 0 = k, looking like ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
1 0 ··· −a1,0 1 .. . . 0 .. .. .. . . −an−1,0 0 · · ·
0
.. 0
.
0 .. . .. . .. .
⎞ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎠
1
−1 A satisfies b0 1A b0 = 1 and A simple computation shows that 1A := 0TA 0DA 1 bj A b0 = 0 whenever 0 < j < n. In other words, we have achieved that the first column equals b0 . Clearly, the matrices 0DA and 0TA depend continuously on the (entries of −1 the) matrix A ∈ U0 . Moreover, 0DA depends continuously on 0DA because we only have to invert a single entry. Therefore, the product 1A depends continuously on A, as well. Since we obtain 1A = E if we start with A = E, there exists a neighborhood U1 ⊆ U0 of E such that every element A ∈ U1 satisfies b1 1A b1 ∈ K × . Therefore, we can repeat the process with the sub-matrix obtained by deleting the leftmost column and the uppermost row from 1A .
52
B Topological Groups
Formally, assume that k−1TA , k−1DA and kA have been defined for each A in Uk−1 , and that a neighborhood Uk ⊆ Uk−1 of E has been chosen such that every element A ∈ Uk satisfies bk kA bk ∈ K × . Now let kDA be the diagonal matrix whose entry at position k is the corresponding diagonal coefficient bk,k of A, while all other coefficients equal 1. Thus kDA = bk bk kA bk bk + j =k bj bj , and bk kA bk is invertible. Further, put kTA := E − j >k bj bj Abk bk , and compute
−1 k := kTA kDA A . It turns out that the k-th column of k+1A contains only zeros below the diagonal. For starting points A in some neighborhood Un−1 , our process stops with a matrix nA which is an upper triangular matrix (which comes as no surprise: good old Gauss elimination is still working), such that every diagonal entry equals 1. Note that n−1TA = E. A similar, but much simpler procedure leads to (upper triangular) matrices nTA , . . . , 2n−2TA depending continuously on A such that n 2n−2T . . . nT A A = E. We obtain that, for each A ∈ Un , the product A k+1A
2n−2
TA . . . nTA
n−1
TA
DA . . . 0TA 0DA
n−1
is the inverse of A, and depends continuously on A. Thus we have proved most of the following assertions: 5.3 Theorem. Let K be a topological ring such that the set K × of invertible elements is open in K, and inversion is continuous in K × . Then GL(n, K) is a topological group, and open in K n×n . Proof. It remains to show that GL(n, K) is open in K n×n : this follows from the facts that GL(n, K) contains the open set Un−1 , and that multiplying with X ∈ GL(n, K) is a homeomorphism of K n×n leaving GL(n, K) invariant and mapping E to X. 2 5.4 Corollary. For each Hausdorff topological field K (irrespective of commutativity) and each n ∈ N, the group GL(n, K) is a topological group, and open in K n×n . 2 Since finite powers and open subspaces of locally compact spaces are locally compact again, we also have the following. 5.5 Corollary. If K is a locally compact Hausdorff field and n is a natural number 2 then GL(n, K) is a locally compact group. 5.6 Example. The ring R := (Z/(2Z))N is a compact non-discrete topological Hausdorff ring such that R × is closed, but not open.
53
6. Quotients
6 Quotients 6.1 Definition. Let (G, T ) be a topological group, and let H be a subgroup of G. Then we have the natural surjection πH : G → G/H := {Hg | g ∈ G} defined by g πH = H g. In the sequel, we will always endow the set G/H with the quotient topology T /πH . 6.2 Lemma. Let G be a topological group, and let H be a subgroup of G. Then the following hold. (a) The map πH : G → G/H is open. (b) The map μ¯ : G/H × G → G/H defined by (H x, g)μ¯ = H xg is continuous and open. (c) If H is a normal subgroup then G/H is a topological group (with the natural operations, as defined in 3.29 (c)). Proof. We abbreviate π := πH . If U is open in G G×G ← then U π π = H U is open as well, and thus U π is α open. This proves assertion (a). The map α : G × G → G/H × G : (x, g) → G/H × G (H x, g) is an open surjection, and therefore a quotient β map. Since α μ¯ = μπ is continuous, we have that μ¯ is continuous. Because α is a continuous surjection G/H × G/H and μπ is open, the map μ¯ is open.
μ
/G πH
μ¯
/ G/H 8
μ
If H is a normal subgroup then G/H is a group with multiplication μ defined by (H x, Hy)μ = H xy. The surjection β : G/H × G → G/H × G/H defined by β (H x, y) = (H x, Hy) is a quotient map (being open). Thus continuity of μ¯ = βμ implies continuity of μ . Inversion ι in G/H is obtained as (H x)ι = H x ι . Thus continuity of ι follows from the observations that π ι = ιπ is continuous, and that π is a quotient map. 2 6.3 Remark. Assertion 6.2 (b) implies that the map ρ¯g : G/H → G/H defined by (H x)ρ¯g = H xg = (H x, g)μ¯ is a homeomorphism (with inverse ρ¯g ι ). This is the reason why G/H is called a homogeneous space of G. In general, the quotient map πH : G → G/H is not closed. 6.4 Example. Consider the group G = R2 and the subgroup H = R × {0}. The hyperbola X = {(x, x −1 ) ∈ R2 | x ∈ R {0}} is closed in G, but XπH is not closed in G/H .
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B Topological Groups
However, Lemma 3.19 implies the following. 6.5 Lemma. If G is a topological Hausdorff group and H is a compact subgroup of G then πH is a closed map. 2
Separation Properties 6.6 Proposition. Let G be a topological group, and let H be a subgroup of G. Then both G and G/H belong to the class T3 . Moreover, the following are equivalent: (a) G/H ∈ T0 . (b) G/H ∈ T2 ∩ T3 . (c) H is closed in G. Proof. Since G/{1} and G are homeomorphic, it suffices to consider G/H . We abbreviate π := πH . In the space H π , only the sets ∅ and H π are closed. Thus H = H implies / T0 and therefore G/H ∈ / T0 . If H is closed then H x is closed for each that H π ∈ x ∈ G, and G/H ∈ T1 . In particular, we have already shown that assertions (a) and (c) are equivalent. Of course, assertion (b) implies (a) and (c). In view of T1 ∩ T3 = T2 ∩ T3 , it remains to show G/H ∈ T3 . ← Let U be a neighborhood of 1π in G/H . Then U π is a neighborhood of 1 ← ι in G, and we find a neighborhood V of 1 such that V V ⊆ U π . We claim that V π π← is contained in U ; this yields that G/H ∈ T3 . In fact, for each a ∈ V π we have that the neighborhood (aV )π of a π meets V π . Thus there are elements v, w ∈ V such that (av)π = wπ . Now av ∈ H w, and a ∈ H wv −1 ⊆ H V V ι ⊆ H U . 2
Extension Properties A property P of topological groups is called an extension property if the following holds: whenever G is a topological group and N is a normal subgroup of G such that both N and G/N have the property P , then G has the property P . 6.7 Theorem. Let (G, T ) be a topological group, and let H be a subgroup of G. (a) If G is connected then G/H is connected. Conversely, if both H and G/H are connected then G is connected.
6. Quotients
55
(b) If G is compact then G/H is compact. Conversely, if both H and G/H are compact then G is compact. (c) If G is locally compact then G/H is locally compact. Conversely, if both H and G/H are locally compact then G is locally compact. (d) If G is discrete then G/H is discrete. Conversely, if both H and G/H are discrete then G is discrete. Proof. If the space G is connected then its continuous image G/H is connected as well. If H is a connected subgroup of G, we have that H is contained in G1 . Then the quotient map πG1 factors through πH ; that is, there is a continuous map α : G/H → G/G1 such that πG1 = πH α. Thus the quotient G/G1 is a continuous image of G/H . If G/H is connected, we obtain that G/G1 is connected as well. But G/G1 is totally disconnected by 2.9 (d), which means that G = G1 . Thus assertion (a) is established. For the sake of readability, put π = πH . If the group G is compact then its continuous image G/H is compact. If G is locally compact, then every π -preimage of a neighborhood of 1π contains a compact neighborhood. The image of this compact neighborhood under π is a compact neighborhood, since π is open, see 6.2. Thus G/H is locally compact. Now assume that both H and G/H are locally compact. Pick a neighborhood U of 1 in G such that U ∩ H is compact. Then there is a closed neighborhood T of 1 such that T T ι ⊆ U . For each x ∈ T , the set T ∩ H x = (T x ι ∩ H )x is a closed subset of (U ∩ H )x and therefore compact. As T π is a neighborhood of 1π in the locally compact space G/H , there exists a compact neighborhood C of 1π such that C ⊆ T π . Pick a closed neighborhood R of 1 in G such that RR ι ⊆ T and R π ⊆ C. In order to see that R is compact, consider a family (Vα )α∈A of open sets such that R is contained in α∈A Vα . For each x ∈ T , we have T ∩ H x ⊆ G = (G R) ∪ α∈A Vα . As T ∩ H x is compact, there exists a finite subset Fx of A such that T ∩ H x ⊆ (G R) ∪ α∈Fx Vα . We abbreviate Vx := α∈Fx Vα . Pick an open neighborhood Wx of 1 in G such that Wx ⊆ R and (T ∩ H x)Wx ⊆ (G R) ∪ Vx ; this is possible since T ∩ H x is compact, see 3.18. Because the family ((xWx )π)x∈T forms an open covering of C,we find a finite subset E of T ← such that C ⊆ e∈E (eWe )π . This means C π ⊆ e∈E H eWe , and we obtain ) ) ← R = R ∩ Cπ ⊆ R ∩ H eWe ⊆ (RWeι ∩ H e)We e∈E
⊆
)
e∈E
e∈E
(T ∩ H e)We ⊆ (G R) ∪
) e∈E
Ve .
Thus R ⊆ e∈E Ve ⊆ e∈E α∈Fe Vα is covered by a finite subfamily of (Vα )α∈A . We have proved that R is a compact neighborhood, and assertion (c) is established.
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In order to complete the proof of assertion (b), we assume that H and G/H are even compact and observe that we can choose U = G = T = R and C = G/H in the proof above. Assertion (d) follows from the observation that G/H discrete implies that H is open in G. 2 6.8 Theorem. In every locally compact group G the connected component G1 is the intersection of all open subgroups of G. If G is compact then G1 is the intersection of all open normal subgroups of G. Proof. We know from 2.9 that G/G1 is totally disconnected. Applying 4.13, we have the assertions for G/G1 . Now the πG1 -pre-images of open (normal) subgroups in G/G1 are open (normal) subgroups in G. 2 6.9 Proposition. Let G be a locally compact group, let H be a closed subgroup of G, and let π : G → G/H be the natural map. (a) The connected component (G/H )1π coincides with the closure of (G1 )π . (b) If G is totally disconnected then G/H is totally disconnected. (c) Conversely, if both H and G/H are totally disconnected then G is totally disconnected. Proof. Assume first that G is totally disconnected, and consider an element x in G/H {1π }. As G/H is Hausdorff, there is a neighborhood U of 1π in G/H such ← that x ∈ / U . Then U π is a neighborhood of 1 in G, and we find an open compact ← subgroup S of G such that S ⊆ U π , see 4.13. As π is an open map, the set S π is open and compact in G/H . Thus S π contains the connected component C of 1π in G/H . Since x ∈ G/H {1π } was arbitrary, this means C = {1}, and G/H is totally disconnected. We have thus established assertion (b). In order to prove assertion (a), consider an arbitrary locally compact group G, and a closed subgroup H of G. We put S := G1 H , then G/S is homeomorphic to (G/G1 )/(S/G1 ), and totally disconnected by assertion (b). As H ≤ S, we have a map κ : G/H → G/S such that πS = π κ, where πS : G → G/S is the natural map. For the connected component C of 1π in G/H , we obtain C κ ⊆ 1πS . Thus π C ⊆ S π = H G1 ⊆ (G1 )π . The reverse inclusion is obvious, and assertion (a) is established. Finally, assume that H is totally disconnected. If G1 = {1} then (G1 )π is a connected subset of G/H containing more than one point, and G/H is not totally disconnected. This proves assertion (c). 2 If one forms the quotient by a subgroup that is not closed, then the quotient is not Hausdorff and cannot be totally disconnected.
6. Quotients
57
6.10 Theorem. Let G be a topological group, and let H be a subgroup of G. (a) If G is σ -compact then G/H is σ -compact. (b) If G is σ -compact and H is closed in G then H is σ -compact. (c) If both H and G/H are locally compact and σ -compact then G is locally compact and σ -compact. Proof. Let π : G → G/N be the natural surjection. Assume first that subG is σ -compact, and let (Cn )n∈N be a sequence of compact sets such that G = n∈N Cn . Then the sets Cnπ are compact, and G/H = n∈N Cnπ is σ -compact. If H is closed in G then Cn ∩ H is compact, and H = n∈N Cn ∩ H is σ -compact. Now assume that both H and G/H are locally compact and σ -compact. From 6.7 we know that G is locally compact. Let (Dn )n∈N be a sequence of compact subsets of G/H such that G/H = n∈N Dn , and let Cn be a sequence of compact subsets of H such that H = n∈N Cn . Let n ∈ N. As H is locally compact, ← each element f ∈ Dnπ possesses a compact neighborhood Uf in H . As π is an open map, the collection of images Ufπ contains a neighborhood for each elπ ← such ement of D F n . Since Dn is compact, there exists n ⊆ Dn a finite πsubset ← ← π π π = f ∈Fn H Uf ⊆ that Dn ⊆ f ∈Fn Uf . Now we find Dn ⊆ f ∈Fn Uf Ck Uf . The sets Ck Uf are compact, and we have found a covering k∈N f ∈Fn π ← G = n∈N Dn = n∈N k∈N f ∈Fn Ck Uf of G by countably many compact sets, as required. 2 The compactly generated locally compact groups will turn out to form a very manageable class of topological groups. At present, we observe some rather simple properties. 6.11 Theorem. Let G be a topological group, and let N be a normal subgroup of G. (a) If G is compactly generated then G/N is compactly generated. (b) If G is locally compact and both N and G/N are compactly generated then G is compactly generated. Proof. If C ⊆ G is compact and G = C then C πN is compact and G/N = C πN . This proves assertion (a). Now assume that G is locally compact, that N = C for some compact subset C of N , and that G/N = D for some compact subset D of G/N . The πN pre-image of D will be denoted by E. We choose a compact neighborhood U of 1 in G. Then D ⊆ e∈E (U ◦ e)πN , and we find a finite subset F of E such that D ⊆ (U F )πN . Now the set CU F is compact, since it is the image of the compact 2 set C × U F under μ, and CU F = G. This proves assertion (b).
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6.12 Corollary. (a) Every locally compact group contains an open subgroup which is compactly generated. (b) A locally compact group G is compactly generated exactly if G/G1 is compactly generated. In particular, every connected locally compact group is compactly generated. Proof. If G is locally compact, we take a compact neighborhood C and find that H := C is a compactly generated open subgroup, and assertion (a) follows. As a connected group has no proper open closed subgroup, we infer that every connected locally compact group is compactly generated. Thus assertion (b) follows from 6.11. 2 6.13 Definition. Let G be a topological group. We say that G has no small subgroups if there is a neighborhood of the neutral element in G containing no subgroup except the trivial one. This property plays a crucial role in the theory of locally compact groups. Prominent examples of groups having no small subgroups are the (additive) groups Rn , where n is a natural number. At the present stage, we only note some rather trivial aspects. We leave the easy proof as an exercise. 6.14 Lemma. Let G be a topological group. (a) If G is discrete then G has no small subgroups. (b) If G has an open subgroup H having no small subgroups then G has no small subgroups. (c) If G has no small subgroups and ϕ : H → G is a continuous injective homomorphism then H has no small subgroups. 6.15 Theorem. The properties of being connected, totally disconnected, compact, locally compact, discrete, compactly generated locally compact, σ -compact locally compact, or having no small subgroups, respectively, are extension properties. Proof. We have seen in 6.7, 6.9, 6.11, and 6.10 that the properties of being connected, compact, locally compact, discrete, totally disconnected, compactly generated locally compact, or σ -compact locally compact, respectively, are extension properties. Assume that G is a topological group, and that N is a normal subgroup of G such that both N and G/N have no small subgroups. This means that there are neighborhoods U and V of 1 in G such that U ∩ N contains no subgroup except the trivial one, and every subgroup contained in V is also contained in N . Then U ∩ V contains only the trivial subgroup. 2
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59
Topological Aspects of the Isomorphism Theorems As a first step towards a complete understanding of split extensions (which we will achieve in 10.14), we prove the following. 6.16 Lemma. Let G and H be topological groups, and assume that σ : H → G and π : G → H are continuous homomorphisms such that σ π = idH . Then H is isomorphic to G/ ker π as a topological group. Moreover, the map σ is an embedding, and π is a quotient map. Proof. Let κ : G → G/ ker π be the natural map, and let γ : G/ ker π → H be the unique map such that κγ = π. Then γ is a bijective continuous homomorphism, and we observe (σ κ)γ = σ π = idH . In order to establish that γ is a two-sided inverse for σ κ, we compute κγ = κγ (σ κ)γ and deduce idG/ ker π = γ (σ κ) using the facts that γ is injective and κ is surjective. As γ is a homeomorphism, we have that π = κγ is a quotient map. The map σ induces a bijection τ : H → H σ whose inverse τ −1 is just the restriction of π to H σ , and therefore continuous. This shows 2 that σ is an embedding. 6.17 Theorem. Let H be a subgroup of a topological group G, and let π : G → Q be a quotient morphism with kernel N . Then ((N ∩ H )h)β := N h defines a continuous bijective homomorphism β from H /(N ∩ H ) onto N H /N, and (N h)α := hπ defines an isomorphism α of topological groups from N H /N onto H π . Proof. Let γ : H → NH be the inclusion, and let πN ∩H : H → H /(N ∩ H ) and πN : N H → N H /N be the natural maps. Then πN ∩H β = γ πN is continuous, and thus β is continuous since πN ∩H is a quotient map. H
γ
πN∩H
_ H /(N ∩ H )
/ NH
/G
πN
β
_ / N H /N
π
α
/ Hπ
_ /Q
The restriction ϕ of π to NH is a quotient map, since N H is π -saturated, see 1.35. Thus ϕ is continuous and open. Now ϕ = πN α implies that α is continuous and 2 open. In general, the continuous bijection β in 6.17 is not open. 6.18 Example. For every real number r, the subgroup rZ is closed in R. Consider ,2 R/Z . If r is not rational, then the restriction q|rZ is the natural map q : R a continuous injection, but its image (rZ)q is not homeomorphic to rZ.
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Recall that a topological space is called σ -compact if it is the union of a countable family of compact subspaces. Every compactly generated group (and, a fortiori, every connected locally compact group) is σ -compact. 6.19 The Open Mapping Theorem. Assume that G is a locally compact, σ compact group. Then every surjective continuous homomorphism from G onto a locally compact Hausdorff group is an open map. Proof. Let (Cn )n∈N be a sequence of compact subsets of G whose union is G. Consider a surjective continuous homomorphism ϕ from G onto a locally compact Hausdorff group H , and put K := ker ϕ. We can factor ϕ = πK β with a continuous homomorphism β : G/K → H . Since πK is open, it suffices to show that β is open; that is, it suffices to consider the case where ϕ is injective. Since ϕ −1 is a homomorphism between topological groups, it remains to show that ϕ −1 is continuous at 1. Let U be a neighborhood of 1 in G. We find a compact neighborhood V of 1 in G ◦ such that V = V ι and V V ⊆ U . For each natural number n, we have C n ⊆ Cn V ◦ and find a finite n ⊆ Fn V . Now G = n∈N Fn V set Fn ⊆ Cn such that C ϕ and H = n∈N (Fn V )ϕ . The set A := n∈N Fn is countable, and we have H = a∈A aV ϕ . According to 1.29, the space H is not meager. Thus at least one of the sets aV ϕ has nonempty interior. This means that V ϕ is a neighborhood of v ϕ for some v ∈ V , and we obtain that W := V ϕ V ϕ is a neighborhood of 1 in H −1 with W ϕ = V V ⊆ U . 2 6.20 Example. Consider the group R with the discrete topology D and the usual topology U. Then (R, D) and (R, U) are locally compact groups, but (R, D) is not σ -compact. The identity is a continuous bijective homomorphism from (R, D) onto (R, U), but not open. Even a continuous bijective homomorphism of a topological group onto itself need not be open, as the following example shows. 6.21 Example. Let c be an infinite set. As in 3.23 (a), we construct a topology on A := Qc such that A is a topological group, the subgroup Zc is open, and the topology induced on Zc is the product topology. Now the homomorphism nA : A → A defined by a nA = na is continuous and bijective for each positive integer n. However, the fact that nZc is not open for n ≥ 2 shows that nA is not an open map. 6.22 Example. Let F be a nontrivial finite discrete group, and consider the product P := F c , where c is an arbitrary infinite cardinal number. Then there are two different topologies (at least) that turn P into a topological group: firstly, the discrete topology D, and secondly, the product topology P . Leaving the details for an exercise, we claim that there are isomorphisms of topological groups
6. Quotients
61
α : (P , D)2 → (P , D) and β : (P , P )2 → (P , P ). We define γ : P 3 → P 3 by stipulating ((u, v)α , y, z)γ = (u, v, (y, z)β ). Then the map γ is a continuous bijective homomorphism from (P , D) × (P , D) × (P , P ) onto itself which is not open. 6.23 Theorem. Let G be a topological group, and assume that A and B are subgroups of G such that AB = G and A ∩ B = {1}. Then the following hold. (a) The map ϕ : A × B → G defined by (a, b)ϕ = ab is a continuous bijection (but not necessarily a homomorphism). (b) If, and only if, both A and B are normal in G then ϕ is a homomorphism. (c) If B is closed and normal in G, both A and G/B are locally compact and A is σ -compact, then ϕ is a homeomorphism. Proof. The map ϕ is a restriction of the continuous multiplication in G, and therefore continuous. Surjectivity of ϕ is granted by our assumption AB = G. As equality ab = (a, b)ϕ = (c, d)ϕ = cd implies c−1 a = bd −1 ∈ A ∩ B = {1}, the map ϕ is injective. If both A and B are normal subgroups, we have a −1 b−1 ab ∈ A∩B = {1} for all a ∈ A and all b ∈ B. This implies that ϕ is a homomorphism. Conversely, if ϕ is a homomorphism (and thus an isomorphism of groups), we have that A = (A × {1})ϕ and B = ({1} × B)ϕ are normal subgroups. Now assume that B is closed and normal in G, both A and G/B are locally compact and A is σ -compact. As B is closed in G, the quotient G/B is Hausdorff. Restricting the natural map π : G → G/B to the subgroup A we obtain a continuous bijective homomorphism ψ : A → G/B, and we know from 6.19 that ψ is an open −1 −1 −1 map. This means that π ψ −1 is continuous. Writing g ϕ = (g π ψ , g(g π ψ )−1 ) we see that ϕ −1 is continuous. Thus ϕ is a homeomorphism. 2 6.24 Definition. Let G be a topological group, and assume that A and B are normal subgroups of G such that AB = G and A∩B = {1}. If the map ϕ defined as in 6.23 is an isomorphism of topological groups, we write G = A ⊕ B and say that G is the interior direct product of A and B. 6.25 Example. Consider the subgroups A = Z, B = bZ for an irrational real number b, and G = A+B in R. Then A, B and A×B are discrete and A∩B = {0}, but G is not discrete. Thus the general assumptions of 6.23 are satisfied, but ϕ is not an isomorphism of topological groups.
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Cyclic Subgroups 6.26 Weil’s Lemma. For each locally compact Hausdorff group G, the following hold. (a) Let ϕ : Z → G be a homomorphism. Then either ϕ induces an isomorphism of topological groups from Z onto Zϕ , or Zϕ is a compact Abelian group. (That is, for each g ∈ G, the cyclic subgroup C generated by g is either discrete and infinite, or has compact closure in G.) (b) Let ϕ : R → G be a continuous homomorphism. Then either ϕ induces an isomorphism of topological groups from R onto Rϕ , or Rϕ is compact. Proof. We treat both assertions simultaneously. Assume H ∈ {Z, R}, and let ϕ : H → G be a homomorphism. If H = R, assume in addition that ϕ is continuous (for H = Z this is anyway the case). Without loss, we replace G by H ϕ . Then G is commutative. We use additive notation. If ϕ is not injective, we pick a nontrivial element k ∈ ker ϕ, and factor ϕ = πβ, where π : H → H /k is the natural map and β : H /k → G is the continuous homomorphism mapping k + h to hϕ . As H /k is compact, we obtain that H ϕ = H πβ is compact, and H ϕ = G. Assume now that ϕ is injective. As H ϕ is dense in G, every neighborhood in G ← has nonempty ϕ-pre-image. If there exists a neighborhood U in G such that U ϕ ← has compact closure C in H , then U ∩ H ϕ = U ϕ ϕ has compact closure C ϕ in H ϕ , and H ϕ is locally compact. According to 6.19, the map ϕ induces an open map from H onto H ϕ . As ϕ is a continuous injective homomorphism, we obtain that ϕ induces an isomorphism of topological groups from H onto H ϕ . There remains the case that ϕ is injective and each neighborhood V in G has unbounded ϕ-pre-image. If the pre-image of some V were bounded above, we could pick a point t ∈ H such that t ϕ ∈ V ◦ and observe that 2t ϕ − V is a neighborhood of t ϕ whose pre-image is bounded below. This yields a neighborhood V ∩ (2t ϕ − V ) with bounded pre-image, contradicting our assumption. Thus we obtain that for each h ∈ H the image of [h, ∞[ under ϕ meets each neighborhood V in G, and is therefore dense in G. We pick a compact neighborhood U of 0 in G. Using 3.17, we infer G = [h, ∞[ϕ = U ◦ + [h, ∞[ϕ . As U is compact, we find a finite subset F of ]0, ∞[ such that U ⊆ U + F ϕ . Let m denote the biggest element of F . ← For an arbitrary element g of G, the set [0, ∞[ ∩ (U + g)ϕ has a smallest element s. From s ϕ − g ∈ U we infer the existence of f ∈ F such that s ϕ − g ∈ U + f ϕ . This means that (s − f )ϕ ∈ U + g, and s − f < 0 by the minimality of s. Thus we have s < f ≤ m, and we obtain g ∈ s ϕ − U ⊆ [0, m]ϕ − U . As m does 2 not depend on g, this means that G ⊆ [0, m]ϕ − U is compact. Assertion (a) of Proposition 6.26 will be used in order to derive a first step towards a deeper understanding of locally compact commutative Hausdorff groups
6. Quotients
63
in 6.31 below. We introduce the notion of free Abelian group, which will be used in a lemma crucial to the proof of 6.31. 6.27 Definition. An Abelian group F is called free Abelian (of rank c) if there are a set X of cardinality c and a map ι : X → F such that for each Abelian group A and every map ϕ : X → A there is a unique group homomorphism ϕ : F → A such that ϕ = ι ϕ. It is easy to see that ι has to be an injective map. From ι idF = ι and the uniqueness of ι one infers ι = idF . Therefore, it is no loss to assume X ⊆ F and ι = idF |X , and we will do so in the sequel. The free Abelian groups of finite rank are easy to describe: 6.28 Example. For every natural number n, the group Zn is free Abelian of rank n. In fact, we put X := {1, . . . , n} and let j ι be the function that maps j to 1, and every x ∈ X {j } to 0. If ϕ : X → A is any map, we observe that the map ϕ : Zn → A defined by ϕ ϕ ϕ (z1 , . . . , zn ) = z1 1 + · · · + zn n is the unique homomorphism required in 6.27. The reader should be aware that Zc need not be free Abelian, if c is allowed to be infinite. See [23], A1.65; compare also the remarks on Whitehead’s Problem in Appendix 1 of [23]. 6.29 Lemma. Let A be an Abelian topological group. If B is an open subgroup of A such that A/B is a free Abelian group then A is the interior direct product A = B ⊕ C of B and some discrete subgroup C isomorphic to A/B. Proof. Pick a set X and a map ι : X → F := A/B as in 6.27. For each element x ∈ X we pick x ϕ ∈ A such that B + x ϕ = x ι ; this is possible since ι is injective. This defines a map ϕ : X → A, and we have ϕκ = ι, where κ : A → F is the natural map. The homomorphism ϕ κ satisfies ι( ϕ κ) = ϕκ = idX . Therefore, we ϕ have ϕ κ = ι = idF . We put C := F and observe A = B + C (since C κ = F ) and B ∩ C = {0} (since the restriction of κ to C is injective). Mapping (b, c) to b + c we obtain thus a bijective homomorphism from B × C onto A, whose restriction to the open subgroups B × {0} and B is a homeomorphism. Thus it is an isomorphism of topological groups; see 3.33. 2 6.30 Corollary. Let A be an Abelian topological group. If B is a subgroup of A such that A/B is discrete and isomorphic to Zn for some natural number n then A is isomorphic to B × Zn as a topological group, where the cartesian product carries the product topology. Proof. This follows from 6.29 and the fact that Zn is a free Abelian group.
2
6.31 Proposition. Let A be a locally compact commutative Hausdorff group. Assume that V is a compact neighborhood of 0 in A, and let B denote the subgroup
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generated by V . Then there exists a discrete subgroup D of B such that B/D is compact and D ∩ V = {0}. Moreover, one can choose the group D such that it is isomorphic to Zd for some natural number d. Proof. As the group B is open in A, it is locally compact. Since V is a neighborhood of 1, the set W := V ∪ (−V ) is a compact neighborhood of 1 which generates B and satisfies W = −W . Put W0 = {0} and define Wn inductively by Wn+1 = Wn + W . Then B = n∈N Wn . Since W2 is compact, there exists a finite subset F of B such that W2 ⊆ F + W ◦ ⊆ F + W . Let C denote the subgroup generated by F . From W1 ⊆ W2 ⊆ C + W and the fact that Wn ⊆ C + W implies Wn+1 ⊆ C + W2 ⊆ C + C + W = C + W we infer B = C + W . As C is finitely generated, the set of all natural numbers n with the property that there exists an injective homomorphism from Zn to C is bounded. Thus the set of all natural numbers n with the property that C contains a discrete subgroup isomorphic to Zn has a maximal element, say d. Pick a discrete subgroup D of C isomorphic to Zd . Then D ∩ W is finite, and passing to mD for a suitable positive integer m we can achieve {0} = D ∩ W ⊇ D ∩ V . Let π : B → B/D be the natural map. Being discrete, the subgroup D is closed in B, and B/D is a locally compact Hausdorff group. According to 6.30, the group C π does not contain any discrete infinite cyclic groups: otherwise, we could find a discrete subgroup isomorphic to Zd+1 in C, contradicting our choice of d. We claim that C π is compact. Indeed, for any c ∈ C the cyclic subgroup cπ is either finite or non-discrete. By 6.26 (a), the closure of cπ in B/D is compact, π and we infer that C = f ∈F f π is compact. 2 Finally, we observe that B/D = C π + W π = C π + W π is compact. We close this section with an elementary characterization of finite cyclic groups. 6.32 Theorem. A finite group G is cyclic if, and only if, there is at most one subgroup of order d, for each divisor d of |G|. Proof. We use Euler’s function φ(d) := |Z(d)× | = |{a ∈ Z(d) | ord(a) = d}| counting the number of generators of a cyclic group of order d: the observations ) * {a ∈ Z(n) | ord(a) = d} = n = |Z(n)| = φ(d) d|n
d|n
and ) * {a ∈ Z(n) | ord(a) = d} {a ∈ G | ord(a) = d} = |G| = d|n
lead to the claim.
d|n
2
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6.33 Corollary. Let F be a commutative field, and let G be a finite subgroup of F × . Then G is cyclic. Proof. Every subgroup of order d in G consists of roots of the polynomial Xd − 1. Since a polynomial of degree d has at most d roots in any commutative field, our characterization 6.32 applies. 2
Exercises for Section 6 Exercise 6.1. Show that R/Z and U(1, C) are isomorphic as topological groups. Exercise 6.2. Show that the natural map from R onto R/Z is not closed. Exercise 6.3. Find examples of quotient maps πH : G → G/H where H is a non-compact subgroup but πH is a closed map. Exercise 6.4. Prove 6.14. Exercise 6.5. Show that Rn has no small subgroups, where n is a natural number. Exercise 6.6. Let n be a natural number. Show that the group GL(n, C) has no small subgroups. Hint. Consider a neighborhood U of the neutral element that is so small that the characteristic values of elements of U are close to 1. Exercise 6.7. Let (Gα )α∈A be a family of topological groups all of which are nontrivial and have no small subgroups. Show that the product α∈A Gα has no small subgroups exactly if A is finite. Exercise 6.8. Let G be a topological group, and let H be a subgroup of G. Show that H is open in G exactly if G/H is discrete. Exercise 6.9. Show that Rn is σ -compact, for each n ∈ N. Exercise 6.10. Show that, in any topology, the group (Q, +) is σ -compact. Exercise 6.11. On the group Q, let D be the discrete topology, and let T be the topology induced by the usual topology on R. Show that the group (Q, +, D) is not compactly generated, but the group (Q, +, T ) is. Hint. Use that a convergent sequence (plus its limit point) yields a compact subspace of R. Exercise 6.12. Prove the claims made in 6.22. Exercise 6.13. A group is called a torsion group if it is the union of its finite subgroups. Show that being a torsion group is an extension property, and that every quotient of a torsion group is a torsion group. Exercise 6.14. A topological group is called compact-free if it has no nontrivial compact subgroup, and it is called torsion-free if it has non nontrivial finite subgroup. Show that these properties are extension properties. Give examples of compact-free or torsion-free groups with nontrivial quotients that fail to have these properties.
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7 Solvable and Nilpotent Groups In the study of topological groups, one often restricts oneself to the class of Hausdorff groups, for various sensible reasons. In fact, this is not so severe a restriction, for instance, closedness of the trivial subgroup {1} forces a topological group to be Hausdorff, see 6.6. However, one has to be careful with concepts from abstract group theory when passing to non-discrete groups: according to 6.6, the quotient by a normal subgroup is Hausdorff exactly if the normal subgroup is closed.
Hausdorff Solvable Groups We start our investigation with the derived series of a group and the corresponding notion of solvability. 7.1 Definition. Let G be a topological group. (a) The derived group is the subgroup d(G) generated by {x −1 y −1 xy | x, y ∈ G}. We define the Hausdorff derived group D(G) to be the closure d(G) of d(G) in G. Inductively, we obtain dn+1 (G) = d(dn (G)) and Dn+1 (G) = D(Dn (G)). The sequence (dn (G))n∈N{0} is called the derived series of G, while (Dn (G))n∈N{0} is called the Hausdorff derived series. (b) The group G is called solvable if its derived series terminates at {1}; that is, there is a natural number s such that ds (G) = {1}. We say that G is Hausdorffsolvable if the Hausdorff derived series terminates at {1}; that is, there is a natural number h such that Dh (G) = {1}. The least possible values for s and h are called the derived length and the Hausdorff derived length of G, respectively. 7.2 Remark. Note that Dn (G) is closed in G, for each natural number n. In particular, every Hausdorff-solvable group is Hausdorff. 7.3 Example. With the topology {∅, G}, every group G becomes a topological group. If G is Abelian, we thus obtain d(G) = {1} but D(G) = G. 7.4 Lemma. For each n ∈ N {0}, the subgroups dn (G) and Dn (G) are fully invariant in the topological group G; that is, every continuous homomorphism ϕ : G → G satisfies dn (G)ϕ ≤ dn (G) and Dn (G)ϕ ≤ Dn (G). Proof. As ϕ maps x −1 y −1 xy to (x ϕ )−1 (y ϕ )−1 x ϕ y ϕ , we obtain d(G)ϕ ≤ d(G), and D(G)ϕ ≤ D(G) since ϕ is continuous. Restricting ϕ to dn (G) and Dn (G), we 2 inductively obtain the assertion.
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As a consequence of Lemma 7.4, we know that the derived series and the Hausdorff derived series of a topological group G consist of normal subgroups of G. The importance of the notion of solvability lies in the fact that the class of solvable groups is the smallest class of groups that contains all Abelian groups and is closed with respect to extensions. We obtain an analogous characterization of Hausdorff-solvable groups in Proposition 7.8 below. 7.5 Lemma. For each subgroup H of a topological group G, we obtain the inequality d(H ) ≤ d(H ). Proof. The map κ : G × G → G defined by (x, y)κ = x −1 y −1 xy is continuous. Inductively, we define Xn by putting first X1 := {(x, y)κ | x, y ∈ H } and then Xn+1 := {uv | u ∈ Xn , v ∈ X1 }. As (y, x)κ is the inverse of (x, y)κ , these sets are closed under inversion, and n∈N{0} Xn is the subgroup generated by X1 ; that is, coincides with d(H ). We form Yn analogously, only replacing H by H , and obtain Y1 ⊆ X1 since κ is continuous. Thus d(H ) = n∈N{0} Yn ⊆ n∈N{0} Xn ⊆ d(H ). 2 7.6 Lemma. For each topological group G and every natural number n, we have dn (G) ≤ dn (G) = Dn (G). Proof. We proceed by induction on n, starting with the trivial case n = 1. As the operators d(.) and D(.) are monotone, our induction hypothesis dn (G) ≤ Dn (G) implies dn+1 (G) ≤ d(Dn (G)), and dn+1 (G) ≤ d(Dn (G)) = Dn+1 (G) follows. The induction hypothesis Dn (G) ≤ dn (G) implies d(Dn (G)) ≤ d(dn (G)), and Lemma 7.5 yields d(dn (G)) ≤ dn+1 (G). Therefore, the assertion Dn+1 (G) = d(Dn (G)) ≤ dn+1 (G) follows. 2 Summing up our discussion, we obtain: 7.7 Theorem. Let G be a topological group. Then G is a solvable Hausdorff group exactly if G is Hausdorff-solvable. In this case, the Hausdorff derived length equals the derived length of G. 2 7.8 Proposition. Let G be a topological group. (a) For every natural number n, the quotient group dn (G)/ dn+1 (G) is Abelian, and Dn (G)/ Dn+1 (G) is an Abelian Hausdorff group. (b) The group G is solvable exactly if there are subgroups S0 , . . . , Sk of G such that for each n ∈ {1, . . . , k} the group Sn is a normal subgroup of Sn−1 with Abelian quotient Sn /Sn−1 , and S0 = G, Sk = {1}.
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(c) The group G is Hausdorff solvable exactly if there are subgroups S0 , . . . , Sk of G such that for each n ∈ {1, . . . , k} the group Sn is a normal subgroup of Sn−1 with Abelian Hausdorff quotient Sn−1 /Sn , and S0 = G, Sk = {1}. 7.9 Remark. Proposition 7.8 shows that the class of Hausdorff-solvable groups is the smallest class of Hausdorff groups that contains all Abelian groups and is closed with respect to extensions. Moreover, the class of Hausdorff-solvable groups is closed with respect to the forming of quotients by closed normal subgroups. Proof of Proposition 7.8. It is well known that H / d(H ) isAbelian for each group H . Observing that quotients by closed subgroups are Hausdorff, we obtain assertion (a). In order to prove assertion (c), assume first that G is Hausdorff-solvable. Then Sn := Dn (G) satisfies our requirements. Conversely, assume that S0 , . . . , Sk are subgroups as in c. Then S1 is closed in S0 = G, and G/S1 Abelian implies d(G) ≤ S1 . This yields D(G) ≤ S1 . Replacing Sn by Tn−1 := Sn ∩ D(G) and using induction on k, we obtain that G is Hausdorff-solvable. The well-known assertion (b) now follows from (c) by replacing the topology of G by any topology that renders G a topological group such that dn (G) is closed for each n; for instance, 2 the discrete one.
Hausdorff Nilpotent Groups We now turn to a variation of the theme of derived series: the (ascending and descending) central series, and the corresponding notion of nilpotency. 7.10 Definition. Let G be a topological group. (a) The descending (or lower) central series of G consists of the subgroups zn (G) generated by {x −1 y −1 xy | x ∈ G, y ∈ zn−1 (G)}, starting with z0 (G) = G. The group G is called nilpotent if zc (G) = {1} for some natural number c, the least possible c is called the nilpotency class of G. (b) The Hausdorff descending (or lower) central series of G consists of the subgroups Zn (G) obtained by taking the closure of the subgroup generated by the set {x −1 y −1 xy | x ∈ G, y ∈ Zn−1 (G)}, starting with Z0 (G) = G. The group G is called Hausdorff-nilpotent if Zd (G) = {1} for some natural number d, the least possible d is called the Hausdorff nilpotency class of G. (c) The ascending (or upper) central series (zn (G))n∈N of G is obtained as follows: put z0 (G) = {1}, and let zn+1 (G) be the full pre-image of the center of G/ zn (G) under the natural map from G onto G/ zn (G).
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7.11 Remarks. (a) The central series defined in Definition 7.10 consist of fully invariant subgroups. (b) If G is a Hausdorff group then its center z1 (G) is closed, and inductively one sees that zn (G) is closed in G. (c) Every Hausdorff-nilpotent group is Hausdorff. The following is well known (and proved by a simple induction argument, applied to G/ zn (G)): the set {x −1 y −1 xy | x ∈ G, y ∈ zn (G)} is contained in zn−1 (G). 7.12 Theorem. We have the inequality zn (G) ≤ Zn (G). If G is nilpotent of class c and Hausdorff then Zn (G) ≤ zc−n (G). Consequently, a topological group G is nilpotent Hausdorff if, and only if, it is Hausdorff-nilpotent. In this case, the Hausdorff nilpotency class equals the nilpotency class of G. Proof. The first assertion is proved by an easy induction. Now assume that G is nilpotent of class c. Then Z0 (G) = G = zc−0 (G), and assuming Zn (G) ≤ zc−n (G) we infer {x −1 y −1 xy | x ∈ G, y ∈ Zn (G)} ⊆ zn+1 (G). If G is Hausdorff, we know from 7.11 that zn+1 (G) is closed, and contains Zn+1 (G). 2 The inequalities zn (G) ≤ Zn (G) ≤ zc−n (G) explain the terms “lower” and “upper” central series. 7.13 Proposition. Let G be a nilpotent group. Then the following hold. (a) Every nontrivial normal subgroup of G has nontrivial intersection with the center of G. (b) Every proper subgroup of G is properly contained in its normalizer. Proof. In order to prove assertion (a), consider a nontrivial normal subgroup N, and pick the positive integer i in such a way that M := N ∩ zi (G) = {1} = N ∩ zi−1 (G). Then M is a normal subgroup of G, and the set of commutators {m−1 g −1 mg | m ∈ M, g ∈ G} is contained in M ∩ zi−1 (G) ≤ N ∩ zi−1 (G) = {1}. But this means that M is contained in the center z1 (G) of G, and we obtain i = 1. Assertion (b) is proved by induction on the nilpotency class of G, starting with the trivial case where the group G is Abelian. So assume that H is a proper subgroup of G with normalizer N. The center Z := z1 (G) is contained in N , and our induction hypothesis asserts that the normalizer N/Z of the subgroup H Z/Z of G/Z properly contains H Z/Z if H Z/Z is a proper subgroup of G/Z. Thus H is properly contained in G in that case. In the remaining case G/Z = H Z/Z, we infer G = H Z ≤ N. 2 7.14 Proposition. If G is a connected topological group then dn (G), Dn (G), zn (G) and Zn (G) are connected for each natural number n.
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Proof. We use the continuous map κ : G×G → G defined by (x, y)κ = x −1 y −1 xy. If A, B are connected subsets of G then (A × B)κ is connected. Describing the set dn (G) as in the proof of Lemma 7.5, we obtain it as a union of connected sets, all containing the neutral element. Thus dn (G) is connected. Analogously, we see that zn (G) is connected. The proof is completed by the observation that the closure of 2 a connected set is connected again. 7.15 Remark. As properties like being compact (Hausdorff) or being locally compact (Hausdorff) are preserved under the forming of closed subgroups and quotients by these, our notions of Hausdorff-solvable or Hausdorff-nilpotent also work well if one wishes to restrict oneself to the corresponding subclasses of the class of all Hausdorff groups. 7.16 Proposition. Being Hausdorff-solvable is an extension property, but nilpotency is not. Proof. Let G be an extension of N by Q. If Q is Hausdorff-solvable of derived length s, then Ds (G) ≤ N. Thus Dt (N ) = {1} implies Ds+t (G) = {1}, and G is Hausdorff-solvable of derived length at most s + t. The group of all permutations of a set with three elements has a commutative normal subgroup (namely, the group of all even permutations) with commutative quotient, but is not nilpotent. 2 For every subgroup S of a topological group G, an easy induction shows the inclusions Dn (S) ≤ Dn (G), Zn (S) ≤ Zn (G), and zn (S) ≤ zn (G). For each surjective continuous homomorphism ϕ : G → H onto a Hausdorff group H , we have (Dn (G))ϕ ≤ Dn (H ) and (Zn (G))ϕ ≤ Zn (H ). This yields the following. 7.17 Proposition. Let G be Hausdorff-solvable (nilpotent) group of derived length s (of class c), let S be a subgroup of G, and let ϕ : G → H be a surjective continuous homomorphism onto a Hausdorff group H . Then S and H are Hausdorff-solvable 2 (nilpotent) of derived length at most s (of class at most c). We leave the proof of the following as an exercise. a Hausdorff7.18 Proposition. Let A be a finite set. For each α ∈ A, let (Gα , Tα ) be
solvable topological group. Then the cartesian product Xα∈A Gα , α∈A Tα is a Hausdorff-solvable group. The analogous conclusion holds if we replace “solvable” by “nilpotent”. 7.19 Example. Let F be a commutative field. Then it is easy to see that the subgroup SL(n, F ) of GL(n, F ) contains the derived group d(GL(n, F )): in fact, we have det(g −1 h−1 gh) = 1 for all g, h ∈ GL(n, F ). With some more effort, one can show SL(n, F ) ⊆ d(GL(n, F )), and obtain equality. If F has at least 4 elements, or if n = 2, one can even prove SL(n, F ) = d(SL(n, F )). However, the group SL(2, F ) is solvable if |F | ≤ 3.
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Exercises for Section 7 Exercise 7.1. Let G be any group. Verify that the set {x −1 y −1 xy | x ∈ G, y ∈ zn (G)} is contained in zn+1 (G). Exercise 7.2. Prove that G/ d(G) is Abelian, for each group G. Exercise 7.3. Find groups G satisfying d(G) = G. Exercise 7.4. Determine dn (G) for G ∈ {GL(n, F), SL(n, F)} and F ∈ {R, C}. Exercise 7.5. Determine as well as the upper and lower central series of the derived series the group G = a0 b1 a, b ∈ F, a = 0 , where F is any commutative field. Exercise 7.6. We consider the group ⎧⎛ 1 a ⎪ ⎪ ⎪ ⎪ 0 1 ⎨⎜ ⎜ 0 0 G= ⎜ ⎜ ⎪ ⎪ ⎪⎝0 0 ⎪ ⎩ 0 0
c b 1 0 0
0 0 0 1 0
⎞ 0 0⎟ ⎟ 0⎟ ⎟ d⎠ 1
⎫ ⎪ ⎪ ⎪ ⎪ ⎬ a, b, c, d ∈ R , ⎪ ⎪ ⎪ ⎪ ⎭
endowed with the topology induced by GL(5, R). Show that the subset ⎫ ⎧⎛ ⎞ 1 0 c 0 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎟ ⎪ ⎪ 0 1 0 0 0 ⎬ ⎨⎜ ⎟ ⎜ ⎟ ⎜ Nf = ⎜0 0 1 0 0 ⎟ (c, d) ∈ Z(1, f ) + Z(0, 1) ⎪ ⎪ ⎪ ⎪ ⎝0 0 0 1 d ⎠ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ 0 0 0 0 1 forms a closed normal subgroup (for each f ∈ R). Compare the derived series and the Hausdorff derived series, as well as the upper and lower central and Hausdorff central series of G and of the quotient G/Nf . Hint. The results differ drastically in the cases where f is a rational number, or an irrational one, respectively. Use the fact that the set {1, f } generates a dense proper subgroup of R if f is irrational in order to find a sequence in d(G/Nf ) converging to an element outside d(G/Nf ). Exercise 7.7. Verify in detail the assertions in 7.17. Exercise 7.8. Prove 7.18. Exercise 7.9. Show that the cartesian product over an infinite family of solvable (nilpotent) groups may fail to be solvable (nilpotent). Exercise 7.10. Determine the derived series of SL(2, F ) in the cases where F is a field with 2 or 3 elements. 1x z 1y Exercise 7.11. Let H := x, y, z ∈ R , and let T be any Hausdorff topology on 1
H such that (H, T ) is a topological group. Show that d((H, T )) and D((H, T )) coincide.
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8 Completion This section contains a discussion of Cauchy filterbases (with the necessary concepts regarding uniformities), completeness, and completions. Special attention is paid to completions of topological groups, rings, and fields. The results will be needed mainly for the classification of locally compact fields, while we have tried to keep the rest of the text free of uniformities. The present section also contains some hints where a uniform point of view may lead to deeper understanding of results proved elsewhere. 8.1 Definition. Let X be a set. A nonempty set V of subsets of X2 is called a uniformity on X if the following conditions are satisfied: (a) Each V ∈ V contains the diagonal X := {(x, x) | x ∈ X}. (b) For all V , W ∈ V, there exists S ∈ V such that S ⊆ V ∩ W . (c) For each V ∈ V, there exists R ∈ V such that the set R ◦ R := {(x, z) | ∃y ∈ X : (x, y) ∈ R (y, z)} is contained in V . ↔
(d) For each V ∈ V, there exists T ∈ V such that T := {(y, x) | (x, y) ∈ T } is contained in V . If V is a uniformity on X, we call (X, V) a uniform space; the elements of V are also called entourages. A uniformity is called a uniform structure if it satisfies (e) If W ⊆ X contains some entourage V ∈ V then W itself belongs to V. Clearly, the set V uni := {W ⊆ X | ∃V ∈ V : V ⊆ W } is the smallest uniform structure containing a given uniformity V on X. Uniformities V and W on X are called equivalent if V uni = W uni , that is, if for each (V , W ) ∈ V × W there are S ∈ V and T ∈ W such that T ⊆ V and S ⊆ W. 8.2 Examples. There are two fundamental constructions: (a) Let (X, d) be a metric space. For ε > 0, put Vε := {(x, y) ∈ X2 | d(x, y) < ε}. Then Vd := {Vε | ε > 0} is a uniformity on X, called the uniformity defined by the metric d. (b) Let (G, T ) be a topological group. For any open neighborhood U of 1 in G, we put RU := {(x, y) ∈ G2 | yx −1 ∈ U } and LU := {(x, y) ∈ G2 | x −1 y ∈ U }, and obtain the right uniformity R := R(G,T ) := {RU | U ∈ T1 } and the left
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uniformity L := L(G,T ) := {LU | U ∈ T1 } on G. Continuity of multiplication and inversion imply 8.1 (c) and 8.1 (d), respectively. There is a minimal uniform structure containing both the left and right uniform structures on (G, T ), generated by the bilateral uniformity S(G,T ) := {RU ∩ LV | U, V ∈ T1 }. Quite obviously, these uniformities coincide if the group G is Abelian. In general, however, they are not equivalent, see Exercise 8.8. 8.3 Definition. Let (X, V) be a uniform space. For V ∈ V and Y ⊂ X we write := {z ∈ X | ∃y ∈ Y : (y, z) ∈ V }, and abbreviate x V := {x}V . There is a topology TV on X such that {x V | V ∈ V} is a neighborhood basis at x; namely
YV
TV := {U ⊆ X | ∀x ∈ U ∃V ∈ V : x V ⊆ U } . We call TV the uniform topology induced by V. A uniform space is called Hausdorff (compact, connected, etc.) if the uniform topology has this property. 8.4 Example. For V = RU ∈ R(G,T ) and Y ⊆ G, we have YV
= {z ∈ G | ∃y ∈ Y : zy −1 ∈ U } = U Y.
Analogously, we have Y (LU ) = Y U . 8.5 Remark. The basic idea behind the notion of uniformity is the concept of uniform continuity from calculus: what we introduce here is a way to compare the “size” of neighborhoods, even in cases where no metric is available: neighborhoods x V and y W may be thought of the same radius if V = W . Moreover, condition 8.1 (c) allows to pick an entourage of “half the radius”, and repeated application gives entourages R with R ◦ R ◦ R := {(w, z) ∈ X2 | ∃x, y ∈ X : (w, x), (x, y), (y, z) ∈ R} contained in any given entourage. Further iteration of this construction allows to “divide the radius of neighborhoods” by any positive integer: a procedure that is crucial for many arguments from calculus. 8.6 Lemma. A uniform space (X, V) is Hausdorff exactly if V ∈V V = X . Proof. We have S := V ∈V V = {(x, y) ∈ X 2 | ∀V ∈ V : y ∈x V } and X ⊆ S by 8.1 (a). If T := TV ∈ T2 , we infer {x} = T ∈Tx T = V ∈V x V for each x ∈ X, and conclude S = X . Conversely, assume T ∈ / T2 , then there are x, y ∈ X with x = y such that ↔
∩ y W = ∅ holds for all U, W ∈ V. There exists R ∈ V with R ⊆ U ∩ W , and we obtain (x, y) ∈ R ◦ R. This means (x, y) ∈ V ∈V V ◦ V ⊆ S. 2
xU
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8.7 Lemma. For every uniform space (X, V), we have TV ∈ T3 . Proof. We have to show that every neighborhood of x ∈ X contains a closed one. ↔
For V ∈ V, pick R ∈ V such that R ◦ R ⊆ V . Then z ∈ x R yields z R ∩ x R = ∅, ↔
and any y ∈ z R ∩ x R satisfies (x, y), (y, z) ∈ R. Thus (x, z) ∈ R ◦ R ⊆ V , and z ∈ x V follows. We have proved x R ⊆ x V . 2 Our main interest in uniform structures will be in the applications to topological groups, rings, and fields. The presence of binary operations necessitates the following construction (the details are left for an exercise, see Exercise 8.5). 8.8 Definition. Let (X, V) and (Y, W ) be uniform spaces. Then the sets
PV ×W := (x1 , y1 ), (x2 , y2 ) ∈ (X × Y )2 | (x1 , x2 ) ∈ V , (y1 , y2 ) ∈ W . are the members of a uniformity P := {PV ×W | V ∈ V, W ∈ W }, called the product of V and W , or simply the product uniformity on X × Y . The topology induced by the product uniformity on X × Y coincides with the product of the topologies induced by the uniformities V and W . 2 8.9 Definition. Let (X, V) be a uniform space, and let Y ⊆ X. Then V|Y := {V ∩ Y 2 | V ∈ V} is a uniformity on Y , called the induced uniformity. Note that the induced uniform structure V uni |Y coincides with V|Y uni . 8.10 Definition. Let (X, V) and (Y, W ) be uniform spaces. A map ϕ : X → Y is called uniformly continuous from (X, V) to (Y, W ) if, for each W ∈ W , there ← exists V ∈ V such that V ⊆ W (ϕ×ϕ) := {(u, v) ∈ X2 | (uϕ , v ϕ ) ∈ W }. 8.11 Examples. (a) Uniformities V and W on X are equivalent if, and only if, the identity is uniformly continuous from (X, V) to (X, W ), and also uniformly continuous from (X, W ) to (X, V). (b) Let (G, T ) and (H, U) be topological groups. A map ϕ : G → H is uniformly continuous with respect to the right uniformities if, and only if, for each U ∈ U there exists T ∈ T such that yx −1 ∈ T implies y ϕ (x ϕ )−1 ∈ U .
Uniformities on Groups 8.12 Lemma. In any topological group (G, T ), the topologies induced by the right, left or bilateral uniformity coincide with T . Proof. Compare the neighborhood bases at g ∈ G.
2
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8.13 Remark. The fact (cf. 8.7) that TV ∈ T3 whenever V is a uniformity constitutes a deeper reason for the fact that every topological group belongs to T3 , cf. 6.6. 8.14 Lemma. Let (G, T ) be a topological group. Then inversion ι : G → G : g → g −1 is a uniformly continuous map from (G, R(G,T ) ) to (G, L(G,T ) ). Proof. Simply note that RUι×ι = {(x ι , y ι ) | yx −1 ∈ U } = {(u, v) | v −1 u ∈ U } = 2 {(u, v) | u−1 v ∈ U ι } = LU ι . 8.15 Corollary. In every topological group, inversion is uniformly continuous with respect to the bilateral uniformity. 2 8.16 Proposition. In every Abelian topological group, the group operations are uniformly continuous with respect to the right (left, bilateral) uniformity. Proof. Let (G, +, −, 0, T ) be an Abelian topological group. Inversion is uniformly continuous by 8.14. For T ∈ T0 , pick U ∈ T0 such that U + U ⊆ T . Now ((u, v), (x, y)) ∈ PRU ×RU means (u, x), (v, y) ∈ RU , implying (x +y)−(u+v) = (x − u) + (y − v) ∈ U + U ⊆ T , and it follows that the image of PRU ×RU under 2 addition is contained in RT . Note that multiplication in a general topological group need not be uniformly continuous with respect to the left or right uniformity. In fact, we have (cf. also Exercise 8.8): 8.17 Lemma. Assume that (G, T ) is a topological group, and that V is a uniformity on G with TV = T such that multiplication μ : G2 → G : (g, h) → gh is uniformly continuous. Then V is equivalent to both R(G,T ) and L(G,T ) . In particular, the left and the right uniformities on (G, T ) are equivalent in this case. Proof. According to the definition 8.3, the set {1V | V ∈ V} forms a neighborhood basis at 1 in the topological group (G, TV ). In order to show that the uniformities V and R(G,T ) are equivalent, it therefore suffices to show that for each W ∈ V there exists V ∈ V such that R1V ⊆ W and V ⊆ R1W . Our assumption that μ be uniformly continuous implies that there is V ∈ V such that
PV ×V = (e, f ), (g, h) ∈ (G × G)2 | (e, g), (f, h) ∈ V
← ⊆ W (μ×μ) = (e, f ), (g, h) | (ef, gh) ∈ W . Using the fact that V contains G , we infer R1V = (x, y) ∈ G2 | yx −1 ∈ 1V = (x, y) ∈ G2 | (1, yx −1 ) ∈ V
= (1, x)μ , (yx −1 , x)μ | (1, yx −1 ) ∈ V
⊆ (e, f )μ , (g, h)μ | (e, g), (f, h) ∈ V ,
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and conclude R1V ⊆ W . Conversely, we use (x −1 , x −1 ) ∈ G ⊆ V to conclude that ((x, x −1 )μ , (y, x −1 )μ ) = (1, yx −1 ) ∈ W holds for each (x, y) ∈ V , and V ⊆ R1W is proved. Analogously, one sees that uniform continuity of μ implies 2 that V and L(G, T ) are equivalent, as well. 8.18 Lemma. Every continuous homomorphism ϕ : (G, T ) → (H, U) between topological groups is uniformly continuous from (G, R(G,T ) ) to (H, R(H,U) ). Proof. This follows immediately from the observations that, for each U ∈ U, we ← 2 have U ϕ ∈ T and (RU ϕ ← )ϕ×ϕ = {(x ϕ , y ϕ ) | (yx −1 )ϕ ∈ U } ⊆ RU . 8.19 Remark. It is easy to transfer (the proof of) 8.18 to left uniform structures. One could also use 8.14.
Completeness 8.20 Definition. Let (X, V) be a uniform space. (a) A filter(basis) C on X is called a Cauchy filter(basis) with respect to V if for each V ∈ V there exists C ∈ C such that C 2 ⊆ V . (b) The uniform space is called complete if every Cauchy filterbasis converges. (c) A topological group (G, T ) is called complete if the uniform space (G, R(G,T ) ) is complete. (d) A topological group (G, T ) is called bilaterally complete if the uniform space (G, S(G,T ) ) is complete. The notion of a complete topological group is more symmetrical than the definition appears to be. In fact, we infer from 8.14: 8.21 Lemma. Let (G, T ) be a topological group. Then (G, R(G,T ) ) is complete if, and only if, the uniform space (G, L(G,T ) ) is complete. 2 8.22 Lemma. Let ϕ : (X, V) → (X, W ) be a uniformly continuous map, and let C be a Cauchy filterbasis on X. Then C ϕ := {C ϕ | C ∈ C} is a Cauchy filterbasis, as well. Proof. For W ∈ W , pick V ∈ V such that V ϕ×ϕ ⊆ W . Then pick C ∈ C with 2 C × C ⊆ V , and observe C ϕ × C ϕ ⊆ W . 8.23 Lemma. For any filterbasis C on a uniform space (X, V) and each x ∈ X, the following are equivalent: (a) The filterbasis C converges to x ∈ X.
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C∈C
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C, and C is a Cauchy filterbasis.
Proof. Assume first that C converges to x, that is, every neighborhood U of x contains a member of C. Then every C ∈ C has nontrivial intersection with U , and x ∈ C. For V ∈ V, pick R ∈ V with R ◦ R ⊆ V , and choose C ∈ C such that ↔
C ⊆ x R ∩ x R. Then C 2 ⊆ R ◦ R ⊆ V shows that C is indeed a Cauchy filterbasis. Conversely, assume x ∈ C∈C C, and that C is a Cauchy filterbasis. Every neighborhood of x in the space (X, TV ) contains some x V , with V ∈ V. We pick R ∈ V with R ◦ R ⊆ V and C ∈ C with C × C ⊆ R. As x R is a neighborhood of x ∈ C, we find y ∈ x R ∩ C = ∅. Now (x, y) ∈ R and {y} × C ⊆ C × C ⊆ R 2 imply {x} × C ⊆ R ◦ R ⊆ V , and C ⊆ x V follows. 8.24 Corollary. A Cauchy filterbasis C on a uniform space (X, V) converges whenever the filterbasis {C | C ∈ C} contains a compact element. In particular, every compact uniform space is complete. Proof. Pick a compact element K ∈ {C | C ∈ C}, and apply the fact that every filterbasis of closed sets in a compact space has non-empty intersection (see 1.23) to the filterbasis {C ∩ K | C ∈ C}. 2 8.25 Theorem. Every locally compact Hausdorff group is complete. Proof. Let (G, T ) be a locally compact group, pick a compact neighborhood K of 1, and choose C ∈ C with C 2 ⊆ RK . For any c ∈ C, we thus have Cc−1 ⊆ K, and C is contained in the compact set Kc. 2 8.26 Corollary. Every discrete group is complete.
2
8.27 Remarks. In a Hausdorff uniform space, every complete subspace is closed. Thus 8.25 gives an explanation for the fact that locally compact (and, in particular, discrete) subgroups of Hausdorff groups are closed, cf. 4.7. Using 8.18, 8.22, and the fact that every open set in a topological group is a union of cosets of {1}, one may extend 8.25 to the non-Hausdorff case. We do not pursue this here.
Completion of Hausdorff Uniform Spaces 8.28 Definition. Let (X, V) and (Y, W ) be uniform spaces. (a) A map η : X → Y is called an embedding of uniform spaces (or briefly, a uniform embedding) if η is injective and both the bijection ψ : (X, V) → (Xη , W |Xη ) : x → x η and its inverse ψ −1 : (Xη , W |Xη ) → (X, V) are uniformly continuous.
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(b) A uniform embedding η : (X, V) → (Y, W ) is called a Hausdorff completion of (X, V) if the uniform space (Y, W ) is complete Hausdorff, and X η is dense in (Y, TW ). Briefly, we will just say that (Y, W ) is a completion of (X, V), if η is (supposed to be) clear. For the sake of conciseness (and uniqueness), we concentrate on Hausdorff spaces. The original idea motivating the following construction was to form a space of Cauchy filterbases (or Cauchy nets), and then identify objects that “should have the same limit”. We follow the presentation of [53], § 11, which in turn was inspired by [49]. This approach effectively hides the original quotient process: Instead of identifying Cauchy filterbases, we choose natural representatives in each class, namely minimal Cauchy filters, cf. 8.30 (f) below. 8.29 Definitions. For each filterbasis C on a uniform space (X, V), we write C V := {C V | V ∈ V, C ∈ C}, and abbreviate x V := {{x}}V. We are going to prove . := C V fil and that these are filterbases, the generated filters will be denoted by C xˆ := x V fil . 8.30 Lemma. Let B and C be Cauchy filterbases on a uniform space (X, V). . ⊆ C. . In particular, one has C . = C (a) The inclusion B ⊆ C fil implies B fil . .= C . . ⊆ C fil . (b) The collection C V is a Cauchy filterbasis, and C (c) For each x ∈ X, the filterbasis x V is a neighborhood basis at x in (X, TV ). .=C . ⇐⇒ ∀V ∈ V ∃D ∈ B fil ∩ C fil : D 2 ⊆ V . (d) We have B . = C. . (e) In particular, an inclusion B ⊆ C fil implies B . is the smallest Cauchy filter contained in C fil . (f) The filter C Proof. Consider B ∈ B and V ∈ V. Pick C ∈ C with C ⊆ B, then C V ⊆ B V . and B . ⊆ C. . yields B V ∈ C, In order to show that C V is a filterbasis, consider C, D ∈ C and V , W ∈ V. Pick E ∈ C and U ∈ V such that E ⊆ C ∩ D and U ⊆ V ∩ W , then E U ⊆ C V ∩ D W . Applying 8.1 (c) twice, we find T ∈ V such that T ◦ T ◦ T ⊆ V , and pick S ∈ V ↔
with S ⊆ T ∩ T . As C is a Cauchy filterbasis, there is C ∈ C with C 2 ⊆ T . Now C S × C S ⊆ V follows, and we have shown that C V is a Cauchy filterbasis. As V contains the diagonal X , we have C ⊆ C V , and obtain (C V)/ V ⊆ 0C V fil . Conversely, the inclusion (C T )T ⊆ C V for T ◦ T ⊆ V shows C V ⊆ (C V)V fil , and an application of assertion (a) completes the proof of assertion (b). Assertion (c) follows from the very definition of TV . . .=C . one deduces immediately B . ⊆ B fil ∩ C fil , and the fact that B From B is a Cauchy filterbasis (cf. assertion (b)) yields the existence of D as claimed in
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assertion (d). Conversely, assume that for each V ∈ V there exists D ∈ B fil ∩ C fil such that D 2 ⊆ V . For each C ∈ C we pick d ∈ D∩C = ∅, and {d}×D ⊆ V .⊆B . follows from assertion (b). yields D ⊆ C V . This shows C V ⊆ B fil , and C As this argument is symmetric in B and C, assertion (d) is proved, and assertion (e) follows immediately. Assertion (f) follows from the observation (e). 2 For each x ∈ X, the minimal Cauchy filter containing {x} clearly is the filter of all neighborhoods of x: this fits nicely with 8.30 (c). 8.31 Definitions. Let (X, V) be a uniform space. We write . := (X, . | C is a Cauchy filterbasis on (X, V) X V) := C for the set of minimal Cauchy filters (cf. 8.30 (f)), and .| V ∈V , . := V V where (for any V ⊆ X) we put
. := B, .2 | ∃B ∈ B . C . ∈X .∩C . : B2 ⊆ V . V . : x → x. ˆ Moreover, let η := η(X,V) : X → X . is represented by a Cauchy Note that 8.30 (e) implies that every element in X filter on (X, V), and we have .= C . | C is a Cauchy filter on (X, V) . X We are going to show that η(X,V) is a Hausdorff completion of (X, V) whenever (X, V) is a Hausdorff uniform space. . . is a uniformity on X. 8.32 Lemma. If (X, V) is a uniform space then V ↔ ↔ . .∩W . and V . = V are obvious Proof. For V , W ∈ V, the relations V ∩W ⊆ V . follows from 8.30 (d). consequences of our definitions. The condition X . ⊆V .◦V .⊆V .◦V . we find D . and . C) . ∈V .∈X It remains to show V ◦ V : for (B, 2 2 . . . . (B, C) ∈ (B ∩ D) × (C ∩ D) such that B ⊆ V ⊇ C . Now B ∩ C = ∅ yields B × C ⊆ V ◦ V ⊇ C × B, and V ⊆ V ◦ V then gives (B ∪ C) × (B ∪ C) ⊆ V ◦ V . .∩C . is a filter, it contains B ∪ C, and (B, . C) . belongs to V Since B ◦V. 2 . the filterbasis Cˇ := {B . | B ∈ B} . ∈ X, .∈X . |B∈C . 8.33 Lemma. For any C . T .). . in (X, is a neighborhood basis at C V . | C V ∈ B} .∈X . | V ∈ V, C ∈ C is a neighborhood In fact, even Cˇ := {B basis.
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. is a typical neighborhood of C . T .). We have to . in (X, Proof. Let V ∈ V, then C.V V . := {B . | B ∈ B} . such that B .∈X . is contained in .V find B ∈ C C , and we have to . show that each element of Cˇ is a neighborhood of C. . . Since C is a Cauchy filterbasis, we find B ∈ C with B 2 ⊆ V . From 8.30 (b) we .= C . such that C W ⊆ B. . . ⊆ C fil , and find W ∈ V together with C ∈ C know C 2 .. This means . . ∈V . . . For B ∈ B we have B ∈ B ∩ C, and B ⊆ V yields (C, B) . . . B ∈ C.V , and B ⊆ C.V as required. is a neighborhood of C, . ⊆ B: For . we show .W In order to show that B C 2 . . . . A ∈ C.W , we find A ∈ A ∩ C with A ⊆ W , and for each c ∈ A ∩ C = ∅ we have . yields B ∈ A. . Thus {c} × A ⊆ W . This leads to A ⊆ cW ⊆ C W ⊆ B, and A ∈ A . 2 A ∈ B. . V) . is Hausdorff. 8.34 Lemma. The uniform space (X, . . C) . ∈ . .V . . Proof. Consider (B, V ∈V . For each W ∈ V there exists D ∈ B ∩ C ⊆ 2 . . . . B ∩ C such that D ⊆ W , and 8.30 (d) yields B = C. Thus (X, V) is Hausdorff by 8.6. 2 . V) . is complete. 8.35 Lemma. The uniform space (X, . V). . As in the proof of 8.33, we define Proof. Let F be a Cauchy filterbasis on (X, . . . B := {B ∈ X | B ∈ B} for B ⊆ X. We claim that ∈ F fil C := B ⊆ X | B = ∅, B . Without loss of is a Cauchy filterbasis on (X, V), and that F converges to C. generality, we may replace F by F fil , and thus assume that F is a Cauchy filter. B ∈ F , and A ∩ B ∈ F because F is a For A, B ∈ C, the definition yields A, filter. Now ∩ B ⇐⇒ A, B ∈ B .∈A . ⇐⇒ A ∩ B ∈ B . ⇐⇒ B .∈A B ∩B shows A ∩ B ∈ F , and we have verified that C is a filterbasis. In order to show that C is a Cauchy filterbasis, we consider V ∈ V and construct C ∈ C with C 2 ⊆ V . Pick R ∈ V with R ◦ R ◦ R ⊆ V , and choose S ∈ V with ↔ . V), . we find F ∈ F with F 2 ⊆ . S ⊆ R ∩ R . As F is a Cauchy filter on (X, S. We . . . with B 2 ⊆ S. pick any B ∈ F , then B is a Cauchy filter, and there exists B ∈ B We obtain (B S)2 ⊆ V , and it remains to check that C := B S belongs to C: since F is a filter, it suffices to show F ⊆ C. . ∈ F , we have (A, . B) . ∈ F2 ⊆ . .∩ B . such For each A S, and there exists A ∈ A 2 that A ⊆ S. Pick a ∈ A ∩ B = ∅, then {a} × A ⊆ S yields A ⊆ a S ⊆ B S = C, as required. . implies C ∈ A. . Thus we have proved A . ∈ C, and A ∈ A
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. According to 8.33, it suffices to consider Finally, we show that F converges to C. 1 . | C V ∈ B} . . with C ∈ C and V ∈ V. neighborhoods of the form C V = {B ∈ X Since C is a Cauchy filterbasis, we find D ∈ C with D ⊆ C and D 2 ⊆ V . By our ⊆ C1 ∈ F , and D ⊆ D V ⊆ C V yields D V. 2 definition of C, we know D 8.36 Lemma. For every uniform space (X, V), the map η(X,V) is uniformly con tinuous, and its image is dense in (X, V). Proof. We abbreviate η := η(X,V) . Let W ∈ V, we search for V ∈ V such that ↔ . . Choose V , R, S ∈ V such that V ◦ V ⊆ R ⊆ S ∩ S and S ◦ S ⊆ W . V η×η ⊆ W Then (x, y) ∈ V implies y V ⊆ x R and x R × x R ⊆ W . The first of these inclusions ˆ The second means that x R is a neighborhood of both x and y, and x R ⊆ xˆ ∩ y. . . Thus V η×η ⊆ W , as claimed. inclusion now shows (x η , y η ) = (x, ˆ y) ˆ ∈W . T .). Let C be a It remains to show that Xη is dense in the topological space (X, V . Cauchy filterbasis on (X, V), and consider a neighborhood of C. According to 8.33, := {B . | B ∈ B} .∈X . with we may assume that this neighborhood is of the form B . B = C V ∈ C, where C ∈ C and V ∈ V. Pick c ∈ C, then cV is a neighborhood of c in (X, TV ), and cV ⊆ C V = B yields that B belongs to the neighborhood ∩ X η = ∅, as required. ∩ X η , and B 2 filter c. ˆ This shows that cˆ belongs to B 8.37 Lemma. If (X, V) is Hausdorff then η(X,V) is a completion. Proof. After 8.35 and 8.36, it remains to show that η := η(X,V) is injective, and that the inverse of the co-restriction ψ of η to its image is uniformly / continuous. 0 In the Hausdorff space (X, TV ), equality x V fil = xˆ = yˆ = y V fil of neighborhood filters (cf. 8.30 (c)) implies x = y. Thus η is injective. . we find . ∩ (Xη × X η ) = V η×η : in fact, for (x, ˆ y) ˆ ∈V For V ∈ V, we have V 2 2 B ∈ xˆ ∩ yˆ with B ⊆ V , and (x, y) ∈ B yields (x, y) ∈ V . This shows that ψ −1 is uniformly continuous. 2 In order to prove that the completion that we have constructed is as unique as possible (see 8.39 below), we introduce a lemma that will be of independent value, as well: for instance, it allows to extend a uniformly continuous group multiplication (or inversion) quite easily, see 8.40 below. 8.38 Lemma. Let (X, X), (Y, Y) and (D, D) be uniform spaces, with an embedding ι : (D, D) → (X, X) such that D ι is dense in (X, TX ). Assume that (Y, Y) is complete Hausdorff. For every uniformly continuous map ϕ : (D, D) → (Y, Y) there exists a unique continuous extension : (X, TX ) → (Y, TY ) (that is, a map such that ι = ϕ), and this extension is uniformly continuous. Proof. Without loss, we may identify D = D ι , and assume that ι is the identity. Existence and uniqueness of the extension have been proved in 1.46: recall from 8.7
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that TY ∈ T3 , and note that every convergent filterbasis in (D, TD ) is a Cauchy filterbasis, whose image under ϕ in Y is a Cauchy filterbasis, and converges to a unique point because (Y, Y) is complete Hausdorff. It remains to show that is uniformly continuous. For V ∈ Y, pick T ∈ Y such that T ◦ T ◦ T ⊆ V . Since ϕ is uniformly continuous, there exists S ∈ X such that (S ∩ D 2 )ϕ×ϕ ⊆ T , and we find R ∈ X with R ◦ R ◦ R ⊆ S. We claim that R × is contained in V . For (u, v) ∈ R, the -preimages of the neighborhoods u T and v T are neighborhoods of u and v, respectively, and we find a, b ∈ D such ↔
↔
that a ∈ uR , a ϕ ∈ u T , b ∈ v R, and bϕ ∈ v T . Then (a, b) ∈ (R ◦ R ◦ R) ∩ D 2 yields (a ϕ , bϕ ) ∈ T , and (u , v ) ∈ T ◦ T ◦ T ⊆ V follows, as claimed. 2 8.39 Corollary: Uniqueness of completion. For a uniform Hausdorff space (X, V), let η : (X, V) → (Y, Y) and γ : (X, V) → (Z, Z) be Hausdorff completions. Then there is a bijection : Y → Z such that η = γ , and both
: (Y, Y) → (Z, Z) and −1 : (Z, Z) → (Y, Y) are uniformly continuous. Proof. Apply 8.38 in the case (ι, ϕ) = (η, γ ) to obtain a uniformly continuous extension (that is, η = γ ). Interchanging γ with η gives an extension H of η (that is, γ H = η). Now the uniqueness assertion in 8.38 together with η H = γ H = η and γ H = γ yields H = idY and H = idZ . 2 This uniqueness result allows us to speak (somewhat loosely) of the completion of a uniform Hausdorff space.
Completions of Hausdorff Groups For Hausdorff groups with uniformly continuous multiplication, it is easy to extend the multiplication to the completion: 8.40 Theorem. Let (G, μ, ι, ν, T ) be a Hausdorff group, and assume that the multiplication μ : G × G → G is uniformly continuous with respect to the right uniformity. Then the multiplication μ has a unique extension . R . . . μ : (G, (G,T ) ) × (G, R (G,T ) ) → (G, R (G,T ) ), and inversion ι extends to . R . . ι : (G, (G,T ) ) → (G, R (G,T ) ) . . such that (G, μ,. ι,. ν, TR ) is a complete Hausdorff group. (G,T )
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Proof. First of all, recall from 8.17 that uniform continuity of μ implies that the left and the right uniformity are equivalent. Then 8.14 yields that inversion is uniformly continuous, as well. The candidate . ν for the neutral element is just the image of the neutral element of G under the embedding into the completion. We apply 8.38 to obtain the extensions . μ and. ι. Since associativity can be described by commutativity of the diagram .×G .×G . G MMM q q MMidM ×. . μ×id qq μ MMM qqq q M& q xq . . .×G . G × GM G MMM q q MMM qq MMM qqq. q . μ μ q MM& xqqq . G we obtain associativity of . μ from the uniqueness assertion in 8.38. Similarly, one proves that . ν actually yields a neutral element, and that . ι has the properties that 2 are required for inversion. Using 8.16, we obtain: 8.41 Corollary. Every Abelian Hausdorff group possesses a completion.
2
In the general case, where the multiplication is not uniformly continuous, it is still possible to extend the multiplication. However, we have to construct the extension explicitly. The following lemma will be needed in order to show that the product of Cauchy filterbases is well defined, and continuous. 8.42 Lemma. Let A be a Cauchy filterbasis with respect to the right uniformity R on a topological group (G, T ). Then for each X ∈ A and each U ∈ T1 there exist A ∈ A and S ∈ T1 such that AS ⊆ U X and A ⊆ X. Proof. By continuity of multiplication, we find T ∈ T1 such that T T ⊆ U . Since A is a Cauchy filterbasis, there exists Y ∈ A with Y Y −1 ⊆ T . We choose a ∈ X ∩ Y , then Y a −1 ⊆ T , and there exists S ∈ T1 with aSa −1 ⊆ T . This yields Y ⊆ T a and aS ⊆ T a. As A is a filterbasis, we may pick A ∈ A with A ⊆ X ∩ Y , and obtain AS ⊆ Y S ⊆ T aS ⊆ T T a ⊆ U a ⊆ U X. 2 8.43 Lemma. Let (G, T ) be a topological group, and let V be one of the uniformities R, L, or S. For Cauchy filterbases A and B in (G, V), the collection AB := {AB | A ∈ A, B ∈ B} is a Cauchy filterbasis in (G, V), as well. Proof. For filterbases A, B and any binary operation ∗, the collection A ∗ B := {A ∗ B | A ∈ A, B ∈ B} clearly is a filterbasis.
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We consider V = R first. For U ∈ T1 , we have to find A ∈ A and B ∈ B such that (AB)(AB)−1 ⊆ U . By continuity of multiplication, there exists T ∈ T1 such that T T ⊆ U . We pick X ∈ A with XX−1 ⊆ T . According to 8.42, we find A ∈ A and S ∈ T1 with AS ⊆ T X and A ⊆ X. Choosing B ∈ B such that BB −1 ⊆ S we find (AB)(AB)−1 = ABB −1 A−1 ⊆ ASA−1 ⊆ T XA−1 ⊆ T XX −1 ⊆ T T ⊆ U , as required. The uniformity L is treated analogously. Now consider Cauchy filterbases A and B with respect to V = S. Then A and B are Cauchy filterbases with respect to both the right and the left uniformities, and so is AB, by the preceding remarks. This means that AB is a Cauchy filterbasis with respect to S. 2 8.44 Theorem. Let (G, T ) be a topological Hausdorff group, and let V be one of . = (G, the uniformities R, L, or S. On the respective completion G V), we define . . . . . 2 a multiplication μV : G × G → G : (A, B) → AB. This multiplication is well defined, associative, and continuous. In other words: the completion ((G, V), μV ) is a topological semigroup. Proof. We consider the case V = R first, and abbreviate μ := μR . For Cauchy B . 3 and B 3 , we have to show AB .= A .=B 2 =A filterbases A, A , B, B with A 2 .B . = AB 2 . The inclusions A . ⊆ A fil and B . ⊆ B fil It suffices to show A .B . ⊆ AB fil . Now A .B . fil is a Cauchy filter (cf. 8.43) (cf. 8.30 (b)) yield A 2 ⊆ contained in the Cauchy filter AB fil , and minimality (cf. 8.30 (f)) yields AB 2 .B . fil . Finally, we apply 8.30 (b) to obtain AB 2 = AB 2 =A .B .. A . Our next aim Thus we have defined a binary operation μ on the completion G. . . is to show that μ is continuous at (A, B). According to 8.33, each neighborhood . contains a neighborhood of the form D := {D . | D ∈ D} . ∈ G .∈G . with of C . D = C (RV ) ∈ C, where C ∈ C and V ∈ T1 . Therefore, we have to consider X ∈ A, Y ∈ B together with U ∈ T1 , and find (A, B) ∈ A × B and S ∈ T1 such that μ maps A (RS ) × B (RS ) into the neighborhood XY (RT ). First of all, recall from 8.4 that Z (RU ) = U Z. Since multiplication is continuous on G, there exists U ∈ T1 such that U U ⊆ T . Using 8.42, we find A ∈ A and S ∈ T1 with AS ⊆ U X and A ⊆ X. Without loss, we may also assume S ⊆ U . Putting B := Y , we obtain (SA)(SB) ⊆ SU XB ⊆ U U XB ⊆ T XB = T XY . This implies μ
1 × SB 1 μ= D .E . | SA ∈ D, . SB ∈ E . (R ) × (R ) = SA A S B S ⊆ F. | SASB ∈ F. ⊆ F. | T XY ∈ F. = XY (RT ), as required. Associativity of the continuous operation μ follows from the fact that its restriction to a dense subset is associative.
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The proof for V = L is the same, we just have to reverse all products. For V = S, the arguments are analogous to those that we have given explicitly. 2 8.45 Remarks. The topological semigroup ((G, R), μR ) is called the Weil completion of the topological group G. It is easy to see that . 1 is a neutral element in the Weil completion. In general, however, inversion in G does not map Cauchy filterbases (with respect to R) to Cauchy filterbases, and there is no natural extension to the Weil completion. Analogous remarks apply to the semigroup ((G, L), μL ). The situation is much nicer for the semigroup ((G, S), μS ), called the Raikov completion of (G, T ). In fact, inversion is uniformly continuous with respect to S by 8.15, and extends to the Raikov completion. This means that the Raikov completion is a topological group. For Abelian groups, of course, the Raikov completion coincides with the Weil completion. 8.46 Proposition. Let (G, μ, ι, ν, T ) be a topological group. (a) If (G, R) or (G, L) is complete then (G, S) is complete. (b) If (G, S) is complete and (G, R) and (G, L) have the same Cauchy filterbases, then G is complete with respect to each one of these uniformities. Proof. Clearly, every Cauchy filterbasis with respect to S is also a Cauchy filterbasis with respect to R and L. Thus completeness with respect to any one of the uniformities R or L implies completeness with respect to S. Now assume that (G, R) is complete, and consider a Cauchy filterbasis A in (G, L). Then Aι := {Aι | A ∈ A} is a Cauchy filterbasis in (G, R). By our assumption, the filterbasis Aι converges to some point a ∈ G. Continuity of ι now yields that A converges to a ι . Analogously, one sees that completeness of (G, L) implies completeness of (G, R). Thus completeness with respect to any one of the one-sided uniformities implies completeness with respect to both one-sided uniformities. The assumption that R and L define the same Cauchy filterbases yields that every Cauchy filterbasis in (G, R) is one with respect to S, and completeness with respect to the bilateral uniformity follows from completeness of (G, R). 2 Note that the three uniformities may differ even if they are complete, see Exercise 8.8. Necessity of the extra assumption is shown by an example due to J. Dieudonné [9]. 8.47 Proposition. Let H be a dense subgroup of a complete Hausdorff group (G, T ). Then G is (isomorphic to) the completion of (H, T |H ).
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Proof. It suffices to note that the inclusion η : H → G is a uniform embedding (with respect to the right uniformities, say), and that the operations on G coincide with the operations introduced on the completion because we are dealing with continuous maps on Hausdorff spaces that coincide on dense subsets. 2 8.48 Corollary. Let G, C be Hausdorff groups, assume that C is complete, and let H be a dense subgroup of G. Then every continuous homomorphism from H to C has a unique extension to a continuous homomorphism from G to C. Proof. Apply 8.47 and 8.38, and note that a map between Hausdorff groups is a 2 homomorphism if its restriction to a dense subgroup is a homomorphism. 8.49 Examples. Let p be a prime. (a) Let P := Xn∈N Z(pn ) be endowed with component-wise addition and multiplication, and the product topology T . Then P is a compact Hausdorff ring, and there is a unique ring homomorphism η : Z → P , given by zη = (pn Z + z)n∈N . It is easy to see that η is injective, we will identify Z and Zη . We call Zp := Zη the ring (or the additive group) of p-adic integers, then Zp is the completion of the group (Z, Tp ), where Tp := T |Zη . Alternatively, the topology Tp may be described as the group topology generated by the filterbasis {pn Z | n ∈ N} according to 3.22. See 17.2 and Exercise 17.8 for an alternative definition, and a more detailed study of the compact ring Zp . (b) Every continuous homomorphism from (Z, Tp ) to a complete Hausdorff group G extends to a continuous homomorphism from Zp to G. (c) Let G be a compact (e.g., a finite) Hausdorff group, and let c be any set. Then Gc is the completion of G(c) .
Completion of Hausdorff Rings If (R, T ) is a Hausdorff ring, the additive group (R, +, T ) has a completion .+ ., T.), cf. 8.41. The multiplication on R need not be uniformly continuous, (R, and we cannot refer to a nice general result in order to extend it to a multiplication . We give an explicit construction in the sequel, using the explicit description on R. . that was obtained in 8.44. Note that we now use additive notation for the map of + μ in 8.44. . is a minimal Cauchy filter C, . repreRecall from 8.30 that every element in R sented by a Cauchy filterbasis C on (R, R(R,+,T ) ).
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. ˆ· B . := 8.50 Definition. For Cauchy filterbases A, B on (R, R(R,T ) ), define A A · B, where A · B := {AB | A ∈ A, B ∈ B}. This definition requires that we verify that A · B is a Cauchy filterbasis. To this end, and for the proof that ˆ· is continuous, we need the following: 8.51 Lemma. Let (R, T ) be a topological ring, and let C be a Cauchy filterbasis on (R, R(R,T ) ). Then for each T ∈ T0 there are C ∈ C and S ∈ T0 such that S + S ⊆ T and S · S ⊆ T ⊇ (S · C) ∪ (C · S). Proof. Using continuity of addition and multiplication in R, we pick W, V ∈ T0 such that W + W ⊆ T and V · V ⊆ W . As C is a Cauchy filterbasis, we find C ∈ C with C 2 ⊆ RV . Then C ⊆ V + c holds for any c ∈ C. Using continuity of multiplication again, we find S ∈ T0 such that S ⊆ V ∩ W and (S · c) ∪ (c · S) ⊆ W . Now S + S ⊆ W + W ⊆ T , S · S ⊆ V · V ⊆ W ⊆ T , and S · C ⊆ S · (V + c) ⊆ S · V + S · c ⊆ V · V + W ⊆ W + W ⊆ T . Analogously, we see C · S ⊆ T . 2 8.52 Lemma. The multiplication ˆ· is well defined: for Cauchy filterbases A and . · B. . B, the product A · B is a Cauchy filterbasis, again, and we have A ·B =A Proof. For U ∈ T0 , pick T ∈ T0 such that T +T +T −(T +T +T ) ⊆ U . According to 8.51, there exist S ∈ T0 and (A, B) ∈ A × B such that A2 ⊆ RS ⊇ B 2 with S + S ⊆ T and S · S ⊆ T ⊇ (A · S) ∪ (S · B). We choose (a, b) ∈ A × B, then A ⊆ S +a and B ⊆ S +b yield A·B ⊆ (S +a)·(S +b) ⊆ S ·S +A·S +S ·B +ab ⊆ T + T + T + ab ⊆ U + ab. As (a, b) ∈ A × B was arbitrary, this shows (A · B) × (A · B) ⊆ RU , and A · B is a Cauchy filterbasis. 3 and B 3 , we have to .= A .=B For Cauchy filterbases A, A , B, B with A 3 . In view of 8.30 (b), it suffices to show A 3 ˆ· B .· B . =A . ˆ· B .=A · B. show A . ⊆ A fil and B . ⊆ B fil (cf. 8.30 (b)) yield A .· B . ⊆ A · B fil . The inclusions A . · B . fil is a Cauchy filter contained in the Cauchy filter A · B fil , and Now A . · B . fil . Finally, we apply 8.30 (b) to minimality (cf. 8.30 (f)) yields A · B ⊆ A .· B .. ·B =A obtain A ·B =A 2 . into a topological ring. 8.53 Theorem. The multiplication ˆ· turns the completion R Proof. We have to verify that ˆ· is associative and continuous, that 1ˆ is a neutral element, and that the distributive laws are satisfied. We check distributivity first: for Cauchy filterbases A, B and C and (A, B, C) ∈ A×B×C one easily sees A·(B+C) ⊆ (A·B)+(A·C), and finds A · (B + C) fil ⊇ .· (B .+ . = (Aˆ .· B) . + .· C) . follows from 8.30 (e) . C) . (Aˆ (A · B) + (A · C) fil . Thus Aˆ .+ . ˆ· C . = (A . ˆ· C) . + . B) . and the definition of ˆ· , see 8.52. Analogously, we obtain (A . ˆ· C). . (B
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Now let us check that multiplication is continuous: we will use 3.24 in the version where we do not yet know that multiplication is associative. We have to ˆ 0), ˆ and that for each Cauchy filterbasis show that multiplication is continuous at (0, . → C . ˆ· A . and λ . : C . → A . ˆ· C . are both continuous at 0. ˆ A on R the maps ρA. : C A For T ∈ T0 , pick U ∈ T0 and A ∈ A such that U · U ⊆ T ⊇ (U · A ∪ A · U ), ˆ· U = {C . ˆ· D .| U ∈C . ∩ D} . ⊆W . , and U ˆ· A .= this is possible by 8.51. Then U . . . . . {C ˆ· A | U ∈ C} ⊆ W ⊇ A ˆ· U , as required. Associativity of multiplication now follows from the facts that multiplication is . is Hausdorff. Analogously, continuous, associative on a dense subset, and that R . implies that 1ˆ is a neutral the fact that 1ˆ is a neutral element for a dense subset of R . Alternatively, one could argue as in the proof of the distributive element in R. 2 laws. Proceeding as in 8.48, we obtain: 8.54 Theorem. Let R and S be Hausdorff rings, let ϕ : S → R be a continuous ring . and ηS : S → . S be completions. Then there is homomorphism, and let ηR : R → R .→ . a unique continuous ring homomorphism : R S such that ηR = ϕηS . 2 8.55 Corollary. If R is a subring of a complete Hausdorff ring S then the closure R is isomorphic to the completion of R. 2 8.56 Example. Let R be a compact (e.g., a finite) Hausdorff ring, and let c be any set. Then R c is the completion of R (c) . 8.57 Example. Let R be a locally compact Hausdorff ring, and let D be a dense subring of R. Then R is the completion of D. In fact, we only have to recall from 8.25 that R is complete, and apply 8.47. 8.58 Examples. Let p be a prime, and let Zp be the ring of p-adic integers, as in 8.49. (a) The ring Zp is the completion of the ring (Z, Tp ). Every continuous homomorphism from (Z, Tp ) to a complete Hausdorff ring S extends to a continuous homomorphism from Zp to S. (b) The ideal pZp in Zp is open. Therefore, the quotient Zp /pZp is discrete, and coincides with its dense subring Z/pZ. For x ∈ Zp pZp , we thus find y ∈ Zp such that xy ∈ pZp + 1, and w := 1 − xy ∈ pZp . We put sk := kn=0 wn , and consider the filterbasis T (s) of terminal sets (see 1.38) for the sequence s := (sk )k∈N . Now wn ∈ pn Zp yields that T (s) is a Cauchy filterbasis, and T (s) converges to some element c in the complete space Zp . Continuity of multiplication and the observation (1 − w)sk = 1 − w k+1 yield xyc = (1 − w)c = 1. This shows that every element of Zp pZp has an inverse.
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(c) For a ∈ Zp {0}, we find n ∈ N such that a ∈ Zp p n Zp because the set {pn Zp | n ∈ N} forms a neighborhood basis at 0 in the Hausdorff space Zp . We put na := min{n ∈ N | a ∈ pn+1 Zp }, then a = pna ua with ua ∈ Zp pZp . Since p belongs to the integral domain Z = Zη ≤ Zp , these considerations show that Zp is anintegral domain, as well, and that the field of quotients is obtained as Qp = n∈N p−n Zp . On Qp , the group topology Tp generated by the filterbasis {pn Z | n ∈ N} according to 3.22 also renders multiplication in Qp continuous, cf. 3.24. (It is also not very difficult to check that inversion is continuous.) The field Qp of p-adic numbers is the completion of the ring (Q, Tp |Q ). (d) Every continuous ring homomorphism from (Q, Tp ) to a complete Hausdorff ring S extends to a continuous homomorphism from Qp to S. (e) Every nonzero ring homomorphism defined on a field is injective (the kernel is a proper ideal, and thus equals {0}). In particular, the closure of the image of a continuous nonzero ring homomorphism from (Q, Tp ) to a complete Hausdorff ring S is a field isomorphic to Qp . One might be tempted to conjecture that the ring completion of a topological field is a topological field, again. However, this is not the case: it may even happen that the completion contains divisors of zero. 8.59 Example. The map ε : Q → Q2 × Q3 : q → (q, q) is an injective ring homomorphism. We endow Qε and S := Qε with the induced topologies. Then S is a locally compact ring, hence complete, and in fact a completion of Qε . However, the sequence (2n , 2n ) accumulates at elements (0, x) with x ∈ Z3 3Z3 because Z2 × Z3 is compact. Thus S is a completion of a topological ring algebraically isomorphic to Q, and S contains divisors of zero. As inversion is continuous in the open neighborhood (Z2 × Z3 ) (2Z2 × 3Z3 ), we even see that Qε is a topological field.
Exercises for Section 8 Exercise 8.1. Verify that the right and left uniformities on a topological group are uniformities. Exercise 8.2. Let (X, d) and (X , d ) be metric spaces. Show that a map ϕ : X → X is uniformly continuous (in the sense of 8.10, with respect to the uniformities defined by the metrics) if, and only if, the following condition is satisfied:
∀ε > 0 ∃δ > 0 ∀x, y ∈ X : d(x, y) < δ #⇒ d (x ϕ , y ϕ ) < ε .
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Exercise 8.3. Prove that the uniformity defined by a metric d on a set X induces on X the same topology as the metric. Exercise 8.4. Exhibit examples of metrics on a set X that induce the same topology, but non-equivalent uniformities on X. Show that homeomorphisms between metric spaces need not preserve Cauchy sequences (in the sense of 1.30). Exercise 8.5. Verify the details of 8.8 and 8.9. Exercise 8.6. For metric spaces (X, d) and (Y, e), define m := (X × Y )2 → R : ((x1 , y1 ), (x2 , y2 )) → d(x1 , x2 ) + e(y1 , y2 ). Verify that m is a metric, and that the uniformity Vm induced by m is equivalent to the product of the uniformities Vd and Ve . Exercise 8.7. Let (X, X) and (Y, Y) be uniform spaces. Show that the product uniformity P induces the product topology on X × Y . Exercise 8.8. Consider the following subgroup of GL(2, R), with its usual topology: a b a, b ∈ R, a > 0 . 0 1 Show that the left and right uniformities on this group are not equivalent. What can be said about uniform continuity of the group operations? What about completeness? Exercise 8.9. Let (X, V) be a uniform space. Verify that the uniform topology TV is indeed a topology, and that {x V | V ∈ V} forms a neighborhood basis at x, whenever x ∈ X. Exercise 8.10. Let (X, d) be a metric space. Show that the topology induced by the metric coincides with the uniform topology induced by the uniform structure induced by the metric.
Chapter C
Topological Transformation Groups
9 The Compact-Open Topology 9.1 Definition. Let X and Y be topological spaces. We are going to topologize the set C(X, Y ) of all continuous maps from X to Y . For C ⊆ X and U ⊆ Y we put $C, U % := {ϕ ∈ C(X, Y ) | C ϕ ⊆ U }. Then the compact-open topology Tc-o on C(X, Y ) is the topology generated by the subbasis Sc-o := {$C, U % | X ⊇ C is compact, Y ⊇ U is open} . In the sequel, the set C(X, Y ) will always be endowed with the compact-open topology, unless stated otherwise. 9.2 Lemma. Let X and Y be topological spaces, and consider C(X, Y ) with the compact-open topology. Then the following hold. (a) If Y ∈ T0 then C(X, Y ) ∈ T0 . (b) If Y ∈ T1 then C(X, Y ) ∈ T1 . (c) If Y ∈ T2 then C(X, Y ) ∈ T2 . Proof. Let ϕ and ψ be two elements of C(X, Y ), and pick x ∈ X such that x ϕ = x ψ . If U is a neighborhood of x ϕ in Y such that x ψ ∈ / U then ${x}, U % is an open neighborhood of ϕ in C(X, Y ) that does not contain ψ. If we find a neighborhood V of x ψ disjoint to U , then ${x}, U % and ${x}, V % are disjoint neighborhoods of ϕ and ψ, respectively. This yields the assertions. 2 If both X and Y are nonempty, the implications in 9.2 may be reversed. We leave this as an exercise. 9.3 Lemma. Let X and Y be topological spaces. If C is a compact subset of X and U is an open subset of Y then the closure $C, U % of $C, U % in C(X, Y ) satisfies $C, U % ⊆ $C, U % = $C, U %. Proof. Let ϕ : X → Y be a continuous function. If there exists c ∈ C such that cϕ ∈ / U then ϕ belongs to the open set ${c}, Y U % which is disjoint to $C, U %. 2 Thus $C, U % is closed, and contains the closure of $C, U %.
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9.4 Lemma: Continuity of composition. Let X, Y , and Z be topological spaces, and consider the composition map κ : C(X, Y ) × C(Y, Z) → C(X, Z) : (ϕ, ψ) → ϕψ. (a) For every fixed α ∈ C(X, Y ), the map α : C(Y, Z) → C(X, Z) : ψ → αψ is continuous (that is, the restriction κ|{α}×C(Y,Z) is continuous). (b) For every fixed β ∈ C(Y, Z), the map β : C(X, Y ) → C(X, Z) : ϕ → ϕβ is continuous (that is, the restriction κ|C(X,Y )×{β} is continuous). (c) If Y is locally compact then κ is continuous. Proof. Let α ∈ C(X, Y ), β ∈ C(Y, Z) and assume that C is a compact subset of X and U is open in Z such that αβ ∈ $C, U %. Then C α is compact, and contained in ← ← the open subset U β of Y . The neighborhoods $C, U β % and $C α , U % of α and β are mapped into $C, U % by β and α, respectively. This proves continuity of the restrictions in question. Now assume that Y is locally compact. For each x ∈ C α we find a compact β← neighborhood Vx of x in Y such that Vx ⊆ U C α is compact, there is a . Since ◦ α α finite subset F of C such that C ⊆ W := f ∈F Vf . Then W is open, and D := κ← . 2 f ∈F Vf is compact. Thus we have (α, β) ∈ $C, W % × $D, U % ⊆ $C, U % 9.5 Corollary. If X is locally compact then C(X, X) is a topological semigroup.2 The compact-open topology contains all sets ${x}, U %, where U is open in Y . This yields the following. 9.6 Lemma: Continuity of evaluation. Let X, Y be (arbitrary) topological spaces, and endow C(X, Y ) with the compact-open topology. Then the evaluation map 2 ωx : C(X, Y ) → Y mapping ϕ to x ϕ is continuous for each x ∈ X. 9.7 Lemma. Let X and Y be topological spaces, and let T be a topology on a subset S ⊆ C(X, Y ). If the map ω : X × S → Y defined by (x, ϕ)ω := x ϕ is continuous then T ⊇ Tc-o |S . In 9.6, we consider continuity in only one of the arguments of the map ω. In order to mark the contrast, one speaks of a jointly continuous map in 9.7. Proof of 9.7. Let C be a compact subset of X, and let U be open in Y . We claim that $C, U % ∩ S belongs to T . For every ϕ ∈ $C, U % ∩ S and each c ∈ C there ω exist open neighborhoods Vc of c and Wc ∈ T of ϕ such that (V c × Wc ) ⊆ U . As C iscompact, we find a finite subset F of C such that C ⊆ f ∈F Vf . Now W := f ∈F Wf ∈ T is an open neighborhood of ϕ such that W ⊆ $C, U %. This establishes the claim. The fact that $C, U % was an arbitrary element of a subbasis of Tc-o yields Tc-o |S ⊆ T . 2
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9.8 Lemma: Continuous action. Let X and Y be topological spaces. If X is locally compact then the map ω : X × C(X, Y ) → Y defined by (x, ϕ)ω := x ϕ is continuous. Proof. Let U be open in Y , and consider x ∈ X and ϕ ∈ C(X, Y ) such that x ϕ ∈ U . Then there is a compact neighborhood V of x in X such V ϕ ⊆ U . Now ← (x, ϕ) ∈ V × $V , U % ⊆ U ω . 2 9.9 Proposition. Let X, Y , and Z be topological spaces. We obtain maps and
C(Z, −) : C(X, Y ) → C(C(Z, X) , C(Z, Y )) C(−, Z) : C(X, Y ) → C(C(Y, Z) , C(X, Z))
if we stipulate that the image C(Z, ϕ) of ϕ ∈ C(X, Y ) under C(Z, −) maps δ ∈ C(Z, X) to δϕ, and that the image C(ϕ, Z) of ϕ ∈ C(X, Y ) under C(−, Z) maps γ ∈ C(Y, Z) to ϕγ . (a) If X is locally compact, then C(Z, −) is continuous. (b) If Y is locally compact, then C(−, Z) is continuous. Proof. First of all, we have to convince ourselves that the maps C(Z, ϕ) and C(ϕ, Z) are continuous for each ϕ ∈ C(X, Y ). Let C ⊆ Z be compact, and let U ⊆ Y be ← open. Then U ϕ is open in X, and the pre-image of $C, U % under the map C(Z, ϕ) ← is the open set $C, U ϕ %. The proof for C(ϕ, Z) is similar. We will show assertion (b) in detail, and leave the rest as an exercise. Let be a compact subset of C(Y, Z), and let be open in C(X, Z). Without loss, we may assume that is an element of the subbasis defining the compact open topology; that is, there are a compact subset C of X and an open subset U of Z such that = $C, U %. Consider an element ϕ in the pre-image of $ , % under C(−, Z); that is, we have ϕγ ∈ for each γ ∈ . The composition map κ : C(X, Y ) × C(Y, Z) → C(X, Z) is continuous by 9.4. For each γ ∈ , we find open neighborhoods γ of ϕ and ϒγ of γ suchthat γ ϒγ ⊆ . As is compact, there is a finite subset of such that ⊆ γ ∈ ϒγ . Now := γ ∈ γ is 2 an open neighborhood of ϕ with {C(ω, Z) | ω ∈ } ⊆ $ , %. 9.10 Corollary. Let X, Y , and Z be locally compact spaces. Let πY : Y × Z → Y and πZ : Y × Z → Z be the canonical projections, and define the map α from C(X, Y × Z) to C(X, Y ) × C(X, Z) by ϕ α = (ϕπY , ϕπZ ). Then α is a homeomorphism. Proof. From 9.4 we infer that α is continuous; compare 1.11. It is clear that α is a bijection. In order to see that the inverse α −1 is continuous, consider a compact subset C of X and an open subset U of Y × Z, and fix γ ∈ $C, U %. For
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each c ∈ C, there are open sets Vc and Wc of Y and Z, respectively, such that cγ ∈ Vc × Wc ⊆ U . As γ is continuous, we find a compact neighborhood Dc of c γ such that Dc ⊆ Vc × Wc . Now compactness of C yields C ⊆ D for some c∈F c
α $C, % $D % , V × W ⊆ U which shows finite subset F of C. We obtain c c c c∈F 2 that α −1 is continuous.
Whitehead’s Theorem As an application, we give another criterion for quotient maps. This criterion will be useful when dealing with locally compact semigroups, see 28.11 below. 9.11 Lemma. Let A and B be topological spaces. For x ∈ A, let xˆ : B → A × B be defined by y xˆ := (x, y). Then xˆ ∈ C(B, A × B), and ξ : A → C(B, A × B) : x → xˆ is continuous. Proof. Clearly, the map from the obser xˆ is continuous. Continuity of ξ follows vation that xˆ ∈ $C, j ∈J Uj × Vj %is equivalent to {x} × C ⊆ j ∈J Uj × Vj : without loss, we may assume x ∈ j ∈J Uj . There is a finite index set F ⊆ J such that j ∈F Vj contains the compact set C, and ξ maps the open neighborhood $C, ˆ U of x into the neighborhood 2 j j ∈F j ∈J Uj × Vj % of x. 9.12 Theorem. Let X and Y be topological spaces, and let κ : X → Q be a quotient map. If Y is locally compact then κ × idY : X × Y → Q × Y : (x, y) → (x κ , y) is a quotient map, as well. Proof. We will apply the universal property 1.35 (e). Let ϕ : Q × Y → Z be any map such that γ := (κ ×idY )ϕ is continuous. We have to show that ϕ is continuous. γ ϕ Putting y x := (x, y)γ and y q := (q, y)ϕ , we define maps γ : X → C(Y, Z) and ϕ : Q → C(Y, Z), respectively. Then γ maps x to the composition of xˆ with γ . Thus γ is continuous by 9.11 and 9.4. Since κ is a quotient map, we obtain that ϕ is continuous. Now ϕ can be described as the composition of the continuous map (q, y) → ϕ ) with the continuous action of C(Y, Z) on the (locally compact!) space Y , (y, q cf. 9.8. 2
The Modified Compact-Open Topology 9.13 Lemma. Let (G, μ, ι, ν) be a group, and let T be a topology on G that renders μ continuous. Then (G, μ, ι, ν, T) is a topological group, where the topology T is generated by the subbasis {S ∩ T ι | S, T ∈ T }.
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If (H, μ , ι , ν , T ) is a topological group and ϕ : (H, T ) → (G, T ) is continuous and a homomorphism then ϕ is also continuous with T instead of T . Proof. Obviously, ι is continuous with respect to T. For S, T ∈ T and g, h ∈ G such that (g, h)μ ∈ S ∩ T ι we find U ∈ Tg , V ∈ Th and W ∈ Thι , X ∈ Tg ι such ← ← ← that U × V ⊆ S μ and W × X ⊆ T μ . Then (U ∩ X ι ) × (V ∩ W ι ) ⊆ (S ∩ T ι )μ . The second assertion follows from the observation that for all S, T ∈ T we ← ← ← ← have S ϕ ∈ T and T ιϕ = T ϕ ι ∈ T , whence (S ∩ T ι )ϕ ∈ T . 2 9.14 Definition. The construction of 9.13 is of particular interest if applied to the topology induced by the compact-open topology on Homeo(X) for some locally |Homeo(X) . compact space X. By abuse of language, we write T4 c-o instead of Tc-o 4 We call Tc-o the modified compact-open topology. 9.15 Corollary. Endowed with the modified compact-open topology T4 c-o , the group Homeo(X) of all homeomorphisms of X is a topological group, if X is locally compact. 9.16 Lemma. Let X be a locally compact space, and endow Homeo(X) with the modified compact-open topology T4 c-o . Then the map ω : X × Homeo(X) → X defined by (x, ϕ)ω := x ϕ is continuous. Proof. According to 9.8, the map ω is continuous if we replace T4 c-o by Tc-o . Since Tc-o ⊆ T4 , the assertion follows. 2 c-o 9.17 Application to automorphism groups. Let G be a locally compact group, and let Aut(G) be the group of all topological automorphisms of G. Then Aut(G) is a subgroup of Homeo(G). Endowed with the topology induced from T4 c-o , the group Aut(G) is a topological group, and the map ω : G × Aut(G) → G defined by (g, α)ω := g α is continuous. 2
Other Topologies for Spaces of Maps In a discrete space D, the compact sets are just the finite ones. Thus the compactopen topology on C(D, Y ) is the same as the so-called point-open topology Tp-o , generated by {${x}, U % | x ∈ D, Y ⊇ U is open}. In general, the compact-open topology on C(X, Y ) contains the point-open topology. In fact, the topology on the domain of definition is not used for the construction of Tp-o , and we have met the point-open topology before: 9.18 Lemma. Let X be a set, and let Y be a topological space. Then the pointopen topology on the set Y X of all maps from X to Y is nothing else but the product topology.
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Proof. Let P denote the product topology. The identity is continuous from (Y X , Tp-o ) to (Y X , P ) by the universal property of the product topology 1.11: In fact, for z ∈ X the projection πz : Y X → Y is just the evaluation at z, and the preimage of U ⊆ Y is ${z}, U %. Thus the point-open topology contains the product topology. Conversely, every element ${z}, U % of the defining subbasis for Tp-o is of the form Xz∈X Uz (namely, with Uz = U and Ux = Y for x ∈ X {z}), qualifying it 2 as an element of the product topology. 9.19 Corollary. Let (Cx )x∈X be a collection of compact subsets of a topological space Y . Then the point-open topology on the set Xx∈X Cx ⊆ Y X is compact. 2 We define a topology on spaces of continuous functions into a metric space next. It will turn out that this topology is just a special case of the compact-open topology. 9.20 Definition. Let Y be a topological space, and let (Z, d) be a metric space. For every compact C ⊆ Y , each ε > 0, and every ϕ ∈ C(Y, Z), put BεC (ϕ) := {ψ ∈ C(Y, Z) | ∀x ∈ C : d(x ϕ , x ψ ) < ε}. The topology generated by the collection of all these BεC is called the topology of compact convergence. 9.21 Theorem. Let Y be a topological space, and let (Z, d) be a metric space. Then the compact-open topology Tc-o and the topology of compact convergence coincide on C(Y, Z). Proof. Consider a compact subset C ⊆ Y and an open subset V ⊆ Z. For each %, the image C ϕ is compact, and contained in V . Thus there exists ε > 0 ϕ ∈ $C, U such that c∈C Bε (cϕ ) ⊆ V , cf. Exercise 9.8. Then BεC (ϕ) ⊆ $C, U %, and we have proved that the topology of compact convergence contains the compact-open topology. Conversely, consider ϕ ∈ C(Y, Z) and a neighborhood BεC (ϕ) in the topology of compact convergence. Then every c ∈ C has a neighborhood Vc such that ϕ ϕ ϕ Vc ⊆ Bε/4 (cϕ ), and Vc ⊆ Vc ⊆ Bε/3 (cϕ ) follows. As C is compact, there is a finite subset F ⊆ C with C ⊆ f ∈F Vf . Now Cf := C ∩ Vf is compact, and f ∈F $Cf , Bε/3 (f ϕ )% ⊆ BεC (ϕ) is a neighborhood of ϕ in the compact-open topology. Thus the compact-open topology contains the topology of compact convergence. 2
A Compactness Criterion We conclude this section with the Arzela–Ascoli Theorem, which will be used in 20.4 below to prove that character groups are locally compact.
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9.22 Definition. Let d be a metric on a set Z, and endow Z with the topology defined by this metric. Let Y be a topological space. A subset of Z Y is called equicontinuous at y ∈ Y if for each ε > 0 there exists a neighborhood U of y such that for each ϕ ∈ we have U ϕ ⊆ Bε (y ϕ ). The neighborhood U is called ε-suited for y (with respect to ) in this case. The set is called equicontinuous on Y if it is equicontinuous at each point of Y . We note that every element of an equicontinuous set of maps is continuous. 9.23 Lemma. Let (Z, d) be a metric space, let Y be a topological space, and assume that ⊆ C(Y, Z) is equicontinuous. Then the following hold: (a) The closure in Z Y (with respect to Tp-o ) is equicontinuous on Y , and thus contained in C(Y, Z). (b) On , the topology induced by Tp-o coincides with the one induced by Tc-o . Proof. Consider γ ∈ , some point y ∈ Y , and an ε-suited neighborhood U for y. For u ∈ U , the set Vu := ${u}, Bε (uγ )% ∩ ${y}, Bε (y γ )% ∈ Tp-o is a neighborhood of γ in Z Y . Thus Vu contains some element ϕ of . The triangle inequality d(uγ , y γ ) ≤ d(uγ , uϕ ) + d(uϕ , y ϕ ) + d(y ϕ , y γ ) < 3ε now yields that U is also a 3ε-suited neighborhood of y, with respect to . This proves assertion (a). In general, the topology of compact convergence coincides with Tc-o , and contains the restriction of Tp-o to C(Y, Z). In order to prove the reverse inclusion for the restrictions to , consider ϕ ∈ and a neighborhood BεC (ϕ) ∈ Tc-o . Choose δ > 0 such that 3δ < ε and pick a δ-suited neighborhood Uc (with respect to ) for each c ∈ C. Compactness of C implies that there is a finite set F ϕ⊆ C with C ⊆ f ∈F Uf Now pick yf ∈ Uf , and note that f ∈F ${yf }, Bδ (yf )% ∈ Tp-o is a Tp-o -neighborhood of ϕ contained in BεC (ϕ). Thus Tc-o is contained in the restriction of Tp-o to C(Y, Z). 2 9.24 Theorem. Let Z be a metric space, and let Y be any topological space. Assume that ⊆ C(Y, Z) satisfies the following: (a) The set is equicontinuous on Y . (b) For each y ∈ Y , the set y := {y ω | ω ∈ } has compact closure in Z. Then the closure (with respect to either the point-open or the compact-open topology) is compact. Proof. Let denote the closure with respect to the point-open topology Tp-o . From 9.23 we know that is equicontinuous on Y (and thus contained in C(Y, Z)), and that Tp-o and Tc-o induce the same topology on . However, we have ⊆
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Xy∈Y y ⊆ Z Y , the space Xy∈Y y is compact by Tychonoff’s Theorem 1.33, and so is its closed subspace .
2
If Y is locally compact, one can also prove a converse statement: Every compact subset ⊆ C(Y, Z) is equicontinuous, and y is compact for each y ∈ Y . We do not pursue this here.
Exercises for Section 9 Exercise 9.1. Let X and Y be nonempty spaces. Show that for each a ∈ X the evaluation map ωa : C(X, Y ) → Y defined by ϕ ωa = a ϕ induces a homeomorphism from the subspace of all constants in C(X, Y ) onto Y . Exercise 9.2. Let X and Y be nonempty spaces. Show that for each i ∈ {0, 1, 2} the implication stated in 9.2 can be reversed; that is, if C(X, Y ) ∈ Ti then Y ∈ Ti . Exercise 9.3. Let X, Y , and Z be topological spaces, and assume that Y is locally compact. Prove that C(−, Z) : C(X, Y ) → C(C(Y, Z) , C(X, Z)) is continuous. Exercise 9.4. Consider the set {0, . . . , n − 1} with the discrete topology, and let X be a topological space. Describe the set C({0, . . . , n − 1}, X) and the compact-open topology on it. Compare with Xn . Exercise 9.5. Endow N with the discrete topology, and C := {0} ∪ n1 | n ∈ N {0} with the usual topology. Compare the spaces RN , C(N, R), and C(C, R). Exercise 9.6. Show that a sequence in C([0, 1], R) converges uniformly on [0, 1] exactly if it converges with respect to the compact-open topology. What happens if you replace [0, 1] by R, or by ]0, 1] ? Exercise space. Show that the sub 9.7. Let Z be a metric space, and let Y be a topological basis BεC (ϕ) | ϕ ∈ C(Y, Z) , ε > 0, Y ⊇ C is compact is indeed a basis for the topology of compact convergence on C(Y, Z). Exercise 9.8. Let (Z, d) be a metric space, let A ⊆ Z be compact, and let V ⊆ Z be an open set containing A. Prove that there exists ε > 0 such that a∈A Bε (a) ⊆ V . Hint. For each a ∈ A, choose δa > 0 such that B2δa (a) ⊆ V . Cover A by finitely many sets Bδa (a), and take ε as the minimum over the corresponding δa . Exercise 9.9. Let Y be a topological space, and let Z be a metric space. Show that every finite subset of C(Y, Z) is equicontinuous.
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10 Transformation Groups 10.1 Definition. Let X be a topological space, and let G be a topological group. An action of G on X is a map ω : X × G → X such that the following hold for all x ∈ X and all g, h ∈ G:
ω and (x, 1)ω = x. (x, gh)ω = (x, g)ω , h If the map ω is continuous, we also say that (X, G, ω) is a topological transformation group. A permutation representation of G on X is a homomorphism δ from G into the group Sym(X) of all bijections from X onto X. In our topological context, the case where Gδ ≤ Homeo(X) is most interesting. If X is locally compact, it is reasonable to endow Homeo(X) with the modified compact-open topology. We say that δ : G → Homeo(X) is a representation as topological transformation group if δ is a continuous homomorphism. 10.2 Examples. (a) Let X be a set and define σ : X × Sym(X) → X simply by putting (x, ϕ)σ = x ϕ . Then σ is an action. If X is a locally compact space, endow Homeo(X) with the modified compact-open topology. Then the restriction of σ to X × Homeo(X) is a continuous action by 9.16. (b) Let G be a topological group, and let H be a subgroup of G. Then the map ωH : G/H × G → G/H defined by (H x, g)ωH = H xg is a continuous action by 6.2. (c) Let G be a topological group, and let N be a normal subgroup of G. Then κN : N × G → N defined by (x, g)κN = x ig := g −1 xg is a continuous action. Our next aim is to show that for locally compact spaces X the concepts of topological transformation group and of representation as topological transformation group are equivalent. 10.3 Definition. Let X be a set, and let G be a group. For each action ω : X × G → X we define δ ω : G → Sym(X) by stipulating that g δ ω maps x ∈ X to (x, g)ω . Conversely, for every permutation representation δ : G → Sym(X) we define a map ωδ : X × G → X by putting (x, g)ωδ = (x, g δ )σ , where σ is the natural action as defined in 10.2 (a). 10.4 Lemma. Let X be a set, and let G be a group. (a) For every action ω : X × G → X, the map δ ω is a permutation representation, and ωδ ω = ω. (b) For every permutation representation δ : G → Sym(X), the map ωδ is an action, and δ ωδ = δ.
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(c) If X is a topological space, and ω : X × G → X is a continuous action then δ ω : G → Homeo(X) is a continuous homomorphism, with respect to the compact-open topology. If X is locally compact then δ ω is a representation as topological transformation group; that is, it is continuous also with respect to the modified compact-open topology. Now assume that X is locally compact, and endow Homeo(X) with the modified compact-open topology. (d) If δ : G → Homeo(X) is a representation as topological transformation group, then ωδ : X × G → X is a continuous action. Proof. Assertions (a) and (b) are left as exercises. It remains to prove that δ ω and ωδ in (c) and (d) are continuous. Let X be a topological space, and assume that ω : X × G → X is a continuous action. For each g ∈ G, the map g δ ω : X → X may be considered as the restriction of ω to X × {g}, and is therefore continuous. The inverse (g δ ω )−1 = (g −1 )δ ω is continuous as well. Thus δ ω maps G into Homeo(X). For any compact subset C and any open subset U of X, we have that g δ ω ∈ $C, U % is equivalent to (C×{g})ω ⊆ U . As ω is continuous, we find for each c ∈ C open neighborhoods Vc of c and Wc ω of g such that (Vc × W there exists a finite subset c ) ⊆ U . Since C is compact, F of C such that C ⊆ f ∈F Vf . Now W := f ∈F Wf is a neighborhood of g, and (C × W )ω ⊆ U ; that is, W δ ω ⊆ $C, U %. The rest of assertion (c) follows from 9.13. If δ : G → Homeo(X) is a continuous homomorphism then (x, g)α = (x, g δ ) defines a continuous map α : X × G → X × Homeo(X). According to 9.16, the action ω : X × Homeo(X) → X defined by (x, ϕ)ω = x ϕ is continuous. Thus ωδ = αω is continuous. 2 10.5 Example. If κN denotes the action described in 10.2 (c) then the image of G under δ κN is contained in Aut(N ). Thus δ κN induces a continuous homomorphism from G to Aut(N ), if Aut(N ) is endowed with the topology induced from the modified compact-open topology on C(N, N). For N = G we have that δ κN equals the homomorphism i : G → Aut(G) introduced in 3.26. 10.6 Lemma. Let ω : X × G → X be an action of a group G on a set X, and let Y be another set. For every map ϕ ∈ Y X and every element g ∈ G, we have ϕg ∈ Y X , mapping x ∈ X to (x, g −1 )ωϕ . Then (ϕ, g)ωˆ := ϕg defines an action ωˆ : Y X × G → Y X . If X and Y are topological spaces, and ω is a continuous action of a topological group G, then ωˆ restricts to an action of G on C(X, Y ), which is continuous if X is locally compact.
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Proof. An easy computation (using the fact that inversion is an anti-automorphism of the group G) shows that ωˆ is in fact an action. Now assume that X and Y are topological spaces, and that ω is a continuous action of a topological group G. In order to see that ωˆ restricts to an action of G on C(X, Y ), we simply have to remark that (ϕ, g)ωˆ = (g −1 )δ ω ϕ is continuous exactly if ϕ is continuous. This equation also shows that δ ωˆ may be written as the composition of inversion, the continuous representation δ ω and the map C(−, Y ) introduced in 9.9. If X is locally compact then continuity of ωˆ follows from 9.9, 9.4, and 10.4. 2 10.7 Lemma. Let C be a compact group, let X be a topological space, and let ω : X × C → X be a continuous action. (a) For each subset A ⊆ X and each open subset U ⊆ X with (A × C)ω ⊆ U there exists an open set V ⊆ X such that (V × C)ω ⊆ U . (b) If the action has a fixed point a (that is, a point a ∈ X with ({a} × C)ω = {a}) then for every neighborhood U of a in X there is a neighborhood V of a in X such that (V × C)ω ⊆ U . Proof. Consider a ∈ A. For each c ∈ C, we have (a, c)ω ∈ U . Continuity of ω yields the existence of open neighborhoods Va,c and Wa,c of a and c, respectively, such that (Va,c × Wa,c )ω ⊆ U . Compactness of C yields the existence of a finite subset Fa ⊆ C such that C ⊆ f ∈Fa Wa,f . We put Va := f ∈Fa Va,f , then Va is an open neighborhood Va is open in X and of a, and Vω := a∈A ω = ω (V × C) = contains A. Now (V × C) a a∈A a∈A f ∈Fa (Va × Wf ) ⊆ ω a∈A f ∈Fa (Vf × Wf ) ⊆ U , as required. Assertion (b) follows by specializing A := {a}. 2 10.8 Remark. An important special case of 10.7 is the case where C acts by linear transformations on a vector space X: that is, the bijection cδ ω is a linear map, for each c ∈ C. 10.9 Definition. Let ω : X × G → X be an action. For x ∈ X, the subset x G := {(x, g)ω | g ∈ G} of X is called the orbit of x under G (or under ω), and the subset Gx := {g ∈ G | (x, g)ω = x} is called the stabilizer of x in G (with respect to ω). Putting (Gx g)ϕ := (x, g)ω defines a map ϕ : G/Gx → x G ; in fact, the image of Gx g under ϕ does not depend on the chosen representative g, as Gx g = Gx h implies the existence of f ∈ Gx such that h = f g, and therefore (x, h)ω = (x, f g)ω = (x, g)ω . It is easy to see that ϕ is in fact a bijection; we leave this as an exercise. 10.10 Theorem: Open action. Let ω : X × G → X be a continuous action. Then the following hold. (a) The map ϕ is continuous.
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(b) If x G belongs to T1 then Gx is closed in G. (c) If G is σ -compact and locally compact, and if x G is locally compact Hausdorff, then ϕ is a homeomorphism. Proof. The natural map κ : {x} × G → {x} × G/Gx is continuous and open, and thus a quotient map. As κϕ is continuous (being a restriction of ω), we conclude that ϕ is continuous. If x G ∈ T1 then {x} is closed in x G , and the ϕ-pre-image Gx of {x} is closed in G. Now assume that G is σ -compact and locally compact, and that x G is locally compact Hausdorff. It remains to show that ϕ −1 is continuous. For each g in G, the map αg : G/Gx → G/Gx defined by (Gx h)αg = Gx hg is a homeomorphism of G/Gx . Similarly, the map βg : x G → x G defined by (x, h)ωβg = (x, hg)ω is a homeomorphism of x G . As αg ϕ = ϕβg , we only have to check continuity of ϕ −1 at a single point; say x. We proceed similarly to the proof of 6.19. Let (Cn )n∈N be a sequence of compact subsets of G whose union is G, and let π : G → G/Gx be the natural map. Consider a neighborhood U of 1π = Gx in G/Gx . We pick a compact neighborhood V of 1 ◦ in G such that (V V ι )π ⊆ U . For each natural number n, we have C n ⊆ V Cn ◦ and find a finite set Fn ⊆ Cn such that Cn⊆ V Fn . Now G = n∈N V Fn and x G= n∈N (V Fn )π ϕ . The set A := n∈N Fn is countable, and we have x G = a∈A (V a)π ϕ . According to 1.29, the space x G is not meager. Thus at least one of the sets (V a)π ϕ = V π ϕβa has nonempty interior. This means that V π ϕ is a neighborhood of v π ϕ for some v ∈ V , and we obtain that W := (V V ι )π ϕ is a −1 neighborhood of x in x G such that W ϕ = (V V ι )π ⊆ U . 2
Applications to Actions by Automorphisms 10.11 Definition. Let C and N be groups, and let δ : C → Aut(N ) be a homomorphism. For the sake of readability, we write cˆ = cδ for each c ∈ C. Define ˆ μ : (C × N)2 → C × N by ((c, m), (d, n))μ = (cd, md n). Then (C × N, μ) is called the semidirect product of C and N with respect to δ, and denoted by C δ N or C N if no confusion is possible. If δ is the trivial homomorphism then C δ N is just the (direct) product C × N . 10.12 Proposition. Let C and N be groups, and let δ : C → Aut(N ) be a homomorphism. Then the following hold. (a) The semidirect product C δ N is a group. The set C × {1} is a subgroup, and {1} × N is a normal subgroup of C δ N .
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(b) If C and N are topological groups and the action ωδ is continuous then C δ N is a topological group (with the product topology). If C and N are Hausdorff then C δ N is Hausdorff, and C × {1} and {1} × N are closed. Proof. In order to see that μ is an associative operation, we compute on the one hand
μ
μ ˆ ((c, l), (d, m))μ , (e, n) = (cd, l d m), (e, n)
ˆ ˆ = cde, (l d m)eˆ n = (cde, l d eˆ meˆ n). On the other hand, we have
(c, l), ((d, m), (e, n))μ
μ
μ . = (c, l), (de, meˆ n) = (cde, l de meˆ n).
. = dˆ e, As δ is a homomorphism, we obtain de ˆ and that μ is associative. The −1 c2 −1 −1 . This completes the proof inverse of (c, m) is easily computed as c , m of assertion (a). The multiplication of (c, m) and (d, n) in C δ N can be described as mapping (c, m, d, n) first to (c, d, (m, d)ωδ , n) and afterwards to (cd, (m, d)ωδ n). As multiplication in C and in N is continuous, continuity of ωδ yields continuity of μ. Inversion in C δ N is obtained by mapping (c, n) first to (c, (n, c−1 )ωδ ) and then inverting the components in C and N , respectively. The rest of assertion (b) is clear. 2 The condition that ωδ is continuous is satisfied if N is locally compact, and δ is a representation as topological transformation group; see 10.4 (d). 10.13 Characterization of semidirect products. Let G be a topological group, and assume that there are a normal subgroup N and a subgroup C of G with the properties G = CN and C ∩ N = {1}. Consider the action κN of C on N as in 10.2 (c), and put δ = δ κN . Then the following hold. (a) The map α : C δ N → G defined by (c, n)α = cn is a continuous bijective homomorphism. (b) If N is locally compact then δ is continuous with respect to the (modified) compact-open topology on Aut(N ). (c) If both C and G/N are locally compact, C is σ -compact, and N is closed in G, then α is an isomorphism of topological groups. Proof. The map α is the restriction of the continuous multiplication in G to the set C × N, and thus continuous. In the group G, we have (cm)(dn) = cd(d −1 md)n = (cd)(mid n). Thus α is a homomorphism. This homomorphism is surjective since
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G = CN, and it is injective since ker α ≤ C ∩ N = {1}. Thus assertion (a) is established. Assertion (b) follows from 10.2 (c) and 10.4 (c). For assertion (c), it remains to show that α is a homeomorphism; this has been done in 6.23. 2 We now can solve the problem of determining split extensions, up to isomorphism, as promised in 3.34. 10.14 Theorem. Let G and H be topological groups, and let ϕ : G → H be a continuous homomorphism. Assume that there exists a continuous homomorphism σ : H → G such that σ ϕ = idH . Then the following hold. (a) The map ϕ is a quotient morphism, and σ is an embedding. (b) Let K be the kernel of ϕ, and define δ : H → Aut(K) by hδ = ihσ |K . Then the semidirect product H δ K is a topological group, and the map α : H δ K → G given by (h, k)α = hσ k is an isomorphism of topological groups. Proof. Assertion (a) was proved in 6.16. In order to see that H δ K is a topological group, we have to verify that ωδ is continuous, see 10.12 (b). For h ∈ H and k ∈ K we have (k, h)ωδ = (hσ )−1 khσ . As σ is continuous and G is a topological group, the action ωδ is thus continuous. In order to see that α is continuous at 1, consider for any neighborhood U of 1 in H δ K an open neighborhood V of 1 such that V V ⊆ U . As ϕσ is continuous, we find an open neighborhood W of 1 such that W ϕσ ⊆ V . Being a quotient homomorphism between topological groups, the map ϕ is open, and W ϕ is open in H . Now the set W ϕ × (V ∩ K) is open in H δ K, and (W ϕ × (V ∩ K))α = {v ϕσ k | v ∈ V , k ∈ V ∩ K} ⊆ V V ⊆ U yields continuity of α at 1. For (h, k), (j, l) ∈ H δ K we compute
μ α (h, k)α (j, l)α = hσ j σ (j σ )−1 kj σ l = (hj )σ k ij σ l = (h, k), (j, l) δ . Thus α is a homomorphism. Obviously, the map β : G → H δ K given by g β = (g ϕ , (g ϕσ )−1 g) is the inverse of α. Both α and β are continuous, and assertion (b) follows. 2
Exercises for Section 10 Exercise 10.1. Let ω : X × G → X be an action of the group G on the set X. For any subset S ⊆ X and each g ∈ G, put (S, g) := {(x, g)ω | x ∈ S}. Show that this defines an action of G on the power set (that is, the set of all subsets) of X. Exercise 10.2. Prove assertions (a) and (b) of 10.4.
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Exercise 10.3. Show that the map ϕ in 10.9 is a well-defined bijection. Exercise 10.4. Consider the subgroup G = R(1, r) of R2 with the topology induced by the natural one, and let ω be the restriction of the action ωZ2 : T2 × R2 → T2 on T2 = R2 /Z2 to T2 × G. Show that the orbits of G in T2 are closed in T2 exactly if r is rational, and that this is also equivalent to the assumption that G/Gx is homeomorphic to x G for some x ∈ T2 . Exercise 10.5. Assume that X is locally compact. Let be the set of all constants in C(X, X). Show that is a subsemigroup of C(X, X) (that is, closed under composition). Fix x ∈ X, and show that mapping σ to x σ gives a homeomorphism α from onto X. Verify (σ ϕ)α = (σ α )ϕ , and interpret this as “translating the action”. Exercise 10.6. Find examples of continuous actions with orbits that are open, closed, not open, or not closed, respectively. Exercise 10.7. Let ω : X×G → X be a continuous action. Let X/G := {x G | x ∈ X} be the space of orbits, endowed with the quotient topology with respect to the map β : X → X/G defined by x β = x G . Show that β is an open map. Exercise 10.8. Exhibit actions where the space of orbits is not Hausdorff.
11 Sets, Groups, and Rings of Homomorphisms In this section, we show that the compact-open topology fits well with various operations on sets of continuous homomorphisms between locally compact groups. The results that we obtain will also be useful for the study of locally compact rings via the action on their additive group.
Groups of Homomorphisms 11.1 Lemma. Let G be a topological group, and let H be a Hausdorff group. Then the set Mor(G, H ) of all continuous group homomorphisms from G to H is closed in C(G, H ). Proof. For all elements g, h ∈ G the map αg,h : C(G, H ) → H defined by ϕ αg,h = ← (gh)ϕ hϕι g ϕι is continuous, compare 9.6. Now Mor(G, H ) = g,h∈G {1}αg,h is 2 closed in C(G, H ), since {1} is closed in H . 11.2 Corollary. If G is a locally compact Hausdorff group then Aut(G) is closed in (Homeo(G), Tc-o ) as well as in (Homeo(G), T4 c-o ).
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11.3 Remark. The set Mor(G, H ) need not be locally compact, even if both G and H are. For instance, let G be a free Abelian group of infinite rank c, and endow it with the discrete topology. Then Mor(G, H ) corresponds to the set H c of all maps from c to H , endowed with the product topology. As H is not compact, the space H c is not locally compact. We leave the details as an exercise. For an arbitrary locally compact group G, the group Aut(G) need not be locally compact, as the following example shows. 11.4 Example. Consider the discrete group Z, and let d be an infinite set. The set ← G := {γ : d → Z | |(Z {0})γ | < ∞} of all functions of “finite support” from d to Z forms a subgroup of C(d, Z). We endow the group G with the discrete topology. Every automorphism α ∈ Aut(G) is determined by its restriction α|d¯ , where d¯ consists of the characteristic functions of singleton subsets of d. Conversely, every map from d¯ to Z extends (uniquely) to a homomorphism from G to G. Therefore, we can identify Aut(G) with a subset of Gd . We claim that Aut(G) is not locally compact. In fact, the product topology on Gd coincides with the compact-open topology on C(d, G), since d is discrete. If Aut(G) were locally compact, we could thus find a finite subset f ⊂ d such that the group H := {ϕ ∈ Aut(G) | ϕ|f = idf } is compact. As d is infinite, we can pick x ∈ d f . We put S := {ϕ ∈ Aut(G) | x ϕ = x}. Now H ⊆ SH and compactness of H yield that there is a finite subset E ⊆ H such that H = SE. Thus the set x H := {x ϕ | ϕ ∈ H } = {x ϕ | ϕ ∈ E} is finite, in contradiction to the obvious fact that there are infinitely many permutations of d f , which all extend to elements of H . 11.5 Theorem. Let X be a locally compact space, and let H be a topological group. Then the multiplication μ : C(X, H ) × C(X, H ) → C(X, H ) defined by (ϕ, ψ)μ := (x → x ϕ x ψ ) renders (C(X, H ) , μ) a topological group. Proof. Let μ : H × H → H denote the multiplication in H . Using the homeomorphism α described in 9.10, we obtain that μ = α C(X, μ) is continuous; compare 9.9. In order to see that inversion in (C(X, H ) , μ) is continuous, we observe that the inverse of ϕ is ϕι, and that inversion maps $C, U % onto $C, U ι %. 2 If G is a topological group and the group H is commutative, then Mor(G, H ) is in fact a subgroup of (C(G, H ) , μ). Thus we obtain the following corollary to 11.1. 11.6 Proposition. If G is a locally compact group and H is a commutative Hausdorff group, then the set Mor(G, H ) is a closed subgroup of (C(G, H ) , μ). For every commutative group G, the space Mor(G, G) carries two operations: the “addition” μ discussed above, and the “multiplication” κ given by composition.
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It is easy to see that these operations turn Mor(G, G) into a ring; distributivity is a consequence of the fact that we deal with group homomorphisms. We obtain another corollary, as follows. 11.7 Proposition. If G is a locally compact commutative group then the operations μ, κ and the compact-open topology turn Mor(G, G) into a topological ring, which is closed in C(G, G) if G is Hausdorff. There is a nice application of the result 9.9. 11.8 Proposition. Let A, B, and C be locally compact commutative groups. We define Mor(C, −) and Mor(−, C) as restrictions of the maps C(C, −) and C(−, C) introduced in 9.9. Then the following hold. (a) The map Mor(C, −) is a continuous group homomorphism from Mor(A, B) to Mor(Mor(C, A), Mor(C, B)). (b) The map Mor(−, C) is a continuous group homomorphism from Mor(A, B) to Mor(Mor(B, C), Mor(A, C)). (c) If A is locally compact and coincides with B, then Mor(C, −) is a continuous ring homomorphism from Mor(A, A) to Mor(Mor(C, A), Mor(C, A)), and Mor(−, C) is a continuous ring anti-homomorphism from Mor(A, A) to Mor(Mor(A, C), Mor(A, C)). In this case, the map Mor(C, −) induces a continuous group homomorphism, and Mor(−, C) induces a continuous group anti-homomorphism from the group Aut(A) of invertible elements in the multiplicative semigroups of the ring Mor(A, A) to the groups Aut(Mor(C, A)) or Aut(Mor(A, C)) of invertible elements in the multiplicative semigroups of Mor(Mor(C, A), Mor(C, A)) or Mor(Mor(A, C), Mor(A, C)), respectively. 11.9 Remarks. If composition in Aut(Mor(C, A)) is continuous, then the antihomomorphism in 11.8 (b) remains continuous if we replace the compact-open topologies by the modified compact-open topologies. Note, however, that Mor(B, C) need not be locally compact, and we cannot apply 9.4 to show continuity of composition in Aut(Mor(B, C)). Group anti-homomorphisms yield homomorphisms simply by composing them with inversion in one of the groups. As inversion is continuous with respect to the modified compact-open topology on Aut(B) if B is locally compact, this does not affect continuity of Mor(−, C).
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Rings and Modules of Homomorphisms In the following discussion, it is possible to interchange left modules with right modules, throughout. 11.10 Definitions. Let R be a topological ring, let X be a topological space, and let M be a topological left R-module. Endowed with pointwise addition, the space C(X, M) is a (commutative) group. If X is a topological group then Mor(X, M) is a subgroup of C(X, M). If X is a topological left R-module, then the set Mor R (X, M) of all continuous R-linear maps from X to M is a subgroup of Mor(X, M), and Mor R (M, M) is a subring of Mor(M, M) (with composition playing the role of multiplication). In any case, we may also regard the additive group C(X, M) as a left R-module: for r ∈ R and ϕ ∈ C(X, M), the product r · ϕ maps x ∈ X to r · x ϕ . Changing our point of view, we obtain that M is a right S-module, where S = Mor R (M, M): multiplication by elements of S means just application of these (from the right!). 11.11 Lemma. Let R be a topological ring, and let M be a topological left Rmodule. We define a map λ : R → C(M, M) by stipulating that r λ maps m to rm. Then λ is continuous. If N is a topological right R-module, we have analogously a map ρ : R → C(N, N) such that r ρ maps m to mr, and ρ is continuous. Proof. As M is a topological R-module, the map μ : R × M → M mapping (r, m) to rm is continuous. In particular, we have r λ ∈ C(M, M). Let Q denote the set R λ , endowed with the quotient topology with respect to the surjection λ. It remains to show that this topology contains the topology induced from C(M, M). Since λ : R → S is open (being a quotient of topological groups), we have that the map π : R × M → R λ × M defined by (r, m)π = (r λ , m) is open, as well. μ = (r, m)μ is continuous. Therefore, the map μ : S × M → M given by (r λ , m) According to 9.7, the topology of S contains the topology induced by the compactopen topology. 2 As an immediate application, we have: 11.12 Theorem. Let R be a topological ring. For the sake of distinction, let A be the additive topological group underlying R, and denote it by M if considering it as a right R-module. Then ρ : R → C(M, M) is an embedding of topological rings; that is, the topology of R is induced by the compact-open topology from C(R, R). This embedding may also be interpreted as an embedding ρ : R → Mor(A, A) of topological rings. The map λ : R → C(M, M) is an embedding of topological groups, as well. However, it yields a ring anti-homomorphism rather than a ring homomorphism.
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Proof. The evaluation ω1 : C(R, R) → R mapping ϕ to 1ϕ is continuous by 9.6. We have ρω1 = idR = λω1 . Applying 6.16, we obtain that ω1 is a quotient map, and ρ and λ are embeddings. The rest is clear. 2 11.13 Lemma. Let R be a topological ring, let M be a topological left R-module. and let X be any locally compact space. Then the following hold. (a) The group C(X, M) is a topological left R-module. (b) If X is a topological group then Mor(X, M) is an R-submodule of C(X, M); this submodule is closed if M is Hausdorff. (c) If X is a topological R-module then Mor R (X, M) is another R-submodule of C(X, M); again, this submodule is closed if M is Hausdorff. (d) If M is locally compact then M is a topological right module over the ring Mor R (M, M). Proof. We already know that C(X, M) is a topological group. The multiplication by scalars on C(X, M) is obtained by mapping (ϕ, r) first to (ϕ, r ρ ) and then to ϕr ρ . As ρ is continuous by 11.11 and composition is continuous by 9.4, we have that C(X, M) is a topological R-module. We have seen in 11.6 that Mor(X, M) is a closed subgroup of C(X, M) if M is Hausdorff. In order to decide which elements of Mor(X, M) belong to Mor R (X, M), we consider the maps ωr,m : Mor(X, M) → N defined by ϕ ωr,m = (rm)ϕ − r(mϕ ); for (r, m) ∈ R × M. As evaluation of ϕ, multiplication by scalars and addition in N are continuous, we have that ωr,m is continuous for each pair (r, m) ∈ R × M, and the pre-image Zr,m of 0 under ωr,m is closed in Mor(X, M). Consequently, the subgroup Mor R (X, M) = {Zr,m | (r, m) ∈ R × M} is closed in Mor(X, M). Continuity of multiplication (that is, composition) follows from 9.4. Therefore, Mor R (M, M) is a topological ring. The action of Mor R (M, M) on M is continuous by 9.8. 2 Notice the important special case where R is a topological field (and the Rmodules in question are vector spaces); in particular, the case R = R.
Groups of Automorphisms of Commutative Groups 11.14 Definition. Let A, B, R, S, X, and Y be commutative groups, written additively. For matrices a c r t and b d s u
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with entries that are group homomorphisms a : A → R, b : B → R, c : A → S, d : B → S, r : R → X, s : S → X, t : R → Y , and u : S → Y , the matrix product a c r t ar + cs at + cu = b d s u br + ds bt + du has well-defined entries, namely, the group homomorphisms ar + cs : A → X,
at + cu : A → Y,
br + ds : B → X,
bt + du : B → Y.
Let α, β be the natural maps from A × B onto A, B, and let ρ, σ be those from R × S onto R, S, respectively. Define maps α : A → A × B and β : B → A × B, by a α = (a, 0) and bβ = (0, b). For any homomorphism ϕ : A × B → R × S we put αϕρ αϕσ A R := . [ϕ] B S βϕρ βϕσ X Analogously, we define R S [ψ]Y for each homomorphism ψ : R × S → X × Y . RR X A X A straightforward computation shows A B [ϕ]S S [ψ]Y = B [ϕψ]Y .
The proof of the next assertion is now easy; we leave it as an exercise. 11.15 Lemma. Let A, B, R, and S be commutative topological groups. Endowed with the product of the compact-open topologies, and pointwise addition of matrices, the set a c a ∈ Mor(A, R), c ∈ Mor(A, S), A R := × B, R × S)] [Mor(A B S b d b ∈ Mor(B, R), d ∈ Mor(B, S) becomes a commutative topological group. If A and B are locally compact, then is a topological ring.
A [Mor(A × B, A × B)]A B B
11.16 Proposition. Let A, B, R, S be commutative groups, and let X be a topological space. Using the notation introduced in 11.14, we have the following. (a) Mapping the function ϕ to (ϕρ, ϕσ ) gives an isomorphism of topological groups from C(X, R × S) onto C(X, R)×C(X, S); its inverse maps (γ , δ) to γ ρ¯ +δ σ¯ . R (b) Mapping ϕ to the corresponding matrix A B [ϕ]S is an isomorphism of topological A R groups a c from Mor(A×B, R×S) onto B [Mor(A × B, R × S)]S ; its inverse maps ¯ + βbρ¯ + βd σ¯ . b d to αa ρ¯ + αcσ
(c) If A = R and B = S are locally compact, then the isomorphism in assertion (b) is an isomorphism of topological rings; and induces, therefore, an isomorphism of topological groups from Aut(A × B) onto its image.
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Proof. Mapping ϕ to (ϕρ, ϕσ ) is continuous, compare 9.4. Straightforward computation shows ((γ ρ¯ + δ σ¯ )ρ, (γ ρ¯ + δ σ¯ )σ ) = (γ , δ) and ϕρ ρ¯ + ϕ σ¯ σ = ϕ. Thus the inverse map exists, and is continuous; again by 9.4. Similar arguments yield AA A A A assertion (b). For assertion (c), it remains to observe A 2 B [ϕ]B B [ψ]B = B [ϕψ]B . Of course, the matrix constructions just discussed may be iterated, leading to matrices of arbitrary (but finite) numbers of rows and columns. Proposition 11.16 will come quite handy for the determination of automorphism groups of certain locally compact commutative groups; since various decompositions of these groups as direct products exist, see 25.7 below. The reader may have noticed that these matrix constructions generalize the familiar description of elements of Mor F (V , W ) by matrices if V and W are vector spaces of finite dimension over a field F . Note that Mor F (V , W ) is isomorphic to F if V and W have dimension 1. For the case of locally compact vector spaces, the results above provide convenient means to prove that the matrix description contains the information about the topology, as well. We leave the details as exercises. A special case is worth to be stated explicitly. 11.17 Theorem. The additive group R of real numbers, with its usual topology, is isomorphic as a topological group to Mor(R, R). The topological ring Mor(R, R) is isomorphic to the topological field R. 2 11.18 Theorem. For each pair (n, m) ∈ N × N, we have that Mor(Rn , Rm ) and Rn×m are isomorphic as topological groups. If n = m, then the isomorphism may 2 be chosen as an isomorphism of topological rings.
Exercises for Section 11 Exercise 11.1. Let G be a free Abelian group of infinite rank c, endow it with the discrete topology, and let H be any locally compact Abelian group. Prove that Mor(G, H ) corresponds to the set H c of all maps from c to H , endowed with the product topology. Conclude that Mor(G, H )c is not locally compact. Exercise 11.2. Consider the discrete group Z, and let d be an infinite set. ←
(a) Verify that the set G := {γ : d → Z | |(Z {0})γ | < ∞} of all functions of “finite support” from d to Z forms a subgroup of C(d, Z). We endow the group G with the discrete topology. Prove the following: (b) Every automorphism α ∈ Aut(G) is determined by its restriction α|d¯ , where d¯ consists of the characteristic functions of singleton subsets of d. (c) Conversely, every map from d¯ to Z extends to a homomorphism from G to G. (d) The group Aut(G) is not locally compact.
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Exercise 11.3. Prove the claim in 11.15. Exercise 11.4. Let F be a topological field, and let n be a natural number. Endow F n with the product topology. Show that the topologies induced on GL(n, F ) by the product topology on F n×n and the compact-open topology on C(F n , F n ) coincide. Exercise 11.5. Show that R and Mor(R, R) are isomorphic as topological groups; compare 11.17. Hint. Prove that every element of Mor(R, R) is determined by the image of 1. Exercise 11.6. Explain why there are “much more” discontinuous group homomorphisms from R to itself than continuous ones.
Chapter D
The Haar Integral 12 Existence and Uniqueness of Haar Integrals One fundamental tool in the theory of locally compact groups (and their linear representations) is the fact that one can construct an integral on the group which is invariant under right translation. The present chapter includes a proof that such an integral exists on every locally compact Hausdorff group, and discusses uniqueness properties.
Spaces of Real-Valued Functions on Groups We will use methods and results from the theory of topological vector spaces over R. Among many others, the book [56] by H. H. Schaefer is a convenient source. 12.1 Definition. Let X be a topological space. The set RX of all maps from X to R forms a vector space over R; with operations defined by x ϕ+ψ = x ϕ + x ψ and x rϕ = r(x ϕ ) for all x ∈ X, all ϕ, ψ ∈ RX , and all r ∈ R. In addition, we have a multiplication, given by x ϕ·ψ = x ϕ · x ψ . We are interested in several vector subspaces: (a) The subspace R(X) of all functions of finite support; that is, the space of all functions that map all but finitely many elements of X to 0. (b) The set C(X, R) of all continuous maps from X to R. (c) The set B(X) := {ϕ ∈ C(X, R) | supx∈X |x ϕ | < ∞} of all bounded continuous functions forms a vector subspace of C(X, R). On B(X), we have the supremum norm ϕ := supx∈X |x ϕ |, and the corresponding metric d(ϕ, ψ) :=
ϕ − ψ defining the topology of uniform convergence. (d) Defining the support of ϕ ∈ C(X, R) as supp ϕ := {x ∈ X | x ϕ = 0} we obtain the subspace Cc (X) of B(X) consisting of those elements of C(X, R) whose support is compact. For any vector subspace V of RX , we call V + := {ϕ ∈ V | ∀x ∈ X : x ϕ ≥ 0} the positive cone in V . In particular, we have C + (X) := {ϕ ∈ C(X, R) | ∀x ∈ X : x ϕ ≥ 0}, B + (X) := C + (X) ∩ B(X) ,
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and Cc+ (X) := C + (X) ∩ Cc (X) . As usual, the elements of V + are called positive functions (although “nonnegative” would be more appropriate), and we write ϕ ≥ 0 for ϕ ∈ V + . This notation extends to a pair of partial orderings ≤ and ≥ on V , given by ψ ≤ ϕ ⇐⇒ ϕ ≥ ψ ⇐⇒ ϕ − ψ ≥ 0. Let Y be a set, and let λ : V → RY be a linear map defined on some vector subspace V of RX . Then λ is called positive if (V + )λ ⊆ (RY )+ . If we identify Rn with C(n, R), where n := {0, . . . , n − 1} is considered as a discrete space, we have (Rn )+ = C + (n). In particular, a positive linear form λ : V → R = R1 from a subspace V of C(X, R) satisfies ϕ λ ≥ 0 for every ϕ ∈ V + . We leave the easy proof of the next lemma as an exercise. 12.2 Lemma. Let X be a set, let V be a vector subspace of RX , and let λ : V → R be a positive linear form. Assume that, for every ϕ ∈ V , the function |ϕ| mapping x to |x ϕ | belongs to V , as well. Then |ϕ λ | ≤ |ϕ|λ . 2 We use translates of functions defined on a group G: for ϕ ∈ RG and a ∈ G, we have the function ϕa mapping g ∈ G to (ga −1 )ϕ . Compare 10.6 for a discussion of the corresponding action of G on RG . 12.3 Definition. Let G be a topological group. A linear map λ : Cc (G) → R is called an invariant integral on G if (ϕa )λ = ϕ λ holds for all ϕ ∈ Cc (G) and all a ∈ G. An invariant integral λ is called a Haar integral on G if λ is positive and there exists a function ϕ ∈ Cc+ (G) such that ϕ λ = 0 (of course, we have ϕ λ > 0 then). A linear map is of course determined by its restriction to a positive cone P if P − P = V (for instance, the latter condition is satisfied if ϕ ∈ V implies |ϕ| ∈ V ). The following observation allows us to construct a Haar integral first on a positive cone, and extend it afterwards. 12.4 Lemma. Let V , W be vector spaces over R, let P be a subset of V such that V = P − P , and let λ : P → W be a map. Moreover, assume that the following conditions are satisfied: (a) For all x, y ∈ P , we have x + y ∈ P , and (x + y)λ = x λ + y λ . (b) For each x ∈ P and each r ≥ 0, we have rx ∈ P , and (rx)λ = r(x λ ). λ: V → W. Then (x − y)λ := x λ − y λ defines a linear map
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Proof. The crucial point is to see that λ is well defined, it is very easy to see that λ is a linear map. We leave the details for an exercise. 2 12.5 Lemma. Let G be a topological group. Then every function ϕ ∈ Cc (G) is uniformly continuous; that is, for every ε > 0 there is a neighborhood V of 1 in G such that for each g ∈ G we have (V g)ϕ ⊆ ]g ϕ − ε, g ϕ + ε[. Proof. Fix ε > 0. For is an open neighborhood Vg of 1 in G
each g ∈ G, there such that (Vg g)ϕ ⊆ g ϕ − 2ε , g ϕ + 2ε . Pick an open neighborhood Wg of 1 such that Wg W of C such g ⊆ Vg . As C := supp ϕ is compact, we find a finite subset F that C ⊆ f ∈F Wf f . Pick a neighborhood V of 1 such that V V −1 ⊆ f ∈F Wf . Now consider an arbitrary element g ∈ G. If V g and C are disjoint, we have (V g)ϕ = {0}. If V g ∩ C is nonempty, we find some element f ∈ F such that −1 V g ∩ Wf f = ∅. Then g ∈ V −1 Wf f implies V g ⊆ ϕV V Wϕf f ⊆ Wf Wf f ⊆ ε ε ϕ ϕ ϕ Vf f , and we obtain (V g) ⊆ f − 2 , f + 2 ⊆ ]g − ε, g + ε[. 2 12.6 Lemma. Let C be a compact subspace of a locally compact Hausdorffspace X, and let W1 , . . . , Wn be open subsets of X such that C is contained in nk=1 Wk . Then the following hold. (a) There exist compact subsets Dk of X such that C ⊆ nk=1 Dk◦ and Dk ⊆ Wk . (b) There are continuous functions ϕk : X → [0, 1] with supp ϕk ⊆ Wk such that the sum nk=1 ϕk maps every element of C to 1. Proof. For c ∈ C, we find k ∈ {1, . . . , n} and a compact neighborhood Vc of c such that Vc ⊆ Wk . As C is compact, there is a finite subset S of C such that C ⊆ c∈S Vc◦ . Now put Sk := {s ∈ S | Vs ⊆ Wk } and Dk := s∈Sk Vs . Then Dk is a compact subspace of Wk satisfying s∈Sk Vs◦ ⊆ Dk◦ , and S = nk=1 Sk implies C ⊆ nk=1 Dk◦ . This proves assertion (a). In order to prove assertion (b), we first use assertion (a) three times to find ◦ ◦ compact sets k , and Fk , such that Fk ⊆ Ek ⊆ Ek ⊆Dk ⊆ Dk ⊆ Wk nDk , E n ◦ and C ⊆ k=1 Fk . The compact Hausdorff space D := k=1 Dk is normal by 1.20, and using 1.17 we find functions ψk : D → [0, 1] with supp ψk ⊆ Ek such ψ that Fk k ⊆ {1}. Mapping every point of X nk=1 Ek to 0 yields a continuous k : X → [0, 1] of ψk , compare 1.6. There only remains the problem extension ψ n that k=1 x ψk > 1 may happen for some x ∈ X. This problem is resolved easily: we define a continuous function δ : X → R by x δ := max 1, nk=1 x ψk , and put x ϕk :=
x ψk xδ
.
2
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Existence of Haar Integrals We are now going to construct a Haar integral on a given locally compact Hausdorff group G, depending on some arbitrary nonzero function η ∈ Cc+ (G). Later on, we will see that this integral does not depend too much on η, in fact, a different choice of η will only change the integral by a positive real factor. Our existence proof will use the Axiom of Choice, disguised as Tychonoff’s Theorem. In fact, the existence of Haar integrals does not depend on the Axiom of Choice, and we have decided to use it for the sake of elegance only. As my main interest in Haar integrals lies in the theory of compact groups (where Tychonoff’s Theorem appears to be indispensable), we felt free to do so. Our first step towards the integral introduces a (rough) method of comparing two functions. ϕ
12.7 Definition. Let G be a group. For ϕ, ψ ∈ RG let Bψ be the set of all sequences (bk )k∈N ∈ R(N) with the property that there exists a sequence (ak )k∈N ∈ GN such that ϕ ≤ k∈N bk ψak . (Note that only a finite number of summands is different ϕ from zero, since the sequence (bk )k∈N has finite support.) We use the set Bψ to ϕ define (ϕ : ψ) := inf k∈N bk | (bk )k∈N ∈ Bψ . For arbitrary functions ϕ, ψ ∈ RG , the infimum (ϕ : ψ) may well belong to {∞, −∞}. We will see in 12.10 below that this does not happen if G is a topological group, and ϕ, ψ are continuous positive functions of compact support. As we want to use the values (ϕ : ψ) to construct an invariant integral, the following observations will be useful. 12.8 Lemma. For all a, b ∈ G and all ϕ, ψ ∈ RG , we have (ϕ : ψ) = (ϕa : ψb ). ϕ
ϕ
Proof. This follows immediately from the trivial observation Bψ = Bψab .
2
12.9 Lemma. For all functions ϕ, ψ ∈ RG and all positive real numbers r, s, we have (rϕ : sψ) = rs (ϕ : ψ). Proof. It is obvious that ϕ ≤ k∈N bk ψak is equivalent to rϕ ≤ k∈N rs bk (sψak ). ϕ rϕ This means Bψ = rs b | b ∈ Bsψ , and the assertion follows. 2 12.10 Lemma. Let G be a topological group, and consider ϕ, ψ ∈ Cc+ (G) with ψ = 0. Then (ϕ : ψ) is a nonnegative real number. Proof. We know that t := supg∈G g ϕ is a nonnegative real number, and that the real number s := supg∈G g ψ is positive. Because V := x ∈ G | x ψ > 2s is a nonempty open subset of G, we find elements a1 , . . ., an ∈ G such that the compact set supp ϕ is contained in the union nj=1 V aj . We obtain the inequality
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ϕ ≤ nj=1 2ts ψaj ; in fact, positivity of ψ yields that the inequality trivially holds outside supp ϕ, and for x ∈ supp ϕ we find some k such that x ϕ ≤ t < 2ts (xak−1 )ψ . ϕ We have, therefore, that Bψ is nonempty, and (ϕ : ψ) < ∞. In order to show ϕ ϕ that Bψ is bounded below, consider a sequence (bk )k∈N ∈ Bψ . For each x in G, we 2 have x ϕ ≤ k∈N bk (xak−1 )ψ ≤ k∈N bk s, and obtain st ≤ (ϕ : ψ). It will be convenient to remember the last step of the proof of 12.10. 12.11 Corollary. Let G be a topological group. Consider functions ϕ, ψ ∈ Cc+ (G), and assume ψ = 0. Put t := supg∈G g ϕ , and s := supg∈G g ψ . Then st ≤ (ϕ : ψ); 2 in particular, we have (0 : ψ) = 0 and (ψ : ψ) = 1. 12.12 Lemma. Let G be a topological group, and consider functions ϕ, ψ, π ∈ Cc+ (G). (a) If ψ = 0 then (ϕ + π : ψ) ≤ (ϕ : ψ) + (π : ψ). (b) If ψ = 0 = π then (ϕ : π ) ≤ (ϕ : ψ) (ψ : π). ψ
ϕ
Proof. Consider sequences (bk )k∈N ∈ Bψ and (dk )k∈N ∈ Bπ . In order to prove ϕ+π
assertion (a), define a sequence (ek )k∈N ∈ Bψ putting e2k = bk and e2k+1 = dk . This leads to the inequality, as claimed. Taking )k∈N in GN such that ϕ ≤ k∈N bk ψak and sequences (ak )k∈N and (ck ψ ≤ k∈N dk πck , we obtain bn ψan ≤ k∈N bn dk πck an and ϕ ≤ n,k∈N bn dk πck an .
This leads to the inequality (ϕ : π ) ≤ n,k∈N bn dk = n∈N bn k∈N dk , and assertion (b) follows. 2 12.13 Definition. Let G be a topological group. Pick a function η ∈ Cc+ (G) {0}. For functions ϕ, ψ ∈ Cc+ (G) with ψ = 0, we put p(ϕ, ψ) :=
(ϕ : ψ) . (η : ψ) +
This leads to an element pψ := (p(ϕ, ψ))ϕ∈Cc+(G) of RCc (G) . Of course, one may interpret pψ as a function from Cc+ (G) to R. However, the + interpretation as an element of the cartesian product RCc (G) has its advantages, as well. In fact, using 12.10 and 12.12, we obtain 0 ≤ p(ϕ, ψ) ≤ (ϕ : η). This means that pψ may be considered as an element of a compact space, namely, a product of (awfully many) compact intervals: 12.14 Lemma. We have pψ ∈ Xϕ∈Cc+(G) [0, (ϕ : η)].
2
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D The Haar Integral
Our problem at the moment is that the function pψ is invariant and homogeneous (that is, we have (rϕ)pψ = p(rϕ, ψ) = r · p(ϕ, ψ) = r(ϕ pψ ), compare 12.9), but need not be additive. We will use compactness of the product Xϕ∈Cc+(G) [0, (ϕ : η)] for smoothing our function pψ ; this is where Tychonoff’s Theorem (and, therefore, the Axiom of Choice) comes in. The idea is to approximate the Haar integral by functions pψ , where the support of ψ converges to 0 (in the sense of filterbases). 12.15 Definition. Let G be a topological group. For every open neighborhood V of 1, we define PV := {pψ | 0 = ψ ∈ Cc+ (G) , supp ψ ⊆ V }. 12.16 Lemma. Let (G, T ) bea topologicalgroup. If T is locally compact HausPV | V ∈ T1 is not empty. dorff, then the intersection Proof. V , W ∈ T1 , we have PV ∩W ⊆ PV ∩ PW . This means For neighborhoods that PV | V ∈ T1 is a filterbasis, consisting of closed subsets of the compact space Xϕ∈Cc+(G) [0, (ϕ : η)]. According to 1.23, it remains to show that none of the sets PV is empty. But this is an immediate consequence of the fact that locally 2 compact Hausdorff spaces are completely regular, see 1.25. 12.17 Lemma. Every λ ∈ pψ | ψ ∈ Cc+ (G) {0} is invariant, homogeneous, and sub-additive; that is, for each a ∈ G, each real number r ≥ 0 and all ϕ, π ∈ Cc+ (G) we have the following: ϕ λ = (ϕa )λ ,
(rϕ)λ = r(ϕ λ ),
(ϕ + π )λ ≤ ϕ λ + π λ .
Proof. For every choice of ϕ ∈ Cc+ (G), the map evϕ :
X [0, (ϕ : η)] → R : λ → ϕ λ
ϕ∈Cc+(G)
is continuous with respect to the product topology on Xϕ∈Cc+(G) [0, (ϕ : η)]. This means that we have continuous maps α, β, and γ from Xϕ∈Cc+(G) [0, (ϕ : η)] to R, defined by λα = ϕ λ − (ϕa )λ ,
λβ = (rϕ)λ − r(ϕ λ ),
and
λγ = (ϕ + π )λ − ϕ λ − π λ ,
respectively. We see immediately from 12.8 and 12.9 that {pψ | ψ ∈ Cc+ (G) {0}} is mapped to {0} both by α and by β. By continuity, we extend this observation to the closure, and obtain our first two assertions. The last assertion follows from {pψ | ψ ∈ Cc+ (G) {0}}γ ⊆ (−∞, 0], which is a consequence of 12.12 (a). 2 12.18 Proposition. Every λ ∈ PV | V ∈ T1 is invariant, homogeneous, and additive; that is, for each a ∈ G, each real number r ≥ 0 and all ϕ, π ∈ Cc+ (G) we have the following: ϕ λ = (ϕa )λ ,
(rϕ)λ = r(ϕ λ ),
(ϕ + π )λ = ϕ λ + π λ .
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119
Moreover, we have ϕ λ ≥ 0 for every ϕ ∈ Cc+ (G). Proof. The last assertion follows (by a continuity argument as in the proof of 12.17) from the fact that ϕ pψ ≥ 0 holds for all ϕ, ψ ∈ Cc+ (G). After 12.17, it only remains to show that every λ in PV | V ∈ T1 is additive. Fix ϕ, π ∈ Cc+ (G), and choose a continuous function ξ : G → [0, 1] with compact support such that (supp ϕ ∪ supp π )ξ = {1}; this is possible by 1.27. For r > 0, we consider the function σ := σr := ϕ + π + rξ . As σ does not attain the value 0 on supp ϕ ∪ supp π , we have continuous functions ϕˆ and πˆ from G to R satisfying ϕ π x ϕˆ = xxσ and x πˆ = xx σ on supp ϕ ∪supp π and vanishing outside this set. Moreover, we have ϕˆ + πˆ ≤ 1. For every positive real number ε, we use 12.5 in order to find a neighborhood V of 1 in G such that yx −1 ∈ V implies x ϕˆ − y ϕˆ < ε as well as x πˆ − y πˆ < ε. We choose a nonzero function τ ∈ Cc+ (G) such that supp τ ⊆ V . Then we find (bk )k∈N ∈ Bστ in such a way that (σ : τ ) ≤ k∈N bk ≤ (σ : τ ) (1 + ε). For some sequence (ak )k∈N ∈ GN with σ ≤ k∈N bk τak , we observe * * ϕˆ
x ϕˆ bk x τak ≤ bk ak + ε x τak , x ϕ = x ϕˆ x σ ≤ k∈N
k∈N
where the latter inequality follows from the fact that either xak−1 lies outside ϕˆ
V (and then outside supp τ ), or x ϕˆ ≤ ak + ε. Therefore, we have (ϕ : τ ) ≤ ϕˆ k∈N bk (a k + ε). Simply by replacing ϕ by π in the argument, we also get (π : τ ) ≤ k∈N bk (akπˆ + ε), and end up with the inequalities * bk (1 + 2ε) ≤ (σ : τ ) (1 + ε)(1 + 2ε), (ϕ : τ ) + (π : τ ) ≤ k∈N
recall that ϕˆ + πˆ ≤ 1. Dividing by (η : τ ), we obtain p(ϕ, τ ) + p(π, τ ) ≤ p(σ, τ )(1 + ε)(1 + 2ε). It is tempting to argue that this holds for all positive real numbers ε. Alas, our inequality depends on the fact that τ was chosen after ε was fixed: namely, such that supp τ is contained in some neighborhood V depending on ε. However, this simply means pτ ∈ PV , and a continuity argument as in the proof of 12.17 shows that the inequality is still true if we replace pτ by an element λ ∈ PV . Therefore, we have ϕ λ + π λ ≤ σ λ (1 + ε)(1 + 2ε) for all ε, if λ ∈ PV | V ∈ T1 . This means ϕ λ + π λ ≤ σ λ = (ϕ + π + rξ )λ ≤ (ϕ + π )λ + r(ξ λ ). Since r > 0 was arbitrary, we end up with ϕ λ + π λ = (ϕ + π )λ .
2
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D The Haar Integral
12.19 Lemma. Every λ ∈ PV | V ∈ T1 is positive definite; in the sense that for ϕ ∈ Cc+ (G) we have ϕ λ ≥ 0, and ϕ λ = 0 ⇐⇒ ϕ = 0. + λ = 0. Pick (b ) Proof. It only remains implies ϕ k k∈N to show that ϕ ∈ Cλc (G){0} η λ in Bϕ . Then η ≤ k∈N bk ϕak yields η ≤ k∈N bk ϕak = k∈N bk ϕ λ . This is λ 2 impossible if ϕ λ = 0; recall that 1 = (η:ψ) (η:ψ) = η .
Combining 12.18 and 12.19, we reach the first major goal of the present chapter: 12.20 Theorem. Let G be a locally compact Hausdorff group, fix a nonzero function η ∈ Cc+ (G), and construct the sets PV as in 12.13 and 12.15. Then for every PV | V ∈ T1 , the linear extension 2 λ∈ λ is a Haar integral.
Uniqueness of Haar Integrals The Haar integral that we found in 12.20 is somewhat arbitrary, in at least two ways. Firstly, its construction depends on the choice of a function η, compare 12.13. Secondly, it appears to be possible that the intersection PV | V ∈ T1 contains more than one element. Starting with a Haar integral, we obtain a new one if we multiply all values with a fixed positive real number. Our next aim is to show that this already yields all the Haar integrals on a given locally compact Hausdorff group. 12.21 Lemma. Let ϕ, ψ ∈ Cc+ (G). If ψ = 0 then ϕ μ ≤ (ϕ : ψ) ψ μ holds for every Haar integral μ. ϕ Proof. For every sequence (bk )k∈N ∈ Bψ we infer from ϕ ≤ k∈N bk ψak the μ μ inequality ϕ ≤ k∈N bk ψ ; of course, we use additivity, positivity and invariance of the Haar integral here. 2 12.22 Corollary. Every Haar integral μ is positive definite in the sense of 12.19; that is, for ϕ ∈ Cc+ (G) we have ϕ μ = 0 ⇐⇒ ϕ = 0. Proof. Let ϕ be a nonzero element of Cc+ (G). Pick a function ψ ∈ Cc+ (G) such that ψ μ > 0; such a function exists by the definition of a Haar integral. Now 12.21 2 yields 0 < ψ μ ≤ (ψ : ϕ) ϕ μ , and the assertion follows. 12.23 Theorem. Let G be a locally compact Hausdorff group, and fix some positive λ μ function η ∈ Cc+ (G) with η = 0. If λ, μ are Haar integrals on G then ϕηλ = ϕημ holds for each ϕ ∈ Cc (G). Consequently, there is a positive real number r = rλ,μ such that λ = rμ.
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121
Proof. We proceed in several steps. Pick a function ϕ ∈ Cc+ (G), and a positive real number ε. According to 12.5, we can choose a neighborhood Uε of 1 in G such that xy −1 ∈ Uε implies |x ϕ − y ϕ | < ε as well as |x η − y η | < ε. From now on, we treat the functions ϕ and η simultaneously. For the sake of readability, we write ∈ {ϕ, η}. We put Vε := Uε ∩ Uε−1 , and choose a nonzero function ψ ∈ Cc+ (G) such that supp ψ ⊆ Vε . Defining π ∈ Cc+ (G) by x π = x ψ + (x −1 )ψ , we obtain a nonzero function with support in Vε that is symmetric: x π = (x −1 )π . Now let δ > 0, and choose an open neighborhood W of 1 in G such that W π | < δ. Since the set S := supp ϕ ∪ is compact, and xy −1 ∈ W implies |x π − y supp η is compact, it is covered by a union m k=1 W ak of finitely many translates of W . According to 12.6, we find continuous functions ψk : G → [0, 1] such that supp ψk ⊆ W ak , and that m k=1 ψk maps every element of S to 1. Then we have (A)
=
m *
· ψk ,
and
= μ
k=1
m *
bk , where bk := ( · ψk )μ ≥ 0.
k=1
We claim that the following holds for each y ∈ G: (y − ε) · π μ ≤ ( · πy )μ =
(B)
m *
( · ψk · πy )μ
k=1
In fact, for xy −1
∈ / Vε = 0, and for xy −1 ∈ Vε , we have x ≥ y − ε. This entails · πy ≥ (y − ε) · πy , and our claim follows since μ is a positive linear form. The next claim is (C)
, we have x πy
m *
( · ψk · πy )μ ≤
k=1
m *
( · ψk )μ · (y πak + δ);
k=1
this follows since x ∈ / W ak yields x ψk = 0, while x ∈ W ak gives (xy −1 )(yak−1 ) ∈ W π and x πy ≤ y ak + δ. Thus we obtain ψk · πy ≤ ψk · (y πak + δ). (D) For every y ∈ G, we have y − 2ε ≤ k∈N πbkμ y πak . In order to verify this, pick δ sufficiently small, such that μ δ < π μ ε. Then (B) and (C) yield (y − ε) · π μ ≤
m *
( · ψk · πy )μ ≤
k=1
=
m * k=1
m *
( · ψk )μ · (y πak + δ)
k=1 m *
bk · y πak + μ δ ≤
k=1
bk y πak + π μ ε
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D The Haar Integral
and the claim follows. We define a new function ε ∈ Cc+ (G), mapping x ∈ G to max(x − 2ε, 0). Then we obtain the inequality ( ε : π )π μ ≤ μ ≤ ( : π ) π μ ε In fact, we know from (D) that the sequence πbkμ k∈N ∈ R(N) belongs to B π , and infer ( ε : π) ≤ k∈N πbkμ . Using (A), this gives the first inequality, the second one follows from 12.21. Now we choose a continuous function ξ : G → [0, 1] with compact support, mapping every element of supp to 1; this is possible since G is completely regular, see 1.27. We obtain the inequality ≤ ε + 2εξ . Using 12.12, we conclude (E)
( : π) ≤ ( ε +2εξ : π ) ≤ ( ε : π )+(2εξ : π ) ≤ ( ε : π )+2ε (ξ : ) ( : π ) , and we end up with (F)
(1 − 2ε (ξ : )) ( : π ) π μ ≤ μ ≤ ( : π ) π μ .
In other words, we have shown that for every γ > 0, there is a nonzero function μ π ∈ Cc+ (G) such that (1 − γ ) ( : π ) ≤ π μ ≤ ( : π ). This holds for ∈ {ϕ, η}, and we conclude (1 − γ )
ϕμ (ϕ : π ) ≤ μ η (η : π )
as well as
ϕμ (ϕ : π ) . ≤ ημ (1 − γ ) (η : π )
These inequalities hold for every Haar integral μ, and we obtain ϕλ ϕμ ϕλ 1 1 ≤ ≤ . ηλ (1 − γ )2 ημ (1 − γ )4 ηλ Thus Theorem 12.23 is proved.
2
It is high time to introduce a bit of standard notation: 12.24 Definition. Let G be a locally compact Hausdorff group, and choose a Haar 5 5 integral λ : Cc (G) → R. For ϕ ∈ Cc (G), we write ϕ λ = G ϕ dλ = ϕ dλ. If the 5 Haar integral is fixed by the context, it is also usual to write ϕ λ = G g ϕ dg.
Exercises for Section 12 Exercise 12.1. Let X be a set, let V be a vector subspace of RX , and let λ : V → R be a positive linear form. Assume that, for every ϕ ∈ V , the function |ϕ| mapping x to |x ϕ | belongs to V , as well. Show that |ϕ λ | ≤ |ϕ|λ holds for every ϕ ∈ V .
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123
Exercise 12.2. Fill in the details of the proof of 12.4. Exercise 12.3. Show that the Lebesgue (or Riemann) integral is a Haar integral on Rn , for any natural number n. Exercise 12.4. Determine all Haar integrals on Rn . Exercise 12.5. Consider the natural map π : R → T := R/Z. For every continuous function 51 ϕ : T → R put ϕ μ := 0 x π ϕ dx. Show that μ is a Haar integral for T. Hint. Integrate the translate ϕt π using π(ϕt ) = (π ϕ)t . 51 Exercise 12.6. Why does ϕ ρ = 0 x ϕ dx not define a Haar integral ρ : Cc (R) → R ? 5 Exercise 12.7. Show that putting ϕ λ := GL(n,R) γ ϕ | det γ |−n dγ for each ϕ ∈ Cc (GL(n, R)) defines a Haar integral λ : Cc (GL(n, R)) → R for the group GL(n, R). Exercise 12.8. Let G be a finite discrete group. Determine Cc (G), and find a Haar integral. Exercise 12.9. What happens if you drop the condition of finiteness in the previous exercise, and consider any discrete group? Exercise 12.10. Find a Haar integral on the group a0 b1 | a, b ∈ R, a > 0 , endowed with the topology induced from GL(2, R).
13 The Module Function In this section, we give an important application of the uniqueness property for Haar integrals derived in 12.23. Let G be a locally compact Hausdorff group, and consider a Haar integral λ on G. The group Aut(G) of all automorphisms of G acts on G (as a subgroup of the group of all homeomorphisms of G), and in 10.6 we have seen how this induces an action ωˆ : C(G, R) × Aut(G) → C(G, R); mapping the pair (ϕ, α) to the function ϕα := (ϕ, α)ωˆ := α −1 ϕ. It is obvious that the subspace Cc (G) of C(G, R) is invariant under this action. Wishing to understand how application of α ∈ Aut(G) affects the Haar integral λ, α we define a function λα : Cc (G) → R by ϕ λ := (ϕα )λ . Straightforward calculations yield the following. 13.1 Lemma. For every α ∈ Aut(G), the map λα is a Haar integral, again.
2
13.2 Definition. We know from 12.23 that there is a positive real factor rμ,λ satisfying μ = rμ,λ λ whenever λ and μ are Haar integrals on a group G. For μ = λα as above, we thus obtain for every automorphism α of G a positive real numλ ber mod α := rλα ,λ . In fact, picking η ∈ Cc+ (G) {0} we obtain mod α = (ηηαλ) . The function mod : Aut(G) → ]0, ∞[ is called the module function for the group G.
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D The Haar Integral
We want to be able to interpret our construction as a map (in fact, a homomorphism) mod : Aut(G) → ]0, ∞[ into the multiplicative group of positive real numbers. In order to see that this map is well defined, we hasten to remark the following. 13.3 Lemma. The real number mod α does not depend on the particular Haar integral λ that was used in its definition. Proof. Let α ∈ Aut(G), and abbreviate r := rλα ,λ . Replacing λ by another Haar integral μ = sλ, we easily compute that μα maps ϕ ∈ Cc (G) to (ϕα )μ = s(ϕα )λ = α 2 sϕ λ . This shows μα = sλα = srλ = rμ, and the assertion follows. 13.4 Lemma. For every locally compact group G, the map mod is a continuous group homomorphism from Aut(G) (with the modified compact-open topology) to the multiplicative group of positive real numbers. Proof. Pick ϕ ∈ Cc (G) such that ϕ λ = 0. For α, β ∈ Aut(G), we compute mod (αβ) · ϕ λ = (ϕαβ )λ = ((ϕα )β )λ = mod β(mod α · ϕ λ ) because ωˆ is an action of Aut(G). This means that mod is a group homomorphism, since R× is commutative. Now we have to show that mod is continuous. By 3.33, it suffices to show that mod is continuous at the neutral element. So let ε > 0 be given; we search for finite families (Cj )j ∈J and (Uj )j ∈J of compact setsCj ⊆ G and open sets Uj in G such that Cj ⊆ Uj and that j ∈J $Cj , Uj % ∩ j ∈J $Cj , Uj %−1 is mapped into ]1 − ε, 1 + ε[ by the module function mod . To this end, pick a nonzero function ϕ ∈ Cc+ (G). For every δ > 0 and every s ∈ supp ϕ, we find compact neighborhoods Us and Vs of s such that Vs is contained ϕ δ[. As supp ϕ is compact, there is in the interior of Us , and Us ⊆ ]s ϕ − δ, s ϕ + a finiteset F ⊆ supp ϕ such that supp ϕ ⊆ s∈F Vs◦ . We define the open set W := s∈F $Vs , Us◦ % ∩ s∈F $Vs , Us◦ %−1 . Now for α ∈ W and x ∈ G we have −1 |x ϕ − x ϕα | < 2δ; in fact, this is trivial if both x and x α lie outside supp ϕ, and in the remaining cases, we verify it using a suitable s ∈ F and observing Vs⊆ Us . We pick a function ξ ∈ Cc+ (G) such that x ξ = 1 holds for every x ∈ s∈F Us ; this is possible by 1.27. Now supp(ϕ − ϕα ) ⊆ s∈F Us yields |ϕ − ϕα | < 2δξ , and we have |(ϕ − ϕα )λ | ≤ |ϕ − ϕα |λ < 2δξ λ ; compare 12.2. This means that λ determines a neighborhood W of idG such that mod α ∈ ]1 − ε, 1 + ε[ δ := εϕ λ 2ξ holds for each α ∈ W . 2 Every group G acts on itself via conjugation, and this leads to a homomorphism i from G into Aut(G); compare 3.26. If G is locally compact then this homomorphism is continuous, see 10.5. Composing i and mod , we thus obtain a continuous
13. The Module Function
125
homomorphism from G to the group of positive real numbers. Fearing no confusion, we denote this homomorphism by mod ; that is, we write mod g = mod (ig ) for g ∈ G. This homomorphism can be used to compare integrals that are invariant under translation from the right (as we were considering throughout this chapter) with integrals that are invariant under translations from the left; that is, under the action mapping (ϕ, g) ∈ RG × G to the function g ϕ mapping x ∈ G to (gx)ϕ . 13.5 Lemma. Let G be a locally compact Hausdorff group, and let λ be a Haar integral on G. For every g ∈ G and every ϕ ∈ Cc (G), we have (g ϕ)λ = mod g ·ϕ λ . Proof. It suffices to observe (g ϕ)g = ϕig , since λ is invariant.
2
13.6 Definition. A locally compact group is called unimodular if the homomorphism mod : G → R× is trivial. As an immediate consequence of 13.5, we have the following. 13.7 Lemma. A locally compact group G is unimodular if, and only if, some (and 2 then every) Haar integral is invariant from the left. 13.8 Theorem. The following properties of a locally compact group G imply that it is unimodular: (a) The group G is Abelian. (b) The group G is compact. (c) The closure D1 (G) of the commutator subgroup d1 (G) coincides with G. Proof. If G is Abelian then i is constant. If G is compact then its image under the continuous homomorphism mod is a compact subgroup of the multiplicative group of positive real numbers, and therefore trivial. Finally, the commutator subgroup d1 (G) is contained in the kernel of any homomorphism from G to an Abelian group A. If A is Hausdorff and the homomorphism is continuous, we also have that D1 (G) is contained in the kernel. 2 13.9 Example. The groups SL(n, R) and SL(n, C) are unimodular by 13.8 (c). 13.10 Example. Every R-linear bijection of the group G = Rn is an automorphism of that topological group (in fact, one can show that every continuous group homomorphism from Rn to Rk is R-linear – as long as n and k are finite). For γ ∈ GL(n, R) one can show mod γ = | det γ |. The proof (using the transformation formula for the Riemann or Lebesgue integral) is left as an exercise.
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D The Haar Integral
13.11 Example. It may come as a surprise to the unexperienced that the group GL(n, R) is unimodular, for each natural number n. This is due to the fact that conjugation by any element of GL(n, R) induces a linear bijection S of the space of all n × n matrices over R such that det S = 1. Again, the details are left for an exercise.
Computing the Module Function in Locally Compact Totally Disconnected Groups Let G be a locally compact Hausdorff group, and let U be an open compact subset of G. Then the characteristic function ! 1 if g ∈ U , χU : G → R : g → 0 if g ∈ / U, 5 λ 5of U is continuous, and has compact support. The Haar integral (χu ) = G d λ =: U d λ will then serve as a measure for the size of U . If U is not empty, we may use χU to evaluate the module function: for α ∈ Aut(G) and any Haar integral λ, we λ −1 U )α ) α have mod α = ((χ (χU )λ . Observing (χU )α = α χU = χU we obtain mod α = 5 dλ 5U α U dλ
. In several cases (in particular, if G is totally disconnected), this allows to compute mod α rather explicitly. 13.12 Lemma. Let G be a locally compact totally disconnected group, and let B be a neighborhood basis at 1 consisting of open compact subgroups (see 4.13). (a) Every open compact subset of G is the union of finitely many cosets of some suitable B ∈ B. (b) If C and D are open compact subsets, then there exists B ∈ B such that C and D are unions of cosets of B. Proof. Let C be an open compact subset of G. For each c ∈ C, we find Bc ∈ B such that Bc c ⊆ C, andC = c∈C Bc c. As C is compact, there is a finite subset F ⊆ C such that C = f ∈F Bf f . Putting BC := f ∈F Bf ∈ B we obtain the first assertion. If D is another open compact subset of G, we find BD ∈ B in the same way. Now B := BC ∩ BD satisfies our requirements. 2 13.13 Proposition. Let G be a locally compact group, and let λ be a Haar measure on G. Fix an open compact subgroup A. Let C be an open compact subset of G,
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13. The Module Function
and choose any open compact subgroup B ≤ A such that C is the union of k cosets of B. Then we have 6 6 6 k dλ = k dλ = d λ. |A/B| A C B of k := |F | cosets. Then Proof. We write C = f ∈F Bf as a disjoint union linearity and invariance of the Haar integral and χC = f ∈F χBf yield 6 6 * *
λ * (χB )f = d λ = (χC )λ = (χBf )λ = (χB )λ = k(χB )λ = k d λ. C
f ∈F
f ∈F
f ∈F
B
5 5 5 It remains to express A d λ in the same way: we obtain A d λ = |A/B| B d λ, 2 and the assertion follows. We note the consequence that the module function does not change if we restrict an automorphism to some open invariant subgroup. 13.14 Proposition. Let G be a locally compact group, and let α ∈ Aut(G). For any open compact subgroup A of G, we have 5 |Aα /(A ∩ Aα )| α dλ mod α = 5A = . |A/(A ∩ Aα )| A dλ 5 5 Proof.5 We obtain Aα d λ = |Aα /(A ∩ Aα )|γ and A d λ = |A/(A ∩ Aα )|γ , where γ := A∩Aα d λ, from 13.13. 2 13.15 Theorem. Let G be a locally compact group. If G has an open compact subgroup which is invariant under all automorphisms then G is unimodular. 2 The main application of the following result will be in the case where S is a field, the group U is a right ideal in a subring R of S, and a ∈ R. 13.16 Theorem. Let S be a locally compact totally disconnected ring, and let U be an open compact subgroup of the additive group (S, +). For each invertible element a ∈ S with U a ⊆ U , we have mod ρa = |U/U a|−1 . Proof. If a is invertible then ρa : S → S : x → xa is an automorphism of the additive group (S, +), with inverse ρa−1 = ρa −1 . Applying 13.14 for A := U a and α := ρa−1 , we find mod ρa−1 =
|U aa −1 /(U a ∩ U aa −1 )| |U/U a| = = |U/U a|, −1 |U a/(U a ∩ U aa )| |U a/U a|
and the assertion follows from the fact that mod is a group homomorphism.
2
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D The Haar Integral
Exercises for Section 13 Exercise 13.1. Prove 13.1. Exercise 13.2. Consider G = Rn . For γ ∈ GL(n, R) ≤ Aut(G), show mod γ = | det γ |. Hint. Use the transformation formula for the Riemann integral, and the fact that a linear map coincides with its derivative at each point. Exercise 13.3. Show that the group a0 b1 a, b ∈ R, a > 0 , endowed with the topology induced from GL(2, R), is not unimodular. Exercise 13.4. Show that GL(n, R) is unimodular.
⎛a
Hint. Show first that it suffices to consider elements of the form Ma := ⎝
⎞ 1
..
⎠
. 1
2
Then show that conjugation by Ma , considered as a linear map on Rn , has determinant 1. Finally, apply the transformation formula. Exercise 13.5. Let G be a locally compact Hausdorff group, and assume that every element of G is contained in some compact subgroup of G. Show that such a group is unimodular. Exercise 13.6. Exhibit an example of a locally compact Hausdorff group G with a closed normal subgroup N such that both N and G/N are unimodular, but G is not. Exercise 13.7. Show that every discrete group is unimodular. Exercise 13.8. Let S be a locally compact ring, let R be a subring, and let J = R be an open compact ideal in R. Let a ∈ R be invertible in S, and put |a| := mod ρa . Prove the following: (a) If a ∈ J then the sequence (|a n |)n∈N converges to 0. (b) If a ∈ J and (a n )n∈N converges to some element b ∈ S, then b is not invertible. (c) If R is compact and the inverse a −1 ∈ S belongs to R then |a| = 1.
14 Applications to Linear Representations In this chapter, we use the Haar integral in order to generalize certain concepts from the theory of complex linear representations of finite groups to a corresponding theory for (locally) compact groups. We will write Re(a + ib) := a and Im(a + ib) := b to denote the real and imaginary parts of a + ib.
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Complex Haar Integrals First of all, we have to extend our Haar integrals, that is, certain linear forms on real vector spaces, to complex linear forms with adequate properties. This will be done in the natural fashion. As usual, we write a + ib = a − ib, if a, b ∈ R. For a topological space X, we need the vector space CcC (X) consisting of all continuous functions from X to C with compact support. As usual, we identify C with the direct sum R ⊕ iR; this induces a direct sum CcC (X) = Cc (X) ⊕ iCc (X). C For any map λ : Cc (X) → R we obtain a map λC : CcC (X) → C by (ϕ + iψ)λ = ϕ λ +i(ψ λ ). If λ is R-linear then λC is C-linear. If X is a topological group, we have C C C (ϕ + iψ)a = ϕa + iψa for every a ∈ X, and ((ϕ + iψ)a )λ = (ϕa )λ + i (ψa )λ . We leave the details for an exercise, see Exercise 14.2. We obtain the following. 14.1 Lemma. Let G be a locally compact Hausdorff group, and let λ : Cc (G) → R be a Haar integral. Then λC : CcC (G) → C is a complex linear form, with the following properties: C
C
(a) For each ϕ ∈ CcC (G) and every a ∈ G, we have (ϕa )λ = ϕ λ . C
C
(b) For every ϕ ∈ CcC (G), we have (ϕϕ)λ ≥ 0, and (ϕϕ)λ = 0 ⇐⇒ ϕ = 0. Properties (a) and (b) are equivalent to: (c) The restriction of λC to Cc (G) is a Haar integral. We can reverse the construction in 14.1, as follows. If μ : CcC (G) → C is a complex linear form, with the additional property that Cc (G) is mapped to R by μ, then the restriction of μ to Cc (G) is, of course, a real linear form λ : Cc (G) → R. Moreover, we have μ = λC . If λ is a Haar integral, then of course μ satisfies assertions (a) and (b) of 14.1. 14.2 Definition. Let G be a locally compact Hausdorff group. A complex linear form μ : CcC (G) → C is called a complex Haar integral if assertions (a), (b), and (c) of 14.1 are satisfied. We have seen that every Haar integral λ gives rise to a complex Haar integral μ = λC , and that, conversely, every complex Haar integral is obtained in this way. This means that we can extend Theorem 12.23: 14.3 Theorem. For every locally compact Hausdorff group G there exists a complex Haar integral. If λ and μ are complex Haar integrals, then there exists a positive 2 real number r such that λ = rμ. Easy computations yield the following. 14.4 Lemma. If G is a locally compact Hausdorff group then every complex Haar integral λ satisfies ϕ λ = ϕ λ , for each ϕ ∈ CcC (G). 2
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Some Results from Functional Analysis 14.5 Definitions. Let V be a vector space over F, where F ∈ {R, C}. A map (·|·) : V × V → F is called a scalar product if it satisfies (for all x, y, z ∈ V and all c ∈ F) (a) (x|y) = (y|x), (b) (x + cy|z) = (x|z) + c(y|z), (c) (x|x) ≥ 0, (d) (x|x) = 0 ⇐⇒ x = 0 . If (·|·) is a scalar product on V , we call (V , (·|·)) a pre-Hilbert space. A linear map ϕ : V → V satisfying (x ϕ |y ϕ ) = (x|y) for all x, y ∈ V is called a unitary transformation. The set of all unitary transformations of a pre-Hilbert space H = (V , (·|·)) forms a group called the unitary group of H , and denoted by U(H ). √ Putting d(x, y) := (x − y|x − y), we obtain a metric d on V . If the metric space (V , d) thus obtained is complete, we call (V , (·|·)) a Hilbert space. A pre-Hilbert space is a special example of a normed vector space; that is, a vector space V over F ∈ {R, C} together with a map · : V → [0, ∞) satisfying (for all x, y, z ∈ V and all c ∈ F) the homogeneity condition |cx| = |c| · x , the triangle inequality x + y ≤ x + y , and √ x = 0 ⇐⇒ x = 0. A scalar product (·|·) yields a norm by putting x := (x|x). Via d(x, y) := x − y , the norm yields a metric d on V . If this metric turns V into a complete metric space, we call V a Banach space. The proof of the following is straightforward, and left as exercise Exercise 14.4. 14.6 Lemma. Let G be a locally compact Hausdorff group, and let λ be a complex Haar integral on G. Then (ψ|ϕ) := (ψϕ)λ introduces a scalar product on CcC (G), turning CcC (G) into a pre-Hilbert space. For every a ∈ G, the transformation mapping ϕ to ϕa is unitary. 2 In general, the space CcC (G) is not a Hilbert space. In order to make methods from functional analysis work, we therefore have to consider the completion L2 (G). Although it is obtained by an abstract construction, many of its elements are represented by functions from G to C. For instance, if S ⊆ G is a compact subset then the characteristic function χ : G → {0, 1} defined by x χ = 1 ⇐⇒ x ∈ S is an element of L2 (G), but χ ∈ / CcC (G), in general. 14.7 Remarks. We will briefly indicate a possibility to obtain the completion of a normed vector space (V , · ), leaving the details for exercise Exercise 14.4. One may proceed as follows: first, form the set C of all Cauchy sequences in V , this
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is in fact a vector subspace of the cartesian product V N . Next, extend the norm
· : V → [0, ∞) to a “semi-norm” |·| : C → [0, ∞) by putting |(vn )n∈N | := limn∈N vn . This semi-norm has all the properties of a norm, except that |s| = 0 may also occur for a sequence s ∈ C that is not zero. Now all that remains is to observe that the quotient space C/N by the subspace N := {s ∈ C | |s| = 0} is a complete normed vector space, with norm N + s := |s|. The original space V is embedded into C (and thus into C/N ) as the space of constant sequences. Then every element of the completion may be obtained as the limit of a suitable sequence in V . If the norm · on V stems from a scalar product, the extension of the norm can also be described by an extension of this scalar product. 14.8 Definition. Let G be a locally compact Hausdorff group, and let λ be a complex Haar integral. Then (ψ|ϕ) := (ψϕ)λ defines a scalar product (·|·) : CcC (G) × CcC (G) → C. The completion of the pre-Hilbert space (CcC (G) , (·|·)) will be denoted by L2 (G). As every element ξ ∈ L2 (G) is the limit of some suitable sequence (ϕn )n∈N of elements of CcC (G), we could try to extend the Haar integral to a linear form ˆ λˆ : L2 (G) → C by putting ξ λ = limn∈N ϕnλ . However, this limit need not exist if G is not compact. (If G is compact then one can show that this limit actually exists. We leave this as an exercise.) 14.9 Lemma. Let (V , (·|·)) be a pre-Hilbert space over F ∈ {R, C}, and assume that S is a vector subspace of V . Then the following hold. (a) The orthogonal space S ⊥ := {y ∈ V | ∀x ∈ S : (x|y) = 0} is a vector subspace of V . (b) The space S ⊥ is closed in the metric space (V , d). (c) We have S ≤ (S ⊥ )⊥ , and S ∩ S ⊥ = {0}. (d) If S is complete then V = S ⊕ S ⊥ . Proof. Assertions (a), (b), (c) are left as exercises, see Exercise 14.12. Assume that S is complete, and consider an arbitrary element x ∈ V . We have to find s ∈ S and t ∈ S ⊥ such that x = s + t. The idea is to search for “the best approximation” s to x in S. Put m := inf {d(x, y) | y ∈ S}, and pick a sequence (yn )n∈N with limn∈N d(x, yn ) = m. For u := x − yj and v := x − yk we use the formula (u + v|u + v) + (u − v|u − v) = 2(u|u) + 2(v|v)
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to obtain 7 72
yk − yj 2 = 2 x − yj 2 + 2 x − yk 2 − 4 7x − 21 (yj + yk )7 . Since 21 (yj + yk ) belongs to S, the right hand side is less than or equal to
2 x − yj 2 + x − yk 2 − 2m2 . This last expression converges to 0 as j and k grow large, and we see that the sequence (yn )n∈N is a Cauchy sequence. Therefore, there exists s = limn∈N yn in the complete space S. Note that d(x, s) = m is minimal among all distances from x to elements of S. We claim that t := x − s belongs to S ⊥ . Assume, to the contrary, that there exists y ∈ S such that (t|y) = 0. Without loss, we may also assume y ≤ 1. Computing
t 2 = d(x, s)2 ≤ d(x, s + (t|y)y)2 = x − s − (t|y)y 2 = t − (t|y)y 2 = t 2 − (t|y)(y|t) − (t|y)(t|y) + (t|y) y 2 (y|t) we obtain 2(t|y)(y|t) ≤ y 2 (t|y)(y|t), a contradiction.
2
We will use the following result, due to Fréchet and Riesz. 14.10 Lemma. Let (H, (·|·)) be a Hilbert space over F ∈ {R, C}. Then, for each x ∈ H , the function x mapping h ∈ H to (h|x) is a continuous linear form, and the resulting mapping from H to the space H of all continuous linear forms is a semi-linear bijection; that is, a bijection satisfying (x + y) = x + y and (cx) = c(x ) for all x, y ∈ H and all c ∈ F. Proof. It follows easily from the properties of the scalar product that is a semilinear map. Continuity of x can be seen using Schwarz’s inequality. The details will be treated by exercises, see Exercise 14.10. From x = y we infer (h|x) = (h|y) for all h ∈ H , and putting h = x − y we infer (x − y|x − y) = 0. But this means x − y = 0, and we have shown that is injective. In order to show that is also surjective, we have to work a little harder. Let λ : H → F be a continuous linear form. Clearly, we have 0 = 0 , so assume λ = 0. Then the kernel K of λ is a proper closed vector subspace. According to 14.9, we have H = K ⊕ K ⊥ . Moreover, we find v ∈ H such that K ⊥ = Fv. For h ∈ H we write h = k + f v with k ∈ K and f ∈ F, and compute hλ = f · v λ . On the other vλ hand, we have (h|v) = (f v|v) = f · (v|v), and infer λ = (v|v) v ∈ Fv . As is semi-linear, this means that λ belongs to the image of . 2
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The bijection will also be used, as usual, to transfer the scalar product (and thus the metric, and the topology) from H to H . Moreover, it may be used to identify the semigroups End(H ) and End(H ) consisting of all continuous linear maps from H to H , or from H to H , respectively. 14.11 Corollary. Mapping ϕ ∈ End(H ) to −1 ϕ is an isomorphism of semigroups from End(H ) onto End(H ). 2 The following characterization is a special case of the remarkable result proved in 26.40. See 25.5 and 25.6 for a different approach to this characterization. 14.12 Lemma. Let V be a topological vector space over F ∈ {R, C}. If V is locally compact Hausdorff, then its dimension is finite. 2 In 14.20 below, we will only need a special case of 14.12, namely the fact that every locally compact Hilbert space over R has finite dimension. This case will be treated in Exercise 14.14, while the full assertion will be proved in 26.40. An alternative approach (for real vector spaces) may be found in 25.5 and 25.6 (using Pontryagin duality and thus 14.20 via 14.31, 14.34, 19.8, 21.16, 22.5). 14.13 Definition. A subset A of a real vector space V is called convex if x, y ∈ A implies that tx + (1 − t)y belongs to A whenever t ∈ [0, 1]. It is easy to see that arbitrary intersections of convex sets are convex, again. This means that for every nonempty subset X ⊆ V there is a smallest convex subset conv(X) containing X, called the convex hull of X in V . In normed vector spaces, closed convex sets have a nice geometric characterization in terms of “half spaces”, as follows. The norm plays its role in the proof via the fact that it yields a neighborhood basis consisting of convex sets. A proof can be found in [56], II 9.2. 14.14 The Hahn–Banach–Mazur Theorem. Let V be a normed vector space over R. Then each closed convex subset A of V satisfies 8 ← A= 2 (−∞, r]ϕ | r ∈ R, ϕ ∈ V , Aϕ ⊆ (−∞, r] . The following reasoning allows to interpret the Hahn–Banach–Mazur Theorem for vector spaces over C. 14.15 Lemma. Let V be a topological vector space over C. If λ : V → C is a Clinear form then Re λ := 21 (λ + λ¯ ) is an R-linear form. Conversely, let μ : V → R μ := v μ − i(iv)μ . Then be an R-linear form, and define μ : V → C by v μ is C-linear, we have μ = Re μ, and μ is continuous exactly if μ is. Proof. We leave the details (which involve nothing but straightforward calculations) for an exercise. 2
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14.16 Lemma. Let (V , · ) be a Banach space, and let C ⊆ V be a compact set. Then the closure conv(C) of the convex hull of C is compact. Proof. According to 1.31 and 1.32, it suffices to show that conv(C) is pre-compact. For ε > 0, we put U := v ∈ V | v < 2ε . Because C is compact, we find a finite subset F of C such that C ⊆ F + U . We claim that S := conv(F ) is compact. In fact, the set S is the continuous image of the convex hull T of the natural basis of RF under the linear map from RF to V obtained by extending the inclusion of F . Now the assertion follows from the fact that T is compact. Next, we pick a finite subset E of S such that S ⊆ E + U . An easy computation (using that both S and U are convex) yields that S + U is convex. Therefore, we have conv(C) ⊆ conv(F + U ) ⊆ conv(S + U ) = S + U ⊆ E + U + U , and the proof is complete. 2 14.17 Definition. Let V be a normed vector space. A linear map ϕ : V → V is called a compact operator if B ϕ is pre-compact (in the sense of 1.30) whenever B is a bounded subset of V . One can show that every continuous linear map with finite-dimensional image is a compact operator. Recall from 1.31 that the closure of a pre-compact set in a metric space is pre-compact again, and from 1.32 that closed pre-compact sets in complete metric spaces are in fact compact. Thus, in a Banach space, the closure of the image of a bounded set under a compact operator is compact. 14.18 Definition. Let (V , (·|·)) be a pre-Hilbert space. A linear map ϕ : V → V is called a positive operator if (x ϕ |y) = (x|y ϕ ) and (x ϕ |x) ≥ 0 hold for all x, y ∈ V . 14.19 Remark. It is easy to see that all the eigenvalues of a positive operator are all real, and nonnegative (whence the name). 14.20 Lemma. Let (H, (·|·)) be a Hilbert space, and let ϕ : H → H be a compact positive operator. Then there exists a maximal eigenvalue μ of ϕ. If ϕ = 0 then the corresponding eigenspace Eμ := {x ∈ H | x ϕ = μ · x} has finite dimension. Proof. Without loss of generality, we may assume ϕ = 0. Then s := sup x ϕ | x ∈ H, x ≤ 1 is a positive real number. We are going to show that s is an eigenvalue of ϕ; of course, this implies that s is the largest of all eigenvalues of ϕ. Using Re(x ϕ |y) | x , y ≤ 1 ⊇ Re x ϕ | x1ϕ x ϕ 0 = x ≤ 1
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and Re(x ϕ |y) ≤ |(x ϕ |y)| ≤ x ϕ · y we get s = sup {Re(x ϕ |y) | x , y ≤ 1}. Now
0 ≤ (x − y)ϕ |x − y = (x ϕ |x) − 2 Re(x ϕ |y) + (y ϕ |y) leads to Re(x ϕ |y) ≤
1 ϕ (x |x) + (y ϕ |y) ≤ max (x ϕ |x), (y ϕ |y) 2
and s = sup {(x ϕ |x) | x ≤ 1}. Therefore, we find a sequence (xn )n∈N in H such ϕ that s − n1 < (xn |xn ) ≤ s and 21 < xn ≤ 1. Since ϕ is a compact operator, we find ϕ an infinite subset J ⊆ N such that the sequence (xn )n∈J converges to some z ∈ H . For every n ∈ J , we have 0 ≤ xnϕ − sxn 2 = (xnϕ − sxn |xnϕ − sxn ) = (xnϕ |xnϕ ) − 2(xnϕ |sxn ) + (sxn |sxn ) 2s 2s 2 2 ≤ s − 2s − + s2 = . n n ϕ
Thus the sequence (sxn )n∈J converges to z, as well. This means zϕ = limn∈J sxn = sz, and we obtain z ∈ Es . It remains to show that E1 has finite dimension. To this end, we remark that the image B1 (0)ϕ contains the intersection B1 (0) ∩ E1 , and compactness of B1 (0)ϕ yields that E1 has finite dimension, compare 14.12. 2
Hilbert Modules for Locally Compact Groups 14.21 Definition. Let (V , d) be a metric space which, at the same time, is a vector space over F (where F ∈ {R, C}). Let ω : V × G → V be an action of the group G on V , and consider the corresponding homomorphism δ = δ ω : G → Sym(V ), as in 10.3. We say that ω is an action by linear transformations, or that δ is a linear δ δ representation, if g δ is a linear map, for each g ∈ G. If d(x g , y g ) = d(x, y) for all x, y ∈ V and all g ∈ G, we say that ω is an action by linear isometries, and δ is called an isometric linear representation. If the metric d is given via a scalar product via d(x, y) = (x − y|x − y), actions by linear isometries are also called actions by unitary transformations, and the corresponding representations are called unitary representations. It will also be convenient to interpret a Hilbert space together with an action of a group by linear isometries as a single algebraic object. 14.22 Definition. Let G be a group. By a G-module, we mean a vector space H over F ∈ {R, C}, together with a “multiplication by elements of G”; that is, an
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action ω : H × G → H of G on H by linear transformations. (The name module is due to the fact that this multiplication can be extended to a multiplication by elements of the “group ring” of G over F.) If (H, (·|·)) is a Hilbert space and G acts by unitary transformations, we speak of a Hilbert G-module. If G is a topological group, and ω is continuous, we speak of a topological G-module, or a topological Hilbert G-module, respectively. A submodule of a G-module H is a vector subspace S of H such that (S × G)ω = S. A (Hilbert) G-module is called faithful if the corresponding (unitary) representation is injective; in other words: for every g ∈ G {1} there exists some h ∈ H such that (h, g)ω = h. It should be clear that there is no actual difference between actions by unitary transformations, unitary representations, and Hilbert modules. The change in notation just indicates a change in the point of view that we adopt. We have to be a little bit more careful if topology enters the stage. However, applying 10.4 we obtain the following. 14.23 Lemma. Let G be a topological group, let V be a topological vector space, and let ω : V × G → V be an action by linear transformations. If ω is continuous then the corresponding linear representation δ ω : G → L is continuous, where L is the group of all linear homeomorphisms from V onto itself, equipped with the compact-open topology. 2 Note, however, that multiplication in the group L of 14.23 need not be continuous with respect to the compact-open topology. If it is, then the modified compact-open topology turns L into a topological group, and δ ω is also continuous with respect to this topology; compare 10.4. In particular, this happens if V is a locally compact vector space; which means that V has finite dimension over R (compare 14.12). 14.24 Lemma. Let V be a normed vector space, and let V be the completion of V . (a) If ϕ : V → V is a linear isometry then there exists an isometry ϕ : V → V that extends ϕ. This isometry is uniquely determined. (b) Let ω : V × G → V be an action by linear isometries, and consider the corresponding isometric linear representation δ = δ ω . Mapping g ∈ G to g δ := g δ , we obtain an isometric linear representation δ of G on V , and a corresponding action by linear isometries ω = ωδ . Proof. As every element x ∈ V is the limit x = limn∈N vn of a suitable sequence of ϕ vectors in V , it is clear how ϕ has to be defined: put x ϕ := limn∈N vn . The details are left as an exercise. In order to verify assertion (b), we observe that g δ hδ = (gh)δ implies g δ hδ = (gh)δ , since x δ is determined uniquely by its restriction x δ . 2
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14.25 Lemma. Let (V , d) be a normed vector space, and let G be a topological group. Assume that ω : V × G → V is an action by linear isometries. Then ω is continuous exactly if, for each x ∈ V , the map ωx : G → V defined by g ωx = (x, g)ω is continuous at 1. In this case, the extension ω : V × G → V of the action according to 14.24 is continuous, as well. Proof. If ω is continuous then ωx , as a restriction of ω, is continuous. Conversely, we are going to show that ω is continuous at (x, g) if ωx is continuous at 1. For ε > 0, pick a neighborhood W of 1 in G such that w ∈ W implies 2ε > d(x, wωx ) = d (x, (x, w)ω ). Then d(x, y) < 2ε and hg −1 ∈ W entail
d (x, g)ω , (y, h)ω ≤ d (x, g)ω , (x, h)ω + d (x, h)ω , (y, h)ω
≤ d x, (x, hg −1 )ω + d(x, y) < ε, and we see that ω is continuous. In order to see that ωz is continuous at 1, if z is an arbitrary element of V , we pick v ∈ V such that d(v, z) < 3ε , and a neighborhood U of 1 in G such that u ∈ U implies d (v, (v, u)ω ) < 3ε . Then we obtain
d z, (z, u)ω ≤ d(z, v) + d v, (v, u)ω + d (v, u)ω , (z, u)ω
= d(z, v) + d v, (v, u)ω + d(v, z) < ε, and the proof is complete.
2
14.26 Theorem. Let G be a locally compact Hausdorff group, choose a complex Haar integral λ : CcC (G) → C, and define the scalar product (ψ|ϕ) = (ψϕ)λ . Then the action ω : CcC (G) × G → CcC (G) given by (ϕ, g)ω = ϕg is continuous. Proof. According to 14.25, it suffices to show that ωϕ is continuous at 1, for every ϕ ∈ CcC (G). Let ε > 0. Using 12.5, we pick a neighborhood U of 1 in G such that, for each u ∈ U , we have su := supx∈G |x ϕ − x ϕu | < ε. We fix ϕ ∈ CcC (G), and make sure that U is contained in some fixed compact neighborhood V of 1 in G. Then the set W := SV is compact, where S := supp ϕ. According to 1.25, we find a continuous function ξ : G → [0, 1] with compact support, mapping every element of W to 1. For each u ∈ U , we have supp ϕu ⊆ W , and the function ψ defined by x ψ = |x ϕ − x ϕu |2 also satisfies supp ψ ⊆ W . Therefore, we obtain 2 d(ϕ, ϕu )2 = ψ λ ≤ su2 · ξ λ < ε2 · ξ λ . Returning to the completion L2 (G) of CcC (G), we obtain the following. 14.27 Corollary. Let G be a locally compact Hausdorff group. Then the continuous action ω : CcC (G) × G → CcC (G) via (ϕ, g)ω = ϕg extends to a continuous action ω : L2 (G) × G → L2 (G) by unitary transformations; and δ ω : G → U(L2 (G)) is a continuous unitary representation, with respect to the compact-open topology on U(L2 (G)). 2
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In fact, the unitary representation of G on L2 (G) is injective: for every g ∈ G {1}, we find by 1.25 a function ϕ ∈ Cc (G) such that g ϕ = 1ϕ , and obtain ϕg = ϕ. We rephrase this in terms of Hilbert modules: 14.28 Corollary. For every locally compact Hausdorff group, there exists at least one faithful topological Hilbert module, namely L2 (G). We remark that L2 (G) has infinite dimension, whenever G is infinite. For a locally compact Hausdorff group there are, in general, no faithful topological Hilbert modules of finite dimension.
Hilbert Modules for Compact Groups We are now going to harvest a series of rather impressive results about compact Hausdorff groups. In particular, we are going to show that for such a group, there are “sufficiently many” finite-dimensional topological Hilbert modules: although it may well happen that none of these is faithful, the intersection over all the kernels of the corresponding representations is trivial. This result is the key to many results about the structure of compact groups, since it opens possibilities to apply the theory of compact Lie groups. For a compact group, the constant maps have compact support, of course. Every compact Hausdorff group K has, therefore, a somewhat special Haar integral, 5 namely, the Haar integral that satisfies K 1 dk = 1. Throughout this section, we will use this Haar integral, and denote it by λ. Whenever we consider a Hilbert K-module in this section, the corresponding action ω will be written as x.k := (x, k)ω . 14.29 Lemma. Let K be a compact Hausdorff group, and let (H, (·|·)) be a topological Hilbert K-module. We consider a continuous linear map : H → H . (a) For x, y ∈ H , define a function ϕx,y : K → C by k ϕx,y = (x.k|(y.k) ). Then ϕx,y belongs to CcC (K) = C(K, C). 5 (b) For each y ∈ H , mapping x to K (x.k|(y.k) ) dk = (ϕx,y )λ gives a continuous linear form from H to C. (c) There is exactly one continuous linear map : H → H with the property that 5 (x|y ) = K (x.k|(y.k) ) dk holds for all x, y ∈ H . (d) If
is a compact operator then is a compact operator, as well.
Proof. Assertions (a) and (b) are easy, and left as exercises. Corollary 14.11 gives assertion (c).
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Now assume that is a compact operator. We are going to show that the image B of the closed unit ball B := {x ∈ H | x ≤ 1} has compact closure. To this end, we first note that B is invariant under the action of K. As the operator is compact, we have that A := B is compact, and so is A.K := {a.k | a ∈ A, k ∈ K}; in fact, the set A.K is the image of the compact set A × K under the continuous action ω. According to 14.16, the closure C of the convex hull of A.K is also compact. Fix a real number r, and consider an element y ∈ H such that Re(y|x) ≤ r holds for each x ∈ C. Then b ∈ B implies (b.k) .k −1 ∈ C, and we obtain 6 6 6 Re( y.k|(b.k) ) dk = Re( y|(b.k) .k −1 ) dk ≤ r dk = r. Re( y|b ) = K
K
K
ϕ←
| r ∈ R, ϕ ∈ H , C ϕ ⊆ (−∞, r]}. According to This shows b ∈ {(−∞, r] the Hahn–Banach–Mazur Theorem 14.14 in its complex interpretation 14.15, this intersection equals C, and we have established assertion (d). 2 14.30 Lemma. Let K be a compact Hausdorff group, and let (H, (·|·)) be a topological Hilbert K-module. Pick x ∈ H {0}, and define a mapping : H → H (h|x) by h = (x|x) x. Then the following hold. (a) The map
is a compact positive operator.
(b) The operator constructed according to 14.29 is also a compact positive operator, and = 0.
(c) For each k ∈ K and each h ∈ H , we have h .k = (h.k) . Proof. The operator is a compact operator since it is continuous, and has finitedimensional image (in fact, H = Cx). We leave it as an exercise to verify that is positive. We have seen in 14.29 that is a compact operator. The construction of immediately yields (x|x ) = 0. For u, v ∈ H , we com5 5 5 pute (u|v ) = K (u.k|(v.k) )dk = K ((u.k) |v.k)dk = K (v.k|(u.k) )dk = 5 (v|u ) = (u |v). The observation (u|u ) = K (u.k|(u.k) )dk ≥ 0 completes the proof of the fact that is positive. Assertion (c) follows from the equality (u|h .k) = (u.k −1 |h ) = (ϕu.k −1 ,h )λ = (ϕu,h.k )λ = (u|(h.k) ). 2 Using the knowledge about compact positive operators collected in 14.20, we infer: 14.31 Corollary. In the situation of 14.30, there is a maximal eigenvalue μ of , the corresponding eigenspace Eμ has finite dimension, and Eμ is invariant under
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the action of K. In other words: every topological Hilbert K-module of a compact Hausdorff group contains a nontrivial submodule of finite dimension. 2 14.32 Theorem. Let K be a compact Hausdorff group, and let H be a topological Hilbert K-module. Consider the set F of all nontrivial K-submodules of finite dimension. Then H is the closure of the subspace F generated by F . ⊥ is a closed subspace of H , and it is Proof. The orthogonal space N := F invariant under the action of K. This means that N is a Hilbert K-module. If N were not trivial, it would contain a nontrivial K-submodule
⊥ F of finite dimension by 14.31, and F ∈ F would imply F ≤ F ∩ F , a contradiction. This 2 shows N = {0}, and F = H follows. 14.33 Peter–Weyl Theorem. If K is a compact Hausdorff group then for every k ∈ K {1} there exist a natural number n and a continuous homomorphism ϕ : K → U(n, C) such that k ϕ = 1. Proof. Let k ∈ K {1}. We know that L2 (K) is a faithful Hilbert K-module. From 14.32 we see that there exists at least one submodule F of finite dimension such that k acts nontrivially on F . The restriction of the action of K to F yields a unitary representation which is the homomorphism we are looking for. 2 If we consider an Abelian group in 14.33, we observe that the submodule F splits as a sum of one-dimensional submodules. This means that we can take the natural number n to be 1. Observing that the groups U(1, C) and R/Z are isomorphic, we obtain the following corollary, which will be crucial in our proof of Pontryagin duality. 14.34 Corollary. Let A be an Abelian compact Hausdorff group, written additively. Then for each a ∈ A {0} there exists a continuous homomorphism ϕ : A → R/Z 2 such that a ϕ = 0. 14.35 Weyl’s Trick. Let K be a compact Hausdorff group, let (H, (·|·)) be a Hilbert space, and assume that H is a topological K-module. Then the scalar product (·|·) on H may be replaced by a scalar product (·|· ) that induces an equivalent norm on H , such that (H, (·|· )) becomes a topological Hilbert K-module. More explicitly, assume H = {0}, put M := sup {(v.k|v.k) | k ∈ K, (v|v) ≤ 1} and 6 (v|w ) :=
(v.k|w.k)dk. K
Then M > 0, and M −1 (v|v) ≤ (v|v ) ≤ M(v|v) holds for each v ∈ H .
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Proof. For each pair (v, w) ∈ H 2 , the function bv,w : K → F : k → (v.k|w.k) is continuous, and the integral defining (v|w ) is defined. Since the Haar integral is positive and the scalar product (·|·) is positive definite, the form (·|· ) is positive definite, as well. Together with linearity of the integral, the observations bcu+v,w = cbu,w + bv,w and bw,v = bv,w now yield that (·|· ) is a scalar product on H . We have to show that the set B := {(v.k|v.k) | k ∈ K, (v|v) ≤ 1} is bounded. We apply 10.7 and 10.8, and see that there is a neighborhood V of 0 in H such that V .K ⊆ U := {v ∈ H | (v|v) ≤ 1}. Choosing ε > 0 so small that εU = {v ∈ H | (v|v) ≤ ε2 } is contained in V , we find U.K ⊆ ε −1 V .K ⊆ ε −1 U = {v ∈ H | (v|v) ≤ ε−2 }, and see that B is bounded above by ε −2 . This shows that the supremum M exists, while any v ∈ H with (v|v) √ = 1 yields M ≥ 1. Now consider v ∈ H {0}, and put c := (v|v). Then w := 1c v satisfies (w|w) = 1, and (w.k|w.k) ≤ M holds for each k ∈ K. This implies (v.k|v.k) = (cw.k|cw.k) = c2 (w.k|w.k) ≤ (v|v)M. Replacing k by k −1 and v by v.k, we obtain (v|v) ≤ (v.k|v.k)M, for each k ∈ K. Positivity of5the Haar integral together with the fact that we have chosen the integral such that K dk = 1 now yields the inequalities 6 6 (v|v) = (v|v)dk ≤ (v.k|v.k)dk = (v|v ) M K
K
and 6 (v|v ) =
6 (v.k|v.k)dk ≤
K
(v|v)Mdk = (v|v)M. K
This gives the equivalence of the scalar products (·|·) and (·|· ), and the equivalence of the corresponding norms. It remains to prove that 5 K acts by unitary5 transformations: that is, we have to verify (v.h|w.h ) = K (v.hk|w.hk)dk = K (v.k|w.k)dk = (v|w ), for each choice of v, w ∈ H and h ∈ K. We use the invariance of the Haar integral here. 2 14.36 Remark. In particular, Weyl’s trick 14.35 shows that every finite-dimensional (continuous) linear representation K → GL(n, F) actually is a unitary representation.
Exercises for Section 14 Exercise 14.1. Show that there is a direct sum decomposition CcC (G) = Cc (G) ⊕ iCc (G), whenever X is a topological space. Hint. Use that, for every function ϕ ∈ CcC (X), the function ϕ mapping x to the complex conjugate x ϕ belongs to CcC (X), again.
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Exercise 14.2. Prove Lemma 14.1. Exercise 14.3. Let G be a locally compact Hausdorff group. Show that every complex Haar integral λ satisfies ϕ λ = ϕ λ , for each ϕ ∈ CcC (G). Exercise 14.4. Prove 14.6, and 14.15. Verify the details of the procedure described in 14.7, and interpret the construction in the light of 6.6. Exercise 14.5. Show that every compact operator is continuous. Exercise 14.6. Let H be a pre-Hilbert space, and let ϕ : H → H be a continuous linear map such that H ϕ has finite dimension. Prove that ϕ is a compact operator. Exercise 14.7. Show that every eigenvalue of a positive operator is a nonnegative real number. Exercise 14.8. Let (V , · ) be a normed vector space over R, and let ε > 0. Show that the ball Bε (v) as well as its closure are convex, for each v ∈ V . Exercise 14.9. Prove that, in every vector space over R, the sum of two convex sets is convex. Exercise 14.10. Prove Schwarz’s inequality: in every pre-Hilbert space (V , (·|·)) over F ∈ {R, C}, we have |(x|y)| ≤ x · y , for all x, y ∈ V . Conclude that (·|·) : V × V → F is continuous. Hint. Consider x + cy , for c ∈ F. Exercise 14.11. Show that every action ω : H × G → H by unitary transformations on a Hilbert space H is completely reducible; that is, for every closed vector subspace S ≤ H with (S × G)ω = S there is a closed vector subspace T such that H = S ⊕ T and (T × G)ω = T . Exercise 14.12. Prove 14.9. Exercise 14.13. Let H be a normed vector space over R. Show that every finite-dimensional subspace of H is closed. Hint. Use the fact that any two norms on Rn are equivalent. Then use 4.7. Exercise 14.14. Show that every locally compact Hilbert space H over F ∈ {R, C} has finite dimension. (See 26.40 for a stronger result.) Hint. Consider finite orthonormal systems (v1 , . . . , vn ) in H . If there is a maximal one among these, then this forms a basis for H . If none of the orthonormal systems is maximal, construct a convergent orthonormal sequence (wn )n∈N (using the fact that the set of vectors of norm 1 is compact), and derive a contradiction.
Chapter E
Categories of Topological Groups 15 Categories 15.1 Definition. A category C consists of a class ob C of “objects” (often simply denoted by C), sets Mor(X, Y ) = Mor C (X, Y ) of “morphisms” for each pair (X, Y ) ∈ C × C and a law of composition of morphisms; that is, for all X, Y, Z ∈ C we have a map from Mor(X, Y ) × Mor(Y, Z) to Mor(X, Z) mapping each pair (ξ, η) ∈ Mor(X, Y ) × Mor(Y, Z) to its composite ξ η ∈ Mor(X, Z). Moreover, one requires the following. (a) For each X ∈ C there is an identity of X; that is, a morphism idX in Mor(X, X) such that for each Y ∈ C and all morphisms ξ ∈ Mor(X, Y ) and ζ ∈ Mor(Y, X) the equalities idX ξ = ξ and ζ idX = ζ hold. (b) For all W, X, Y, Z ∈ C and all morphisms ω ∈ Mor(W, X), ξ ∈ Mor(X, Y ), η ∈ Mor(Y, Z) the equality (ωξ )η = ω(ξ η) holds. The union X,Y ∈C Mor(X, Y ) is denoted by Mor C. A morphism ϕ ∈ Mor(X, Y ) will often be indicated as ϕ : X → Y . A subcategory S of a category C is given by a subclass ob S of ob C and subsets Mor S (X, Y ) of Mor(X, Y ) for all X, Y ∈ ob S such that Mor S is closed with respect to composition, and Mor S (X, X) contains idX for each X ∈ ob S. A subcategory S is called full in C if Mor S (X, Y ) = Mor C (X, Y ) for all X, Y ∈ ob S. A covariant functor from a category C to a category D is a map F : ob C → ob D of objects together with maps FXY : Mor C (X, Y ) → Mor D (XF , Y F ) for all X, Y ∈ ob C that preserve composites of morphisms. Moreover, we require that FXX maps idX to idXF . A contravariant functor from C to D is a map F : ob C → ob D together with maps FXY : Mor C (X, Y ) → Mor D (Y F , XF ) for all X, Y ∈ ob C that reverse composites of morphisms; that is, for ϕ ∈ Mor C (X, Y ) and ψ ∈ Mor C (Y, Z) we have (ϕψ)FXZ = ψ FY Z ϕ FXY . Moreover, we require that FXX maps idX to idXF . Mostly, we will denote the maps FXY just by F . 15.2 Examples. (a) The category Set of sets and maps; that is, ob Set is the class of all sets, and Mor Set (X, Y ) is the set of all maps from X to Y . (b) The category Top of topological spaces and continuous maps.
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(c) The category TG of topological groups and continuous homomorphisms. The following full subcategories of TG are also of interest: TA Abelian topological groups, HG Hausdorff groups, HA Abelian Hausdorff groups, DG discrete groups, DA discrete Abelian groups, LCG locally compact Hausdorff groups, LCA locally compact Abelian Hausdorff groups, CG compact Hausdorff groups, CA compact Abelian Hausdorff groups.
(d) For the construction of examples of topological groups, the category TR of topological rings and continuous homomorphisms is useful. There are the full subcategories DR, HR and CR of discrete rings, of Hausdorff rings and of compact Hausdorff rings, respectively. 15.3 Definitions. Let C be any category, and let ϕ : A → B be a morphism in C. (a) The morphism ϕ is called an isomorphism if there exists a morphism ψ from B to A such that ϕψ = idA and ψϕ = idB . In this case, ψ is uniquely determined by ϕ, and we write ψ = ϕ −1 . (b) If A = B then ϕ is called an endomorphism. (c) A morphism which is at the same time an isomorphism and an endomorphism is called an automorphism.
Monics and Epics 15.4 Definition. Let C be a category. A morphism ϕ : B → C in C is called monic if for each object A and each pair of morphisms α : A → B and β : A → B in C the equality αϕ = βϕ implies α = β. Dually, a morphism ϕ : B → C is called epic if for each object D and each pair of morphisms γ : C → D and δ : C → D in C the equality ϕγ = ϕδ implies γ = δ. Note that every isomorphism is both monic and epic. Conversely, a morphism may be monic and epic although it is not an isomorphism. For instance, consider
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the category Top, where a continuous bijection surely is monic and epic, but need not be a homeomorphism. Let C be a subcategory of TG such that for each morphism ϕ : G → H in C the kernel ker ϕ is an object of C and the inclusion ε : ker ϕ → G is a morphism in C. Moreover, assume that for any pair (A, B) of objects in C the constant morphism νAB : A → B (mapping everything to the neutral element) belongs to C. If ϕ : G → H is a monic in C with kernel K, we infer from εϕ = νKG ϕ that ε = νKG ; that is, every monic in C is injective. Conversely, every injective morphism is monic, even in the category of sets and maps. The situation just discussed includes the cases where C is one of the categories introduced in 15.2 (c). For the sake of easy reference, we formulate this explicitly. 15.5 Lemma. Let C be one of the categories TG, TA, HG, HA, LCG, LCA, CG, CA, DG, or DA. Then the monics in C are exactly the injective morphisms. Turning to epics, the picture becomes much more involved. 15.6 Lemma. Let G be a topological space, and let H be a Hausdorff topological space. If ϕ : F → G is a map such that F ϕ = G and α, β are continuous maps from G to H such that ϕα = ϕβ then α = β. Proof. The map ψ : G → H × H defined by g ψ = (g α , g β ) is continuous. Thus G = F ϕ implies Gψ ⊆ F ϕψ ⊆ {(h, h) | h ∈ H }. The diagonal {(h, h) | h ∈ H } is closed in H × H since H is Hausdorff. 2 15.7 Corollary. Let C be a subcategory of the category of Hausdorff topological spaces, with continuous maps as morphisms. Then every morphism with dense image is an epic in C. In particular, 15.7 applies to every subcategory of HG. There remains the question whether the morphisms with dense image are the only epics in these categories. In general, this is false. In fact, this question presents a rather hard problem, known as the epimorphism problem for Hausdorff groups. It has been proved that in LCG, CG, LCA, and CA the epics are exactly those morphisms with dense image. In HG, however, there are epics whose image is not dense, see [63]. 15.8 Theorem. Let C be one of the categories HA, CA, or LCA. Then the epics in C are exactly the morphisms whose image is dense. Proof. In view of 15.7, it only remains to show that a morphism in C whose image is not dense is not epic. So assume that ϕ : A → B is a morphism in C, and that the closure C of Aϕ is different from B. Then B/C is an object in C. The natural map πC : B → B/C and the map ζ : B → B/C mapping everything to 0 are morphisms in C such that ϕπC = ϕζ but πC = ζ . 2
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15.9 Notation. When working in a categorical setting, it is an advantage to express assertions and conclusions using morphisms rather than single elements of the objects involved. It is often very helpful to illustrate the situation at hand by socalled commutative diagrams. A diagram exhibits a collection of (names of) objects of some category C and arrows between some of these objects. An arrow pointing from an object A to an object B corresponds to a morphism in Mor C (A, B). We say that the diagram commutes (or that it is commutative) if the following condition is satisfied: whatever way one chooses from one object in the diagram to another (always going along arrows, in the right direction), the corresponding composites of morphisms turn out the same. We have already encountered several commutative diagrams, in particular when formulating the universal properties of products and quotients (a very typical categorical theme). Occasionally, we find it convenient to indicate in these diagrams the information that a given morphism has some special property, as follows. / to indicate monics, one of type / / to We will use an arrow of type / indicate epics. In any subcategory of the category of topological groups, arrows / denote open maps, for embeddings we use / and arrows of type ◦ ◦ / of type for open embeddings. Finally, quotient morphisms (which, in the ,2 . usual subcategories of TG, are just open surjections) are indicated by If the existence of a morphism is not clear from the start, an arrow with dotted / may be employed. These special conventions are meant as additional shaft help along with the diagrams. As a rule, we will also explicitly state the information encoded in this way.
Exercises for Section 15 Exercise 15.1. Verify that the examples given in 15.2 are indeed categories. Exercise 15.2. Let E : TG → Top and F : TG → Set be defined by (G, μ, ι, ν, T )E = (G, T ) and (G, μ, ι, ν, T )F = G. Show that E and F yield covariant functors. Exercise 15.3. Fix an object F in the category TG. For objects G, H and morphisms ϕ : G → H in TG put GM = Mor(F, G) and GL = Mor(G, F ), define ϕ M : GM → H M L M by α ϕ = αϕ, and define ϕ L : H M → GM by β ϕ = ϕβ. Show that this gives a covariant functor M : TG → Set and a contravariant functor L : TG → Set. Exercise 15.4. Show that, in any category, each isomorphism is monic and epic. Exercise 15.5. Show that monics and epics in Set are exactly the injective and surjective maps, respectively. Exercise 15.6. Show that in Set and in DG a morphism is an isomorphism exactly if it is both monic and epic.
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Exercise 15.7. Find examples of morphisms in Top and in TG that are monic and epic but not isomorphisms. Exercise 15.8. Show that the inclusion ε : Z → Q is both monic and epic in DR, but not an isomorphism. Exercise 15.9. Show that in the full subcategory TA of all commutative groups in TG the epics are exactly the surjective morphisms.
16 Products in Categories of Topological Groups The notion of a product of a family of topological groups is frequently used. As a categorical notion, it depends on the category under consideration. In the present section, we discuss some of the most important categories of topological groups. The general frame work is set by the categories TG and TR of topological groups and topological rings, respectively, with continuous homomorphisms. 16.1 Definition. Let C be any category, and let (Xj )j ∈J be a family of objects Xj ∈ C. An object P ∈ C together with a family (πj )j ∈J of morphisms πj : P → Xj is called a product of the family (Xj )j ∈J (in the category C), if it has the following universal property: (P)
For every object W and every family (ψj )j ∈J of morphisms ψj : W → Xj : W → P such that ψπ j = ψj for each there is a unique morphism ψ j ∈ J. P W @VPPVPVVV @@ PPPVVVVV P V πi@ πj PP Vπk VVVV @@ PPP VVVV PP' VV* X X Xk · · · j i ψ O > ll6 l l ||| l ψ ψi ψ l | j lll k |l|llll W
The morphisms πj are called canonical projections. In many situations where products appear it is very obvious from the description of P how the canonical projections are defined. In such cases, we will sometimes prefer the inaccuracy of suppressing them to an explicit but messy statement. 16.2 Lemma. The product of a family (Xj )j ∈J in a given category is determined up to isomorphism; that is, if (P , (πj )j ∈J ) and (Q, (κj )j ∈J ) are both products of the family, then there are morphisms α : P → Q and β : Q → P such that αβ = idP and βα = idQ . Moreover, we have ακj = πj and βπj = κj for each j ∈ J .
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Proof. We simply apply the universal property of the product (P , (πj )j ∈J ) to define β := κ and the universal property of the product (Q, (κj )j ∈J ) to define α := π. Then αβ = π κ = idP by the uniqueness part of the universal property, and similarly 2 βα = κ π = idQ . 16.3 Theorem. Let (Xj )j ∈J be a family of sets. Then the cartesian product Xj ∈J Xj , together with the canonical projections πk : Xj ∈J Xj → Xk of 1.8, is a product of the family in the category Set of sets and maps. Proof. This has been proved in 1.9.
2
16.4 Theorem. Let ((Xj , Tj )j ∈J ) be a family of topological spaces. Then the
space Xj ∈J Xj , j ∈J Tj (with the usual canonical projections) is a product of the family in the category Top of topological spaces and continuous maps. Proof. This was proved in 1.11.
2
16.5 Definition. We are mainly interested in certain categories of topological algebras, like topological groups, or topological rings. These algebras are sets with additional structure, and the morphisms are usually those maps between the sets that preserve this structure, with composition in the usual sense. Consequently, we obtain a functor from the category under discussion (throughout, a subcategory of TG or TR), mapping each object to the underlying set, and inducing the identity on the set of all morphisms between two given objects. This functor is called the forgetful functor from C to Set. There are also forgetful functors from TG to the category of all groups, and from TR to the category of all rings. As the discrete topologies do not really mean additional structure on a group or a ring (in particular, continuity with respect to discrete topologies is no restriction to the morphisms at all), we will interpret these as forgetful functors from TG to the full subcategory DG of discrete groups, or from TR to the full subcategory DR of discrete rings, thus avoiding to introduce even more categories without real need. One of the main objects in this section is to decide the question whether the forgetful functors preserve products, that is, whether the product of an arbitrary family ((Xj , Tj ))j ∈J of objects in a category C of topological algebras can be described – if it exists – as ( Xj ∈J Xj , T ) with some topology T (which need not be the product topology!). The forgetful functors from TG to DG and from TR to DR preserve products, but we will see soon that their restrictions to certain subcategories fail to do so. 16.6 Theorem. Let ((Rj , Tj ))j ∈J be a family of topological rings. On Xj ∈J Rj ,
define addition and multiplication “component-wise”. Then Xj ∈J Rj , j ∈J Tj (with the usual canonical projections) is a product of the family in the category TR of all topological rings.
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Proof. We leave as an exercise the easy proof that component-wise addition and multiplication (that is, (xj )j ∈J + (yj )j ∈J = (xj + yj )j ∈J and (xj )j ∈J (yj )j ∈J = (xj yj )j ∈J ) turn Xj ∈J Rj into a ring. For addition and additive inverse, the proof is in fact the same as for the cartesian product of groups. Similarly as in the proof of 3.35, we deduce that the operations are continuous, using the universal property of the product topology. 2 16.7 Theorem. Let ((Gj , Tj )j ∈J ) be a family of topological groups. Then the
topological group Xj ∈J Gj , j ∈J Tj (with the usual canonical projections) is a product of the family in the category TG of all topological groups. Proof. This has been established in 3.35.
2
16.8 Theorem. Let (Gj )j ∈J be a family of discrete groups. Then Xj ∈J Gj , endowed with the discrete topology, is a product of the family in the category DG of discrete groups (with the usual canonical projections). Proof. As in the proof of 3.35, the universal property can be derived from the obtained from universal property of the product Xj ∈J Gj in Set: in fact, the map ψ a family (ψj )j ∈J of maps ψj : W → Gj is a homomorphism if all the maps ψj are homomorphisms. 2 We have thus first examples that show that passage to subcategories may change (the topology of) products. 16.9 Example. Let (Gj )j ∈J be an infinite family of nontrivial discrete groups. Then the products in the categories TG and DG exist, but are not isomorphic as topological groups: in TG, the product is not discrete. In arbitrary subcategories of TG, products may behave arbitrarily bad. For instance, it is fairly easy to cook up examples which show that the product of a family of Hausdorff groups in a subcategory of TG need not be a Hausdorff group. 16.10 Definition. A subcategory C of the category TG or of TR is called kernelsaturated, if for any family (ϕj )j ∈J of morphisms ϕj : A → Bj in C the intersection N := j ∈J ker ϕj of all the kernels is an object of C, and the inclusion morphism from N to A also is a morphism in the category C. Moreover, we require that the (unique) constant morphism from A to A is a morphism in C for each object A of C. 16.11 Examples. A full subcategory is kernel-saturated exactly if it is closed with respect to forming arbitrary intersections of kernels of morphisms. The subcategories TG, TA, HG, HA, LCG, LCA, CG, CA, DG, and DA of TG are kernel-saturated. Similarly, the subcategories TR, HR, LCR, CR, and DR of TR are kernel-saturated.
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16.12 Lemma. Let C be a kernel-saturated subcategory of TG or of TR, and assume that P is a product of a family ((Xj , Xj )) j ∈J in C, with canonical projections πj : P → (Xj , Xj ). Then the intersection j ∈J ker πj is trivial, and there is an injective continuous homomorphism (in the larger category) from P to the
product Xj ∈J Xj , j ∈J Xj . Proof. Put N := j ∈J ker πj , and let ε : N → P be the inclusion. Let (ψj )j ∈J be the family of constant morphisms ψj : N → Xj . The equation ψj = επj shows that these constant morphisms belong to C. We have, a priori, two possibilities for the constant morphism, or the inclusion ε. Now uniqueness of ψ shows that ε ψ: is the constant morphism, and N is trivial. In the larger category, we find a continuoushomomorphism π from P to the product Xj ∈J Xj , j ∈J Xj . From ker π π = j ∈J ker πj = {1} we infer that is injective. 2 16.13 Proposition. Let C be a kernel-saturated subcategory of TG or of TR, Moreover, assume and assume that ((Xj , Tj ))j ∈J is a family of objects in C. that there is some topology T containing the product topology j ∈J Tj such that
( Xj ∈J Xj , T ) and the identity ι : ( Xj ∈J Xj , T ) → Xj ∈J Xj , j ∈J Tj belong to C. If the family has a product (P , P ) in C then there is a bijective continuous homomorphism η from (P , P ) onto Xj ∈J Xj , j ∈J Tj . (However, the topology P may differ from the product topology j ∈J Tj .) Proof. We claim that the π from (P , P ) to
continuous homomorphism injective the product Xj ∈J Xj , j ∈J Tj in the proof of 16.12 is surjective.
The canonical projections ϕj : Xj ∈J Xj , j ∈J Tj → (Xj , Tj ) of the product in TG resp. in TR remain continuous if we replace the product topology by the larger topology T . Thus we find in the category C a morphism ϕ : ( Xj ∈J Xj , T ) → (P , P ) with the property ϕ πj = ϕj for each j ∈ J .
( Xj ∈J Xj , T ) id / Xj ∈J Xj , j ∈J Tj RRR QQQ O RRR QQQ ϕ RRR QQQj RRR QQQ π ϕ RRR ( ) / (Xj , Tj ) (P , P ) πj We notice ϕ π πj = ϕ πj = πj for each j ∈ J , and infer that ϕ π is the identity on π is surjective. 2 Xj ∈J Xj . In particular, we have that 16.14 Remarks. In the kernel-saturated subcategories DG, DA, LCG and LCA of TG we can take the discrete topology to play the role of T in 16.13, and obtain that the product of any family in these categories is algebraically isomorphic to the cartesian product. The same applies to the subcategories DR and LCR of TR. In
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the non-discrete case, however, this does not imply that any of the products actually exists, and tells us a priori not much about the topology of the product. 16.15 Definition. Let C be an arbitrary subcategory of TG or of TR. Consider a family ((Xi , Xi ))i∈J of objects of C, and assume that ( Xi∈J Xi , T ) is a product of this family in C, with the usual canonical projections. C Then we write i∈J (Xi , Xi ) := ( Xi∈J Xi , T ). In the subcategories TA, HG, HA, CG, and CA of TG, and in the subcategories HR and CR of TR, products are the same as in TG resp. TR, as the following two simple observations imply. We leave the proofs as exercises. 16.16 Lemma. Let F be a full subcategory of a category C. Assume that the family (Xj )j ∈J of objects of F has a product in C. Then this product is also a product in F, with the same canonical projections. 2 16.17 Lemma. Let C be any one of the subcategories TA, HG, HA, CG, and CA of TG, or of the subcategories HR and CR of TR. Then the product of any family of objects of C, taken in TG resp. TR, belongs to C as well. Summarizing our results so far, we have: 16.18 Theorem. Let C be one of the subcategories TA, HG, HA, CG, CA, LCG, LCA, DG, and DA of TG; or HR, CR, LCR, and DR of TR. If the product of a family ((Xj , Tj ))j ∈J exists in C, then it is isomorphic to ( Xj ∈J Xj , T ), where T is a topology contained in the product topology j ∈J Tj . In particular, the 2 intersection of the kernels of all canonical projections is trivial.
In other words: the forgetful functor from C to the category DG or DR, respectively, preserves products. 16.19 Corollary. Let C be one of the subcategories TA, HG, HA, CG, CA LCG, LCA, DG, and DA of TG; or HR, CR, LCR, and DR of TR. If (P , P ) is the product of the family ((Xj , Xj ))j ∈J in C, then the canonical projections are open. Proof. Pick k ∈ J and let πk denote the canonical projection to (Xk , Xk ). Define a family (ψj )j ∈J of morphisms ψj : (Xk , Xk ) → (Xj , Xj ) by putting ψk = idXk and choosing the constant morphism in all other cases. Then the morphism ψ satisfies ψπk = idXk . Now consider an open neighborhood U of the neutral element is continuous, we have that U ψ← is an open neighborhood of the in (P , P ). As ψ ← ← neutral element in (Xk , Xk ), and U πk ⊇ U ψ ψπk = U ψ is a neighborhood of 2 the neutral element, as well. This yields that πk is an open map, see 3.33.
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16.20 Corollary. Let C be one of the categories TG, TA, HG, HA, CG, CA LCG, LCA, DG, DA; or TR, HR, CR, LCR, DR. Assume that (P , P ) is the product of the family ((Xj , Xj ))j ∈J in C. Then the following hold. (a) The connected component C of the product is the cartesian product of the family of connected components Cj of (Xj , Xj ). (b) The product (P , P ) is connected exactly if for each j ∈ J the object (Xj , Xj ) is connected. (c) The product (P , P ) is totally disconnected exactly if (Xj , Xj ) is totally disconnected for each j ∈ J . Now assume that C is one of the categories LCG or LCA. (d) The product (P , P ) has a nontrivial compact subgroup exactly if at least one member of the family has such a subgroup. (e) If for each j ∈ J the topology Xj is discrete and (Xj , Xj ) has no nontrivial finite subgroups then (P , P ) is discrete. Proof. Let C be the connected componentof (P , P ), and let D denote the connected component of (P , X), where X = j ∈J Xj is the product topology. Then D = Xj ∈J Cj by 2.7. Moreover, we have D ⊆ C since the identity gives a continuous map from (P , X) to (P , P ), preserving connectedness. Conversely, we have that every canonical projection πj : (P , P ) → (Xj , Xj ) maps C onto a subset of Cj , and conclude C = D. This proves assertion (a). Assertions (b) and (c) are immediate consequences. As the intersection over all kernels of canonical projections is trivial by 16.18, a nontrivial compact or connected subgroup of the product entails the existence of such a subgroup in at least one member of the family. If (P , P ) is not connected, we find a proper open subgroup which projects to open subgroups since the projections are open morphisms. This completes the proof of assertion (d). If all members of the family are discrete, we know from assertion (c) that the product is totally disconnected. Thus there are compact open subgroups by 6.8, and assertion (e) follows using assertion (d). 2 The (compact) product of an arbitrary family of finite discrete groups shows that some assumption about finite subgroups is needed in 16.20 (e). 16.21 Theorem. Let (Gj )j ∈J be a family of discrete groups without nontrivial finite subgroups. Then the discrete topology turns the cartesian product Xj ∈J Gj into a product of the family in LCG as well as in DG. However, if J is infinite and all the groups Gj are nontrivial then the product of the family in TG (or in HG) is not discrete.
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Proof. We abbreviate P := Xj ∈J Gj , and denote the discrete topologies on Gj and P by Dj resp. D. The canonical projection πk : P → Gk maps (gj )j ∈J to gk . According to Theorem 16.8, the group (P , D) is a product for the family ((Gj , Dj ))j ∈J in the category DG. If J is infinite and all the Gj are nontrivial then the product topology P is not discrete. As every product of the family in the category TG is isomorphic to (P , P ), this shows that the discrete group (P , D) is not a product in TG if J is infinite, see Lemma 16.2. It remains to show that (P , D) is a product for the family in LCG. So let (W, W ) be a locally compact Hausdorff group, and let (ψj )j ∈J be a family of continuous homomorphisms ψj : (W, W ) → (Gj , Dj ). In the category TG, we find a continuous homomorphism α : (W, W ) → (P , P ) such that απj = ψj for each j ∈ J . We put N := j ∈J ker ψj and observe ker α ≤ N . As the topology P is totally disconnected, we infer that W/N is totally disconnected, as well, see 4.9. According to 4.13, we find an open subgroup C of W containing N such that C/N is compact. The image C ψj is a compact (hence finite) subgroup of the discrete group Gj , and thus trivial for each j ∈ J . This yields C ≤ N, and we obtain that W/N is discrete. This means that α is also continuous if regarded as a map := α. Uniqueness of ψ follows from uniqueness to (P , D), and we can put ψ of α. 2 16.22 Corollary. The group ZJ , endowed with the discrete topology D, is a product for the family of discrete groups ((Gj , Dj ))j ∈J where Gj = Z for each j ∈ J in the category DG of discrete groups as well as in the category LCG of locally compact Hausdorff groups. However, if J is infinite, the group (ZJ , D) is not the product for the family in the category TG of all topological groups. 16.23 Proposition. The family (Gj )j ∈J , where Gj = R (with the usual topology) for each j ∈ J , has a product in the category LCG exactly if J is finite. Proof. If J is finite then the product topology turns RJ into a locally compact Hausdorff group, and we have found a product of the family in LCG; compare 16.16. Now assume that J is infinite, and that (P , T ) is a product in LCG, with canonical projections ψj : P → Gj . With the product topology P the group (RJ , P ) is a product in the category TG, with the usual canonical projections πj . According : (P , T ) → (RJ , P ) to 16.13, we obtain a continuous bijective homomorphism ψ such that ψπj = ψj for each j ∈ J . Without loss, we identify P = RJ and have = idP . ψ The product (P , T ) is connected by 16.20 (b), and has no nontrivial compact subgroup by 16.20 (d). We will see later in 23.11 that this means that (P , T ) is isomorphic to Rn . Taking a subset I ⊂ J with n+1 elements we obtain a surjective continuous homomorphism λ : (P , T ) → RI ∼ = Rn+1 by putting x λ = (xi )i∈I . This is impossible since every continuous homomorphism from Rn to Rn+1 is an 2 R-linear map by 24.6.
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It will turn out that the general situation in LCG is similar to 16.23: the product of a family of connected groups in LCG exists exactly if the product in TG is locally compact. 16.24 Lemma. Let C be a full subcategory of TG or TR, and let ((Xi , Xi ))i∈I and ((Yj , Yj ))j ∈J be families of objects of C. Assume that (P , P ) and (R, R) are products of ((Xi , Xi ))i∈I and ((Yj , Yj ))j ∈J in C, with canonical projections πi : (P , P ) → (Xi , Xi ) and ρj : (R, R) → (Yj , Yj ), respectively. Moreover, assume I ⊆ J and let (αi )i∈I be a family of morphisms αi : (Xi , Xi ) → (Yi , Yi ) in C. : (P , P ) → (R, R) in C such that πi αi = βρ i (a) There is a unique morphism β for each i ∈ I and βρj is the constant morphism for each j ∈ J I . (b) If the intersection j ∈J ker ρj is trivial ( for instance, if C is one of the cate = gories listed in Theorem 16.18), then ker β i∈I ker(πi αi ). (c) If αi is an isomorphism for each i ∈ I and i∈I ker πi is trivial then (P , P ) and
(P β , R|P β) are both isomorphic to the quotient of (R, R) by the intersection i∈I ker ρi . (d) Assume that the forgetful functor from C to DG resp. DR preserves products. If Xiαi is closed in (Yi , Yi ) for each i ∈ I and (Yj , Yj ) is Hausdorff for each j ∈ J I then P β is closed in (R, R). Proof. We denote the neutral elements of the (underlying additive) groups (of rings) in C by 0, irrespective of commutativity. For each i ∈ I , define βi := πi αi . For j ∈ J I , let βj : (P , P ) → (Yj , Yj ) be the constant morphism. The universal property of the product (R, R) then yields assertion (a). In any case, we have ker β ≤ i∈I ker(βρi ) = i∈I ker(π i αi ). Now assume that the intersection i∈I ker ρi is trivial, and consider x ∈ i∈I (ker πi αi ). Then 0 = x πi αi = x βρi yields x β ∈ ker ρi , and x β ∈ ker ρj for each j ∈ J I by the Thus we have x β ∈ construction of β. j ∈J ker ρj = {0}, and x ∈ ker β. This completes the proof of assertion (b). Now assume that αi is an isomorphism, for each i ∈ I . For i ∈ I , put γi = ρi αi−1 . The universal property of the product (P , P ) yields a morphism γ πi = βγ i = βρ i α −1 = πi αi α −1 = πi γ : (R, R) → (P , P ), and we have β i i for each i ∈ I . The uniqueness part of the universal property of the product γ = idP . For y = x β ∈ P β and arbitrary i ∈ I we observe (P , P ) now yields β
γ β = x β γ β = x β = y, and find that the restriction of γ to P β is a two-sided y This shows that (P , P ) and (P β , R| β) are isomorphic. inverse for β. P
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,2 Q := R/ ker γ be the canonical projection. Then η : Q → P Let κ : R γ = idP . On the other hand, we exists with κη = γ , and we find (βκ)η = β = idQ follows because κ is an = = κ = κ idQ , and η(βκ) have κηβκ γ βκ epimorphism. In order to prove the rest of assertion (c), it thus remains to show γ implies 0 = x γ πi αi = x ρi for each i ∈ I , and ker γ = i∈I ker ρi . But 0 = x −1
γ πi , and x γ ∈ ker π . conversely 0 = x ρi implies 0 = x ρi αi = x i Finally, assume that the forgetful functor from C to DG resp. to DR preserves products. Then r ∈ R belongs to P β exactly if r ρj ∈ P βρj = P βj for each j ∈ J . ρ β β Let y ∈ R P , and pick j ∈ J such that y j ∈ / P j . For j ∈ J I this means αj ρ ρ j j y = 0, for j ∈ I we have y ∈ / Yj . In both cases, our assumptions make sure that there is a neighborhood U of y ρj in (Yj , Yj ) disjoint to P βj , and the pre-image ← U ρj is a neighborhood of y in (R, R) disjoint to P β . This proves assertion (d). 2
16.25 Lemma. Let C be a subcategory of TG or TR such that the forgetful functor from C to DG resp. to DR preserves products. Let ( Xj ∈J Xj , T ) be the product of some family ((Xj , Xj ))j ∈J of objects in C. If ((Yj , Yj ))j ∈J is a family of objects in C such that (Yj , Yj ) ≤ (Xj , Xj ) and the topology P induced by T on P := Xj ∈J Yj makes (P , P ) an object of C, then (P , P ) is a product of the family ((Yj , Yj ))j ∈J in C. Proof. The canonical projections will be denoted πj : ( Xj ∈J Xj , T ) → (Xj , Xj ). Let (W, W ) be an object of C, and let (ψj )j ∈J be a family of morphisms ψj from (W, W ) to (Yj , Yj ). Then we can interpret these as morphisms from (W, W ) to (Xj , Xj ), and the universal property of the product ( Xj ∈J Xj , T ) yields a : (W, W ) → ( Xj ∈J Xj , T ) such that ψπ j = ψj for each j ∈ J . morphism ψ ψ ψ j Now W ≤ Yj implies W ≤ P , and we have established the universal property is guaranteed by the fact that the forgetful for (P , P ); note that uniqueness of ψ functor preserves products. 2 16.26 Theorem. A family ((Gj , Tj ))j ∈J of connected locally compact Hausdorff groups has a product in the category LCG exactly if (Gj , Tj ) is compact for all but finitely many of the indices j ∈ J . If a product of the family exists in LCG, it is isomorphic
as a topological group to the product in TG, that is, to Xj ∈J Gj , j ∈J Tj .
Proof. The product Xj ∈J Gj , j ∈J Tj in TG is locally compact exactly if the group (Gj , Tj ) is compact for all but finitely many of the indices j ∈ J . According to 16.16, we have in that case that Xj ∈J Gj , j ∈J Tj is a product in LCG. Conversely, assume that there exists a product (P , P ) for the family in LCG, with canonical projections πj : (P , P ) → (Gj , Tj ). According to 16.13, we can identify P with the cartesian product Xj ∈J Gj such that the canonical projections
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πj coincides with
the usual one, and the identity on P is continuous from (P , P ) to P , j ∈J Tj . By the Malcev–Iwasawa Theorem (see 32.5 for details and references), each locally compact connected Hausdorff group (G, T ) has maximal compact subgroups (forming a single conjugacy class), and for each maximal compact subgroup M there are continuous homomorphisms ρi : R → (G, T ) such that (m, r1 , . . . , rn )ϕ = ρ ρ mr1 1 . . . rn n defines a homeomorphism ϕ : (M, T |M ) × Rn → (G, T ). In particular, this means that every locally compact connected group is either compact or contains a closed subgroup isomorphic to R. Let I be the set of those i ∈ J where (Gi , Ti ) is not compact, and pick a closed subgroup Xi isomorphic to R in (Gj , Tj ) for each i ∈ I . As LCG consists of Hausdorff groups, we can apply 16.24 and obtain that (P , P ) contains a closed subgroup which is a product of the family ((Xi , Ti |Xi ))i∈I in LCG. According to 16.23, this is only possible if I is finite. 2
17 Direct Limits and Projective Limits We start this section with a discussion of some instructive examples, before entering a somewhat abstract and technical discussion. Often, quite complicated groups are obtained as unions of rather easily understandable subgroups. For instance, this is the case with the additive group of rational numbers: this group is a union of infinite cyclic subgroups. The following example is similar, and of great importance in the theory of Abelian groups1 . 17.1 Example. Let p be a prime. The set Z(p∞ ) := Z + pzn | z ∈ Z, n ∈ N forms a subgroup of the (discrete) group /Q/Z. It0 is the union of the ascending sequence of finite cyclic subgroups Dn := Z + p1n = Z + pzn | z ∈ Z ∼ = Z(pn ). ∞ The group Z(p ) is known as Prüfer group. For natural numbers i, j with i ≤ j , let di,j : Di → Dj and δi : Di → Z(p ∞ ) be the inclusion maps. Then the Prüfer group has the following universal property: Let M be an Abelian group. If (αi )i∈N is a family of homomorphisms (DL) αi : Di → M such that αi = di,j αj whenever i ≤ j then there exists a unique homomorphism α : Z(p ∞ ) → M such that δi α = αi .
1 In fact, the groups Q and Z(p ∞ ), where p ranges over the set P of all primes, form the building blocks for divisible discrete Abelian groups, see 4.24.
17. Direct Limits and Projective Limits
Z(p∞bE) hQUj U EE QQUQUQUUUU EE QQQ UUUU E QQ UUUU U δ0 E EE δ1 QQQQQ δ2 UUUUUUU QQQ UUUU EE QQQ UUUU E Q UU / /D d d1,2 D0 D1 0,1 α nn 2 n | n | | nnn || nnn n α0 α1| α | nn 2 ||nnnnn | ~|vn|nnn M
157
/ ···
This example will be generalized by the notion of direct limit below. In a similar way, the study of certain rings can be reduced to the study of manageable subrings. Sometimes, a group or a ring cannot be reduced “from the bottom” (using subgroups) but rather “from the top”: that is, by forming quotients. The following construction provides examples of rings and of groups at the same time.
17.2 Example. Let p be a prime, and let R denote the set of all homomorphisms from Z(p∞ ) to itself. Then pointwise addition (that is, x r+s = x r + x s for x ∈ Z(p∞ ) and r, s ∈ R) and composition as multiplication turn R into a ring. We denote this ring by Zp , it is called the ring of p-adic integers. For each integer z, the map ρz defined by x ρz = zx is an element of Zp . This yields a homomorphism ρ : Z → Zp of rings. As the Prüfer group contains elements of arbitrarily high order, the homomorphism ρ is injective. Because Di is generated by the elements of order p i in Z(p∞ ), every element of Zp maps Di to itself, and we can consider the restriction of Zp to Di . Every homomorphism from Di to Di is of the form ρz for some z ∈ Z; in fact, the integers z ∈ {0, . . . , pi − 1} suffice. For each natural number i, the kernel Ni of the restriction to Di is an ideal of the ring Zp , and Zp /Ni ∼ = Z/pi Z. Thus the restriction map may be regarded as a ring homomorphism from Zp onto the cyclic ring Z/pi Z. We will denote this homomorphism by λi . The intersection i∈N Ni is trivial, because the union i∈N Di equals Z(p∞ ). In this sense (at least), the collection of quotients Zp /Ni ∼ = Z/pi Z carries all information about Zp . However, these bits of information are not isolated from each other: for i ≤ j , we have a homomorphism of rings πi,j : Z/pj Z → Z/pi Z given simply by (pj Z + z)πi,j = p i Z + z. Together with the homomorphisms
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λi : Zp → Z/pi Z, we have the following commuting diagram: Zp RWRWRWW DD RRWRWWWW DD RRR WWWWW RRR WWWW DD RR WWW W λ0 D λ1 RRR DD RRR λ2 WWWWWWWW RRR DD WWWW RRR WWWW D! WW+ ) Z/p0 Z o Z/p1 Z o Z/p2 Z o π0,1
π1,2
···
We combine the information by the following map x ∈ Zp to procedure: i Z. Now x η := (x λi ) the sequence (x λi )i∈N , which is an element of Z/p i∈N i∈N i defines a map η : Zp → i∈N Z/p Z. It is easy to see that η is a homomorphism of rings whose kernel equals i∈N Ni . Thus η is injective, and the rings Zp and (Zp )η πi,i+1 because the are isomorphic. We have (Zp )η = (xi )i∈N | ∀i ∈ N : xi = xi+1 diagram above commutes. (The map η will be generalized in 18.3 below.) Again, we observe a universal property: for each ring M and each family (αi )i∈N of ring homomorphisms αi : M → Z/pi Z such that αj πij = αi whenever i ≤ j there is a unique ring homomorphism α : M → Zp such that α λi = αi for each i ∈ N. In fact, the universal property of the product gives the existence of a ring homomorphism β : M → i∈N Z/pi Z, and our assertion that αj πij = αi whenever i ≤ j yields that M β ≤ (Zp )η . The homomorphism α := βη−1 satisfies our requirements. Zp WISSWSWWW [ II SSSWWWWW II SSSS WWWW II W I SSSSS WWWWWWW W λ0 II λ1 SSS SSS λ2 WWWWWWWWW II SSS WWWWW II SSS WWWWW I$ S) W+ 0Z o 1Z o 2 o π0,1 π1,2 η Z/p Z/p 5 Z/p Z α O < l l y l y lll yy lll l yy l y l α0 α1 lα2 yy lllll y l y yy llll lyll i o M i∈N Z/p Z
π2,3
···
The discussion of Zp as a ring can also be interpreted as a discussion of the underlying additive group. Up to now, we have considered Zp without topology. It is very natural to give the finite sets Z/pi Z the discrete topology. However, the with the topology induced from the image of Zp under η should be considered (compact, non-discrete) product topology on i∈N Z/pi Z. The general setting needs some formal preparation.
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17.3 Definition. Recall from 1.38 that a directed set (J, ) is a set J with a preorder (that is, a reflexive and transitive binary relation on J ) such that for any two elements i, j ∈ J there exists k ∈ J satisfying i k and j k. Every pre-ordered set (J, ) gives a category J with object class J and ! {(i, j )} if i j, Mor(i, j ) = ∅ otherwise, for i j k the composition is defined by (i, j )(j, k) = (i, k). 17.4 Examples. (a) With the usual order relation ≤, we obtain a directed set (N, ≤). (b) Writing n|m if n divides m, we obtain a directed set (N {0}, |). (c) Let B be a set of subsets of some set. If B is a filterbasis (that is, if for all A, B ∈ B there exists C ∈ B such that C ⊆ A ∩ B, and ∅ ∈ / B) then (B, ⊇) is a directed set; note that we have to reverse the inclusion relation. (d) Let S be a semigroup, and define the binary relation on S by x y ⇐⇒ Sx ⊆ Sy. Then is a pre-order, known as the right invariant (pre-)order on S. It is a quite restrictive assumption on the semigroup S that (S, ) be directed.
Direct Limits 17.5 Definitions. Let (J, ) be a directed set, and let C be any category. A directed system over (J, ) in C is a covariant functor D : J → C, where the category J is obtained from the pre-ordered set (J, ) as in 17.3. We will write Di for the image of i ∈ J under D. For i j we write di,j for the image of the morphism (i, j ). The morphisms di,j are called the bonding morphisms of D. If D and E are directed systems over the same directed set, a morphism (of directed systems) α : D → E is a family α = (αi )i∈J of morphisms αi : Di → Ei such that for each morphism (i, j ) in J we have αi ei,j = di,j αj . We have thus constructed the category DSJ (C) of directed systems over (J, ) in C; of course, the composite of α : D → E and β : E → F is αβ = (αi βi )i∈J . A constant in DSJ (C) is a functor that maps each i ∈ J to the same object, and each morphism in J to the identity morphism of that object. If D is a directed system and E is a constant then a morphism from D to E is called a cone over D, with vertex E.
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e1,2 e2,4 / E2 / E4 / ... E E 1 FF E FF E GG
G F F GG FF FF
e1,3F e2,6F
e4,12G GG F F F F GG F F
F# F" # e3,6
e6,12
/ / E12 E E
3 6 E D E
α1 α2 α4
α α
α12
6
3
d1,2 d2,4
/ D2 / D4 / ...
D1 F
FF
FF
FF F F
FF F F
F
Fd
d1,3F d 2,6 4,12 F F FF FF FF
FF FF FF " "
#
/ D6 / D12 / ... D3 d3,6
/ ...
d6,12
A morphism α : D → E between directed systems.
E _@gjPUPUPUUU @@ PPPUUUU @@ PPP UUUU @ PP UUUU U λi @ @@ λj PPPPP λk UUUUUU UUUU PPP @@ UUUU @ PPP UU / / Dk d d Dj Di i,j j,k
/ ···
A cone λ with vertex E over a directed system D.
17.6 Remark. By abuse of language, we will sometimes identify constants in DSJ (C) with objects in C. Morphisms between constants in PSJ (C) should then be interpreted as morphisms between objects of C. This causes no serious problems, in fact, if C and D are constants and α : C → D is a morphism, we have that for each i ∈ J the morphism αi : Ci → Di is the same. 17.7 Examples. (a) Let (J, ) = (N, ≤), and let p be a prime. The definitions of Di and di,j in 17.1 yield a directed system D ∈ DSJ (DA). The maps δi form a cone over D with vertex Z(p∞ ). (b) Let (J, ) = (N, ≤), and let p be an integer. A directed system E ∈ DSJ (DA) is given by the settings Ei = Z/pi Z and (p i Z+x)eij = pj Z+p j −i x for i ≤ j . (c) Let (J, = (N {0}, |). A directed system F ∈ DSJ (DA) is given by ) Fi = zi z ∈ Z and x fij = x for i | j . The morphisms αi : Fi → Q defined by x αi = x yield a morphism α from F to the constant L mapping each i ∈ N to Q.
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(d) More generally, let G be a discrete group, and let J be the set of all finitely generated subgroups of G. For A ∈ J put FA = A. For A, B ∈ J with A ≤ B, define fAB : A → B by x fAB = x. Then F = (FA )A∈J is a directed system in DG. Again, we have a morphism from F to the constant mapping each A ∈ J to G. 17.8 Definition. Let (J, ) be a directed set, and let D ∈ DSJ (C) be a directed system. A direct limit of D in C is a cone λ : D → L over D with vertex L ∈ DSJ (C) with the following universal property: (DL)
For every constant M in DSJ (C) and every cone α : D → M there is a unique morphism α : L → M such that λ α = α.
According to 17.6, we interpret α as a morphism in C. If the constant L maps each element of J to X, and if λ : D → L is a direct limit of D, we say that X models the direct limit of D. X WUkdJj JWUWUWUWUWW JJ UUWUWUWWW W JJ JJ UUUUWUWUWUWWWWWW λj UUU Wλk WWWW λi JJ WW UUUU JJ UUUU WWWWWWWW JJ WWWWW JJ UUUU WWW U/ /D d dj,k D Di i,j j α lll k t l t l t l tt lll tt lll l t l α αi α j lll k tt ttllllll t zttull Mi = Mj = · · ·
/ ···
17.9 Examples. (a) The group Z(p∞ ) models the direct limit of the directed system D given in 17.7 (a); in fact, the cone δ = (δi )i∈N is a direct limit, as we have seen in 17.1. (b) The discrete group (Q, +) models the direct limit of the directed system F in 17.7 (c). In general, a directed system in an arbitrary category will not have any direct limit. If direct limits for a given directed system exist, they are pairwise isomorphic (thus if both X and Y model the direct limit of D ∈ DSJ (C) then X and Y are isomorphic). 17.10 Lemma. Let (J, ) be a directed set, and let λ : D → L and μ : E → M be direct limits of D, E ∈ DSJ (C), respectively. If α : D → E is an isomorphism, then αμ 1 is an isomorphism from L to M.
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−1 λ is the inverse of αμ. Proof. We show that α 1 In fact, for each i ∈ J we have −1 −1 −1 λ = λi αμ 1 = αi μi and μi α λ = αi λi . This yields αi−1 λi αμ 1 = μi and αi μi α −1 λ is the identity morphism λi . The universal property (DL) now asserts that αμ 1 α −1 of L, and α λαμ 1 is the identity of M. 2
As in 17.7 (d), let G be a discrete group, and let J be the set of all finitely generated subgroups of G. For A ∈ J put FA = A. For A, B ∈ J with A ≤ B, let fAB : A → B be the inclusion. Then F = (FA )A∈J is a directed system in DG. Let L ∈ DSJ (DG) be the constant mapping each element of J to G. For each A ∈ J let λA : A → G be the inclusion. Using this notation, we have the following general result, which will play an important role in the proof of Pontryagin’s Duality Theorem (and some applications thereof). 17.11 Theorem. The discrete group G models the direct limit of F ; in fact, the cone λ : F → L is a direct limit of F in DG. Proof. Let M ∈ DSJ (DG) be a constant, and consider a morphism α : D → M. α := a αA . This definition is unambiguous since α is a For A ∈ J and a ∈ A we put a morphism; in fact, if a ∈ A ∩ B then C := a belongs to J , and a αA = a αC = a αB because fCA αA = αC = fCB αB . 2
Projective Limits We now turn to projective limits, a concept which is dual to that of direct limits. 17.12 Definition. Let (J, ) be a directed set, and let C be any category. A projective system over (J, ) in C is a contravariant functor P : J → C. We will write Pi for the image of i ∈ J under P . For i j the image of the morphism (i, j ) is denoted by pi,j . The morphisms pi,j are called the bonding morphisms of P . If P and Q are projective systems over the same directed set, a morphism (of projective systems) α : P → Q is a family α = (αi )i∈J of morphisms αi : Pi → Qi such that for each morphism (i, j ) in J one has αj qi,j = pi,j αi . in J:
i
in C:
Pi o
(i,j )
pi,j
Pj αj
αi
Qi o
/j
qi,j
Qj
17. Direct Limits and Projective Limits
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We have thus constructed the category PSJ (C) of projective systems over (J, ) in C; of course, the composite of α : P → Q and β : Q → R is αβ = (αi βi )i∈J . Again, a constant in PSJ (C) is a functor that maps each i ∈ J to the same object, and each morphism in J to the identity morphism of that object. Constants and morphisms between constants in PSJ (C) will be identified with objects and morphisms in C, respectively; compare 17.6. If P is a projective system and Q is a constant then a morphism from Q to P is called a cone over P , with vertex Q. Note the slight difference between cones over directed systems and cones over projective systems: the arrows point the other way. q1,2 q2,4 ... Q1 Goc Q2 Goc Q4 ocH HH GG GG
HH GG GG
q1,3G
q2,6G
q4,12H
HH G G G G HH
GG
GG
q3,6
q6,12
o o Q3 Q6 Q12 o
α1 α2 α4
α3 α6 α12
p2,4
p1,2 ...
P1 Goc P2 Goc P4 ocG
GGG GGG
GG GG GG
GG
p1,3G p4,12GG
p2,6GG GG GGG GG
GG
G G
... P3 o P6 o P12 o p3,6
...
p6,12
A morphism α : Q → P between projective systems. 17.13 Examples. (a) Let (J, ) = (N, ≤), and let p be a prime. Putting !i := Z/pi Z and defining πi,j as in 17.2, we obtain a projective system ! ∈ PSJ (C), where C ∈ {DA, CA, DR, CR}. The maps λi defined in 17.2 form a cone λ over ! with vertex Zp . (b) Let (J, ) = (N {0}, | ). We obtain P ∈ PSJ (CG) by putting Pi = T for each i ∈ N and defining pij : T → T by (Z + x)pij = Z + ji x if i | j . (c) Let (J, ) = (N {0}, | ). We obtain Q ∈ PSJ (CG) by putting Qj = R/j Z for each j ∈ N and defining qij : R/j Z → R/ iZ by (j Z + x)qij = iZ + ji x if i | j . The morphisms αi : Qi → Pi given by (iZ + x)αi = Z + x yield a morphism α from Q to P .
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(d) Let (J, ) = (N, ≤), and let p be a positive integer. Then Ri := R/p i Z and (pj Z + x)rij = pi Z + x define a projective system R ∈ PSJ (CG). 17.14 Definition. Let (J, ) be a directed set, and let P ∈ PSJ (C) be a projective system. A projective limit of P in C is a cone λ : L → P over P with vertex L in PSJ (C) to P with the following universal property: (PL)
For every constant M in PSJ (C) and every morphism α : M → P there is a unique morphism α : M → L such that α λ = α.
Assume that the constant L maps each element of J to X, and that λ : L → P is a projective limit of P ∈ PSJ (C). Then for each i ∈ J we have that λi is a morphism from X to Pi . We say that X models the projective limit of P . X [ JWTJTWTWTWTWW JJ TTWTWTWWWW JJ TTTTWWWW JJ TTTT WWWWW λj TTT Wλk WWWW λi JJ WW TTTT JJ TTTT WWWWWWWW JJ WWWWW JJ TTTT WW $ T* α o o p pj,k W5+ Pk o i,j PO i l : Pj t lll t lll tt l t l l tt lll αj t αi αk l l l t l tt ll tt lll ttlll Mi = Mj = . . .
···
17.15 Example. In 17.13 (a), take C = DA or DR, and endow Zp with the discrete topology. Then the cone λ is a projective limit of !, and Zp models this projective limit. We will see in the next section how Zp can be equipped with a compact topology such that it models the projective limit of ! in CA and in CR. In general, a projective system in an arbitrary category will not have any projective limit. In 18.1 below we will see, however, that in the categories DG, DA, HG, HA, CG, CA, DR, and CR, every projective system has a limit. If projective limits for a given projective system exist, they are pairwise isomorphic (thus if both X and Y model the projective limit of P then X and Y are isomorphic). We state this in a slightly more general way, the proof is analogous to that of 17.10. 17.16 Lemma. Let (J, ) be a directed set, and let λ : L → P and μ : M → Q be projective limits of P , Q ∈ PSJ (C), respectively. If α : P → Q is an isomorphism, 1 is an isomorphism from L to M. then λα 2
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Exercises for Section 17 Exercise 17.1. Let p be a prime, and let Di := a be a cyclic group of order p i . Determine the ring End(Di ) of all homomorphisms from Di to Di . Hint. Show first that each element of End(Di ) is determined by its value at the generator a. Exercise 17.2. Verify in detail all assertions made in Example 17.2. Exercise 17.3. Verify that each filterbasis B yields a directed set (B, ⊇). Give an example of a filterbasis B such that (B, ⊆) is not a directed set. Exercise 17.4. Let (S, ∗) be a semigroup, and let be the right invariant pre-order on S as defined in 17.4 (d). (a) What can you say about if (S, ∗) is a group? (b) Investigate for (S, ∗) ∈ {(N, +), (Z, +), (Z, ·), (M, ◦)}, where M is the set of all self-maps of a set with at least 2 elements. In which of these cases is (S, ) a directed set? Exercise 17.5. Show that the ring Q of rational numbers models a direct limit of proper subrings such that each of these is a local ring (that is, a ring where the set of all noninvertible elements forms an ideal). Hint. Consider the subsets Lp := nz | z ∈ Z, n ∈ Z pZ , where p is a prime. Exercise 17.6. Let V be the additive group of some vector space. Show that V models the direct limit of a directed system of additive groups of vector spaces of finite dimension. Exercise 17.7. Show that every product models the projective limit of a projective system of products of finitely many of the given factors. Exercise 17.8. Show that the ring Zp is an integral domain; that is, that xy = 0 implies 0 ∈ {x, y} for x, y ∈ Zp . Exercise 17.9. Prove Lemma 17.16. Exercise 17.10. Assume that P , Q ∈ PSJ (C) possess projective limits, modeled by L and M, respectively. Show that every morphism α : P → Q defines a morphism from L to M.
18 Projective Limits of Topological Groups Let C be one of the subcategories TG, TA, HG, HA, CG, CA, LCG, LCA, DG, and DA of TG; or TR, HR, CR, LCR, and DR of TR. Recall from 16.18 that this implies that the product of any family ((Xj , Tj ))j ∈J – if it exists in C – is isomorphic to
Xj ∈J Xj , T , where T is a topology contained in the product topology j ∈J Tj . Note that the product need not exist for C ∈ {LCG, LCA, LCR}.
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Assume that P ∈ PSJ (C) is a projective system over some directed set (J, ). If Pi = (Xi , Xi ), we define p lim P := (xi )i∈J ∈ X Xi | ∀i, j ∈ J : i j #⇒ xi = xj ij ←−
i∈J
C
and write lim P = lim P , T , where T is the topology induced by the topology
←−
of the product
C
←−
i∈J Pi ,
formed in the category C. If the category is fixed by the C
context, we will feel free to abuse notation by writing lim P = lim P . ←−
←−
Let L : J → C be the constant mapping each element of J to lim P . We obtain ←−
a morphism λ : L → P if we define λj : lim P → Pj by ((xi )i∈J )λj = xj . ←−
Using this notation, we have the following. C
18.1 Lemma. The morphism λ is a projective limit of P in C; that is, lim P models ←− the projective limit of P . C C Proof. Proceed as in Example 17.2: let η : lim P → i∈J Pi be the inclusion. Let ←− M be a constant in PSJ (C), and let α = (αi )i∈J be a cone with vertex M over P . C
TNTT @@NNNTNTNTTTT @@ NN TTT @ NN TTTT T λj @ @@ λk NNNNN λl TTTTTT TTTT @@ NNN TTTT @@ NNN TTT) N' o o o p pk,l j,k P η P j i α q8 Pk O A q q qq qqq q q αj α αq k qqq l q q qqqqq C q q β Pi o M
lim P ←− _W
···
i∈J
The universal property of the product gives the existence of a morphism β from C M to i∈J Pi , and our assertion that αj pij = αi whenever i ≤ j yields that M β is C η contained in lim P . The homomorphism α := βη−1 satisfies our requirements; ←− 2 it is unique by the universal property of the product.
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We keep the notation introduced above. In the cases where C is a subcategory of TR, ‘neutral element’ means 0, the neutral element of addition. 18.2 Lemma. Assume that the topology on lim P is induced by the product topology ←− on i∈J Pi . Then for every neighborhood V of the neutral element in lim P , there ←− is some m ∈ J such that ker λm ⊆ V . Proof. Let V be a neighborhood of the neutral element in lim P . As lim P carries ←− ←− the topology induced from the product topology on j ∈J Pj , we can find a finite in Pf for each f ∈ F subset F of J and neighborhoods Uf of the neutral element
such that U := lim P ∩ ( Xf ∈F Uf ) × ( Xj ∈J F Pj ) is contained in V . As (J, ) ←− is a directed set, there is an element m in J such that f m for each f ∈ F . Now 2 λf = λm pf m yields ker λm ≤ ker λf , and we obtain ker λm ⊆ U ⊆ V . In the category TG or its subcategories, projective limits often occur in the following way. 18.3 Definition. Let G be a topological group, and let N be a set of normal subgroups of G. We define the set GN by GN := (NgN )N ∈N ∈ X G/N | ∀L, M ∈ N : L ≤ M ⇒ MgL = MgM N ∈N
and the map ηN : G → GN by g ηN = (Ng)N ∈N . Recall that N is called a filterbasis if for any two elements L, M ∈ N there exists some K ∈ N such that K ≤ L ∩ N. 18.4 Lemma. Assume that C is a subcategory of TG (or of TR) such that the forgetful functor from C to DG (resp. to DR) preserves products. Let G be an object of C, and let N be a set of normal subgroups (resp. ideals) such that for each N ∈ N the quotient G/N belongs to C. Moreover, assume that the family (G/N )N ∈N has C a product P = N ∈N G/N in C. Then the following hold. (a) If N consists of closed normal subgroups (ideals) then GN is a closed subgroup (subring) of P . (b) The map ηN is a continuous homomorphism from G to GN . (c) The kernel of ηN equals N . (d) If N is a filterbasis and P coincides with the product n∈N G/N in TG (resp., in TR) then GηN is dense in GN .
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Proof. The proof that GN is a subgroup (resp. a subring) of XN ∈N G/N is left as an exercise. In order to show that GN is closed in P , we consider the continuous maps pLM : G/L → G/M given by (Lg)pLM = Mg for each pair of elements L, M ∈ N such that L ≤ M. For each N ∈ N , we let ϕN denote the canonical ϕL pLM = x ϕM } is closed projection from P onto G/N. The set ELM := {x ∈ P | x in P by 1.16 (b). Therefore, the intersection GN = {ELM | L, M ∈ N , L ≤ M} is closed in P . This completes the proof of assertion (a). Assertions (b) and (c) are also left as exercises. Under the assumptions of assertion (d), consider an element x = (N xN )N ∈N in GN . If U is an open neighborhood of x, we find a finite subset F of N and πF ϕ ← open subsets UF of G such that F ∈F UF F is a neighborhood of x contained in U . Since N is a filterbasis, we find M ∈ N such that M ⊆ F . Then η F xM = F xF ∈ UFπF implies that xMN = (N xM )N ∈N belongs to U ∩ GηN . We have shown that every neighborhood of an arbitrary element of GN meets GηN . 2 This means that GηN is dense in GN , and assertion (d) is established. 18.5 Remark. Assertion (d) in 18.4 heavily depends on the fact that GN is endowed with the topology induced by the product topology on N ∈N G/N . Let G be a topological group, and let N be a set of normal subgroups that forms a filterbasis. Then (N , ⊇) is a directed set, and we obtain a projective system P ∈ PSN (TG) if we put PN := G/N and define pN M : G/M → G/N by (Mg)pN M = Ng whenever N ⊇ M. Let L : N → TG be the constant mapping each N ∈ N to GN . The morphisms λN : GN → G/N obtained as restrictions of C the projections from N ∈N G/N onto G/N form a morphism λ = (λN )N ∈N from L to P . GN IWSSWSWW _ II SSWSWSWWWW II SSS WWWW II SSS WWWWWW λM I II λK SSSSSSλJ WWWWWWWW WWWW II SSS WWWW I SSS WWW '. ( , % G/J lr r l r l πKJ πMK G/M G/K O > n7 n } n nn }} nnn }} n n } nnn }} }} nnnnn } } nn }}nnnn } n C } nn G/N o V N∈N
Using these definitions, we have the following consequence of 18.1. 18.6 Theorem. The morphism λ : L → P is a projective limit of P .
···
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Exercises for Section 18 Exercise 18.1. Let N be a set of normal subgroups of a group G. Prove that GN is a subgroup of the product N∈N G/N . Exercise 18.2. Let N be a set of normal subgroups of a topological group G. Prove that ηN is a continuous homomorphism, and that ker ηN = N . Exercise 18.3. Let p be a prime. Show that Sp := {(xi )i∈N ∈ TN | ∀i ∈ N : xi = pxi+1 } is a connected compact Hausdorff group. Hint. Show that Sp models the projective limit of a projective system P ∈ PSJ (CA) over (N, ≤), where Pi = T for each i ∈ N. Find a morphism α : R → Sp with dense image. Exercise 18.4. Show that Sp contains no elements of order p. Conclude that Sp is not isomorphic to Tn for any n ∈ N. Exercise 18.5. Try to find a compact connected Hausdorff group which is not trivial but torsion-free; that is, containing no elements of finite order except 0. Hint. Use a projective limit of circle groups over (N {0}, |). The following exercises are concerned with the so-called p-adic topology Tp on Zp , which makes Zp a compact and totally disconnected ring. Throughout, let p be a prime. Exercise 18.6. Consider the projective system ! given in 17.13 as a projective system in CR, and show that there is a unique topology Tp on Zp such that (Zp , Tp ) models the projective limit of !. Exercise 18.7. Show that Tp is compact and totally disconnected. Exercise 18.8. Prove that the natural action of Zp on Z(p∞ ) is continuous, if Zp carries the p-adic topology and Z(p∞ ) is discrete. Hint. Show that for each y ∈ Z(p∞ ) the annihilator {z ∈ Zp | y z = 0} is open in (Zp , Tp ). Exercise 18.9. In the categories DA and DR, find monics from Z to Zp . Exercise 18.10. Endow Zp with the p-adic topology. In the category HA, find a morphism from Z to Zp which is both monic and epic. Exercise 18.11. Give a simple reason why Z and Zp cannot be isomorphic, in any of the categories DA, DR, HA, irrespective of the topologies used.
19 Compact Groups We enclose a section on compact groups in the present chapter since every compact Hausdorff group can be described by a projective limit of (supposedly well understood) subgroups of unitary groups.
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The following result was obtained by Peter and Weyl for compact Hausdorff groups with countable basis at 1 and extended to the case of arbitrary compact Hausdorff groups by van Kampen. We have given a proof in 14.33. 19.1 Peter–Weyl-Theorem. Let G be a compact Hausdorff group. Then for each g ∈ G {1} there exists a natural number n and a continuous homomorphism ϕ : G → U(n, C) such that g ϕ = 1. 2 19.2 Special Case. Let G be a compact Abelian Hausdorff group. Then for each g ∈ G {1} there exists a continuous homomorphism ϕ : G → U(1, C) such that g ϕ = 1. Proof. Without loss of generality, we can assume that Gϕ acts irreducibly on Cn ; that is, there is no nontrivial proper vector subspace that is Gϕ -invariant. In fact, the action of Gϕ is completely reducible: every invariant vector subspace V has an invariant complement, namely V ⊥ . But every irreducible C-linear representation of an Abelian group has rank 1. 2 Using the construction described in 18.3, we obtain the following. 19.3 Corollary. Let C be a compact Hausdorff group, and let N be the set of all kernels of continuous homomorphisms from C to unitary groups. Then ηN is a topological isomorphism from C onto CN . Proof. We have ker ηN = N = {1} by the Peter–Weyl-Theorem 19.1. Thus ηN is injective. We show next that N is a filterbasis. Let α : C → U(n, C) and β : C → U(m, C) be continuous homomorphisms. Embedding U(n, C) × U(m, C) in an obvious way into U(n + m, C), we obtain a continuous homomorphism γ : C → U(n + m, C) by putting cγ = (cα , cβ ). Now ker α ∩ ker β = ker γ ∈ N . As N is a filterbasis, we know that C ηN is dense in CN . Now C ηN is a compact dense subspace of the Hausdorff space CN , and therefore ηN is surjective. Since C is compact and CN is Hausdorff, the bijection ηN is a homeomorphism. 2 If C is a compact Hausdorff group and N is the set of all kernels of continuous homomorphisms from C to unitary groups, then N is a filterbasis, and may therefore be considered as a directed set with respect to reversed inclusion ≥. We put PN := C/N . For N ≤ M, there is a quotient map pN M : C/N → C/M defined by (Nc)pN M = Mc. We have that P : (N , ≤) → CG is a projective system, whose limit is modeled by CN , compare 18.6. Thus we obtain the following corollary to 19.3. 19.4 Compact Hausdorff groups as projective limits of unitary groups. Every compact Hausdorff group models the projective limit (taken in the category CG) of a projective system of closed subgroups of unitary groups, where the bonding morphisms are quotient morphisms.
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If the compact Hausdorff group C is commutative, we infer from 19.2 that the set N1 of all kernels of continuous homomorphisms from C to products of finitely many copies of U(1, C) is a filterbasis with trivial intersection. As U(1, C) is isomorphic to the circle group T, we also obtain the following. 19.5 Compact Abelian Hausdorff groups as projective limits. Every compact commutative Hausdorff group models the projective limit (taken in the category CA as well as in the category CG) of a projective system of closed subgroups of finite powers of the circle group, where the bonding morphisms are quotient morphisms. 19.6 Remark. Using Pontryagin–van Kampen duality, we will see later that every closed subgroup of Tn is topologically isomorphic to a product Ta × F , where F is a finite discrete group and a ≤ n. We note a consequence of 19.5, outside the realm of compact groups. 19.7 Proposition. Let A be a locally compact commutative Hausdorff group, written additively. Then for each a ∈ A {0} there exists a continuous homomorphism χ : A → T such that a χ = 0. Proof. Let a ∈ A {0} and let U be a compact neighborhood of 0 in A. Then V := U ∪ {a} is also a compact neighborhood of 0. Applying 6.31, we find a discrete subgroup D of the subgroup B generated by V such that a ∈ / D and B/D is compact. Let π : B → B/D be the natural map. By 19.2 there is a continuous homomorphism ϕ from B/D to T ∼ = U(1, C) such that (a π )ϕ = 0. As B is open in A, the continuous homomorphism π ϕ : B → T has an extension χ : A → T by 4.21. 2
Approximation of Compact Hausdorff Groups by Lie Groups The closed subgroups of unitary groups are exactly the compact Lie groups. In this sense, Theorem 19.4 provides a method to reduce questions about compact groups in general to (supposedly easier) questions about compact Lie groups. The following is useful in order to avoid handling the complete projective limit in some cases. 19.8 Theorem. Let G be a compact Hausdorff group. (a) For each neighborhood U of the neutral element in G there exists a normal subgroup N of G such that N ⊆ U and G/N is isomorphic to a (closed) subgroup of a unitary group U(n, C) for some natural number n.
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(b) If G is Abelian then for each neighborhood U of the neutral element in G there exists a subgroup N of G such that N ⊆ U and G/N is isomorphic to a subgroup of Tn for some natural number n. Proof. According to the Peter–Weyl Theorem 19.1, we find for each x ∈ G {1} a natural number nx and a continuous homomorphism ϕx : G → U(nx , C) such that x ∈ / ker ϕx . If G is Abelian, we can put nx = 1 for each x. The set K := {ker ϕx | x ∈ G {1}} generates a filterbasis F (consisting just of all in tersections of finitely many members of K), and clearly F = {1}. According to 1.23, this filterbasis converges to 1. This means that for every neighborhood U of the neutral element we find a finite set F ⊆ G{1} such that x∈F ker ϕx ⊆ U . Put n := f ∈F nf . Then f ∈F U(n , C) may be identified with a closed subgroup f defined by g ϕ = (g ϕf )f ∈F is a of U(n, C), and the map ϕ : G → f ∈F U(nf , C) continuous homomorphism whose kernel ker ϕ = f ∈F ker ϕf is contained in U . As G is compact, the image Gϕ is isomorphic to G/ ker ϕ, and assertion (a) is established. Assertion (b) follows from the fact that T and U(1, C) are isomorphic. 2
Totally Disconnected Compact Groups We close this chapter with the observation that compact totally disconnected groups are exactly those that can be described by projective limits of finite groups (that is, the pro-finite groups). Note, however, that a locally compact totally disconnected group may well be simple (for instance, take a simple discrete infinite group), and need not have any nontrivial homomorphisms to finite groups. 19.9 Theorem. Let G be a totally disconnected compact group. Then G models the projective limit of a projective system of finite discrete groups. Proof. According to 4.13, the system N of all open normal subgroups of G has trivial intersection. As the system N is closed with respect to finite intersections, it is a filterbasis. Applying 18.4, we obtain a continuous homomorphism ηN from G to GN with dense image. As G is compact and GN is Hausdorff, we obtain that ηN is an isomorphism, and 18.6 yields the assertion. 2
Exercises for Section 19 Exercise 19.1. Show that for each n ∈ N the groups O(n, R) = {A ∈ Rn×n | AA = 1} and SO(n, R) = {A ∈ O(n, R) | det A = 1} are compact. 2 Hint. Identify the set Rn×n of all n × n matrices over R with Rn . Verify that the product topology is induced by the euclidean metric on Rn . Use the map ϕ : Rn×n → Rn×n defined by Aϕ = AA to show that O(n, R) is closed in Rn×n . Finally, show that O(n, R) is bounded with respect to the euclidean metric.
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Exercise 19.2. Analogously, show that for n ∈ N the groups U(n, C) = {A ∈ Cn×n | AA∗ = 1} and SU(n, C) = {A ∈ U(n, C) | det A = 1} are compact. Exercise 19.3. For n ∈ N and F ∈ {R, C, H}, show that the group SL(n, F) is not compact, except if n ≤ 1. Exercise 19.4. Let p be a prime, and endow Zp with the p-adic topology. For any n ∈ N, endow (Zp )n×n with the product topology, and consider GL(n, Zp ) and SL(n, Zp ) with the induced topologies. Show that (Zp )n×n is a compact ring, and that GL(n, Zp ) and SL(n, Zp ) are compact totally disconnected groups. Exercise 19.5. Exhibit projective systems in CG such that GL(n, Zp ) and SL(n, Zp ) model the respective projective limits.
Chapter F
Locally Compact Abelian Groups 20 Characters and Character Groups Throughout this chapter, we consider the category LCA of locally compact Abelian Hausdorff groups; with continuous group homomorphisms as morphisms. The circle group T = R/Z will play a prominent role. 20.1 Definition. For A ∈ LCA, we denote the set of all morphisms from A to T by A∗ . For a ∈ A and α ∈ A∗ we sometimes use the notation a, α := a α . On the set A∗ , we define an addition by putting a, α + β = a, α + a, β for each a ∈ A and all α, β ∈ A∗ . It is easy to see that this addition turns A∗ into an Abelian group, called the dual (group) of A. The elements of A∗ are also called characters, and A∗ is called the character group of A. On the dual A∗ of A we have the topology induced by the compact-open topology on C(A, T) which makes A∗ a closed subgroup of the topological group C(A, T); see 11.6. For the description of the topology A∗ , the following basis D of (compact) neighborhoods of 0 in T will be convenient. Let π : R → T = R/Z be the natural map. For ε > 0 we put Dε := [−ε, ε]π , and D := {Dε | ε > 0}. For B ∈ LCA and ϕ ∈ Mor(A, B) we define a map ϕ ∗ : B ∗ → A∗ by putting ∗ ϕ β := ϕβ for each β ∈ B ∗ . The map ϕ ∗ is sometimes called the adjoint of ϕ. 20.2 Lemma. For A, B ∈ LCA and ϕ ∈ Mor(A, B), the map ϕ ∗ is a continuous homomorphism from B ∗ to A∗ . ∗
∗
∗
Proof. For β, γ ∈ B ∗ one computes that both (β +γ )ϕ = ϕ(β +γ ) and β ϕ +γ ϕ map a ∈ A to a ϕβ + a ϕγ . Thus ϕ ∗ is a homomorphism. In order to see that ϕ ∗ is continuous, recall from 9.4 that the composition map κ : C(A, B) × C(B, T) → C(A, T) is continuous. Now ϕ ∗ may be regarded as the 2 restriction of κ to the set {ϕ} × B ∗ , and is therefore continuous.
The Topology of the Character Group 20.3 Lemma. Let C be a compact neighborhood of 0 in A, and let δ be a positive real number such that δ < 41 . Then := $C, Dδ % ∩ A∗ is equicontinuous; that is, for every a ∈ A and every ε > 0 there exists a neighborhood W of 0 in A such that for each ω ∈ we have W + a, ω ⊆ Dε + a, ω .
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Proof. It suffices to show that is equicontinuous at 0; in fact, if V is a neighborhood of 0 in A such that V , ⊆ Dε then for each ω ∈ and each a ∈ A the neighborhood V + a satisfies V + a, ω = V , ω + a, ω ⊆ Dε + a, ω . From δ < 41 one infers δ < 21 − δ. We may assume that ε ≤ δ, then ε < 21 − δ, and we find a natural number n such that δ < nε ≤ 21 . Pick a neighborhood V of 0 in A such that V = −V and that every sum of n elements in V belongs to C. In particular, we have V ⊆ C. We claim that V , ⊆ Dε . Assuming that the claim is false, we pick v ∈ V and ω ∈ such that v, ω ∈ / Dε . Then there is γ ∈ ]ε, 1 − ε[ such that v, ω = Z + γ . As V = −V , we may further assume that γ ≤ 21 . From v ∈ V ⊆ C we then infer that γ ≤ δ. As above, we conclude that there exists a natural number m such that δ < mγ ≤ 21 . Now γ > ε means that we can pick m such that m ≤ n, whence mv ∈ C. We reach the contradiction Dδ mv, ω = m v, ω = Z + mγ ∈ / Dδ . 2 20.4 Lemma. The set of all $C, Dε % ∩ A∗ , where C is a compact neighborhood of 0 in A, and 0 < ε < 41 , is a neighborhood basis at 0 of A∗ consisting of compact neighborhoods. Proof. The set := $C, Dε %∩A∗ is equicontinuous, and A, has of course compact closure in the compact set T. According to the Arzela–Ascoli Theorem 9.24, the closure of in C(A, T) is compact. As $C, Dε % and A∗ are closed in C(A, T) by 9.3 and 11.6, we obtain that is compact. It remains to show that each neighborhood U of 0 in A∗ contains a neighborhood of the form $C, Dε %, where C is a compact neighborhood in A, and 0 < ε < 41 . There are finitely many n compact sets C1 , . . . , Cn in A and open sets U1 , . . . , Un in T such that 0 ∈ i=1 $Ci , Ui % ⊆ U. In particular, we have 0 ∈ Ui for each i, and find ε such that 0 < ε < 41 and Dε ⊆ ni=1 Ui . If B is an arbitrary compact neighborhood of 0 in A, we obtain that C := B ∪ ni=1 Ci is a compact neighborhood of 0 in A, and 0 ∈ $C, Dε % ⊆ U. 2 20.5 Theorem. For every locally compact Abelian Hausdorff group A the dual group A∗ is a locally compact Abelian Hausdorff group as well. 20.6 Theorem. For every A ∈ LCA, the following hold. (a) If A is compact then A∗ is discrete. (b) If A is discrete then A∗ is compact. Proof. If A is compact then the open set $A, Dε % consists of 0 alone for each ε < 41 , as D 1 does not contain any nontrivial subgroup of T. Thus A∗ is discrete. : 9 4 If A is discrete, we use 20.4: putting C = {0} we obtain that A∗ = {0}, D 1 5 is compact. 2
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A Natural Transformation We have defined a map ∗ of the object class of the category LCA of all locally compact Abelian Hausdorff groups onto itself; and for each pair (A, B) of objects in LCA a map ∗ from Mor(A, B) to Mor(B ∗ , A∗ ). Let A, B, C be objects of LCA, and consider morphisms ϕ ∈ Mor(A, B) and ψ ∈ Mor(B, C). Then it is easy to see that (ϕψ)∗ = ψ ∗ ϕ ∗ , and that (idA )∗ = idA∗ . This means the following. 20.7 Lemma. We have a contravariant functor
∗:
LCA → LCA.
20.8 Corollary. If ϕ : A → B is an isomorphism in LCA then ϕ ∗ : B ∗ → A∗ is an isomorphism as well. Since the functor ∗ maps the object class of LCA into itself, we can apply it twice. Thus we obtain a covariant functor ∗∗ , defined by A∗∗ = (A∗ )∗ . For every A ∈ LCA, we have a rather natural morphism from A into the double dual A∗∗ , as follows. 20.9 Definition. For every A ∈ LCA we define εA : A → A∗∗ by putting a εA := a, . ; that is, the map a εA maps a character α ∈ A∗ to its evaluation a, α at a. 20.10 Lemma. For each A ∈ LCA, the map εA is a morphism from A to A∗∗ . Proof. Firstly, we have to verify that for each a ∈ A the image a εA is a character of A∗ . The equation a, α + β = a, α + a, β shows that a εA is a homomorphism. The map a εA may be interpreted as the restriction of the map ω : A×A∗ → T given by (a, α)ω = a, α . This map is continuous by 9.8. Thus a εA is continuous, and we have proved that εA is a map from A to A∗∗ . The equation a + b, χ = a, χ +b, χ yields that εA is a homomorphism. In order to show that εA is continuous, it suffices by 3.33 to show that εA is continuous at 0. So let be a compact subset of A∗ , and let U be a neighborhood of 0 in T. For each δ ∈ , we pick an open neighborhood δ of δ in A∗ and an open neighborhood Vδ of 0 in A such that Vδ , δ ⊆ U ; this is possible since the map ω considered above is continuous. As is compact, there exists a finite subset of such that
⊆ ϕ∈ ϕ . Now V := ϕ∈ Vϕ is an open neighborhood of 0 in A, and our choices imply that V , ⊆ U . This means that V εA ⊆ $ , U %, and we have 2 established the continuity of εA . 20.11 Proposition. There is a natural transformation ε = (εA )A∈LCA from the identity on LCA to the functor ∗∗ ; that is, for all A, B ∈ LCA and each morphism ϕ : A → B in LCA, we have εA ϕ ∗∗ = ϕεB . Proof. In order to show that εA ϕ ∗∗ = ϕεB , consider a ∈ A and β ∈ B ∗ . Then on the ∗∗ ∗ ε ε one hand, we have a εA ϕ = ϕ ∗ a εA and β ϕ a A = (ϕβ)a A = a, ϕβ = a ϕ , β . On
20. Characters and Character Groups
the other hand, we have a ϕεB = (a ϕ )εB and β a
ϕεB
177
= a ϕ , β . Thus εA ϕ ∗∗ = ϕεB . 2
Our aim is to prove Pontryagin’s Duality Theorem, which we can now formulate as follows: The natural transformation ε is a natural isomorphism; that is, for each A ∈ LCA, the morphism εA is an isomorphism. The proof of this theorem will be achieved as result of the following two chapters. As a first step, we observe: 20.12 Proposition. For every A ∈ LCA, the following hold. (a) The morphism εA is monic. (b) For each closed subgroup B of A and each a ∈ A B there is a character χ ∈ A∗ such that B ≤ ker χ but a χ = 0. Proof. In view of 15.5, we have to show that εA is injective. So consider a ∈ A{0}. According to 19.7, there is a morphism χ : A → T such that a χ = 0. As χ ∈ A∗ ε and a χ = χ a A , we obtain that a ∈ / ker εA . In order to prove assertion (b), observe that the quotient A/B belongs to LCA. Applying assertion (a) to this quotient, one finds a character ψ ∈ (A/B)∗ such that B+a ∈ / ker ψ. Composing ψ with the natural map πB : A → A/B, we obtain χ = πB ψ with the required properties. 2
Adjoints of Monics, Epics, Embeddings, and Quotients 20.13 Lemma. For each morphism ϕ in LCA the following hold. (a) We have ϕ = 0 exactly if ϕ ∗ = 0. (b) If ϕ ∗ is epic then ϕ is monic. (c) The morphism ϕ is epic exactly if ϕ ∗ is monic. Proof. Consider a morphism ϕ : B → C in LCA. Obviously, ϕ = 0 implies ϕ ∗ = 0. Now assume ϕ ∗ = 0. Then ϕ ∗∗ = 0 yields 0 = εB ϕ ∗∗ = ϕεC . As εC is monic this means ϕ = 0. Now assume that ϕ is not monic. Then we find A ∈ LCA and morphisms α and β from A to B such that α = β but αϕ = βϕ. This means ϕ ∗ α ∗ = (αϕ)∗ = (βϕ)∗ = ϕ ∗ β ∗ . Pick a ∈ A such that a α = a β . By 19.7 we find a character χ ∈ B ∗ such that a αχ = a βχ , and infer α ∗ = β ∗ . Thus we obtain that ϕ ∗ is not epic, and assertion (a) is established.
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Next, assume that ϕ is epic. For χ ∈ ker ϕ ∗ we have ϕχ = 0 = ϕζ , where ζ is the zero in C ∗ . As ϕ is epic, this implies that χ = ζ , and ϕ ∗ is injective and thus monic. Finally, assume that ϕ is not epic. According to 15.7, this implies that B ϕ = C. Then we find χ ∈ C ∗ such that χ = 0 but ker χ ≥ B ϕ . Now χ ∈ ker ϕ ∗ yields 2 that ϕ ∗ is not injective, and therefore not monic. The implication ϕ monic ⇒ ϕ ∗ epic is also true, but will only be deduced as a consequence of Pontryagin duality, see 23.2. We introduce a common generalization of the notions of quotient morphism and of open embedding. 20.14 Definition. Let G and H be topological groups. A continuous homomorphism ϕ : G → H is called proper if it induces an open map from G onto Gϕ ≤ H . A continuous homomorphism ϕ : G → H is proper exactly if it factors as ϕ = πι, where ι : Gϕ → G is the inclusion, and π : G → Gϕ is a quotient morphism (and, therefore, an open map). 20.15 Lemma. Assume that ϕ : B → C is a morphism in LCA. If ϕ is proper, then ∗ ϕ ∗ is proper, as well. Moreover, (C ∗ )ϕ is closed in B ∗ . Proof. Let M be the intersection of B ϕ with a compact neighborhood of 0 in C. We claim that there exists a compact neighborhood N of 0 in B such that M = N ϕ . In fact, we can cover the pre-image of M under ϕ with a collection U of open sets with compact closure. Then {U ϕ | U ∈ U} forms an open covering of the compact set M, recall that ϕ is an open map. Thus some finite subset F of U satisfies {U ϕ | U ∈ F } ⊇ M. Adding at most one element of U to F , we achieve that 0 lies in the interior of F := { U | U ∈ F }. Then F is a compact neighborhood ← of 0 in B such that F ϕ ⊇ M. As M ϕ is a closed neighborhood of 0 in B, the ← intersection N := F ∩ M ϕ satisfies the requirements of the claim. For every ε < 41 , the set X := C ∗ ∩ $M, Dε % is a compact neighborhood of 0 ∗ ∗ in C ∗ . Hence Y := X ϕ = (C ∗ )ϕ ∩ $N, Dε % is a compact neighborhood of 0 ∗ ∗ in (C ∗ )ϕ . Thus Y generates a locally compact open subgroup of (C ∗ )ϕ . By the ∗ Open Mapping Theorem 6.19, the morphism ϕ induces an open map from the open ∗ subgroup generated by X in C ∗ onto the open subgroup generated by Y in (C ∗ )ϕ , 2 and our assertion is proved. 20.16 Corollary. If the morphism ϕ : B → C is a quotient map then ϕ ∗ : C ∗ → B ∗ ∗ is an embedding; that is, it induces an isomorphism from C ∗ onto (C ∗ )ϕ . Proof. Every quotient map is surjective. Thus ϕ is epic, and ϕ ∗ is monic by 20.13, and injective by 15.5. As ϕ is open by 6.2, we obtain that ϕ ∗ induces an open map ∗ from C ∗ onto (C ∗ )ϕ . 2
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20.17 Corollary. Assume that the morphism ϕ : B → C is an open embedding; that is, an open injection. Then ϕ ∗ is a quotient map. Proof. As ϕ is an embedding, we can identify B and B ϕ . Since B is open in C and T is divisible, every character of B extends to a character of C, compare 4.21. This means that every element β ∈ B ∗ is the restriction of some γ ∈ C ∗ ; that is, ∗ β = ϕγ = γ ϕ . Thus ϕ ∗ is surjective. As ϕ is open, we infer that ϕ ∗ is an open surjection; that is, a quotient map. 2 20.18 Corollary. If ϕ : B → C is monic and C is discrete then ϕ ∗ is a quotient map. 20.19 Definition. Let X be a subset of A ∈ LCA. The annihilator of X (in A∗ ) is the subset X⊥A := {α ∈ A∗ | X, α = 0} of A∗ . If no confusion is possible, the annihilator is denoted simply by X⊥ . The set X⊥⊥ = {a ∈ A | a, X⊥ = 0} is called the double annihilator; this construction is almost dual to that of the annihilator. 20.20 Lemma. Let A be an object of LCA, and let X be an arbitrary subset of A. Then the following hold. (a) The annihilator X⊥ is a closed subgroup of A∗ . (b) The double annihilator X⊥⊥ is a closed subgroup of A; in fact, it is the smallest closed subgroup of A containing X. (c) If Y is a subset of A such that Y ⊆ X then Y ⊥ ⊇ X⊥ and Y ⊥⊥ ⊆ X⊥⊥ . Proof. It is easy to see that X⊥ and X ⊥⊥ are subgroups of A∗ and A, respectively. For each x ∈ X, the set {x}⊥ is the pre-image of {0} under the continuous evaluation map ϕ : A∗ → T defined by α ϕ = x, α . Thus X ⊥ = x∈X {x}⊥ is closed in A∗ . Similarly, one sees that the double annihilator X⊥⊥ is a closed subgroup of A. Let B be the smallest closed subgroup of A containing X. Then B ≤ X ⊥⊥ . For each a ∈ A B there is a character χ ∈ A∗ such that B ≤ ker χ but a, χ = 0, see 20.12. Then χ belongs to X⊥ , and we obtain that a ∈ / X⊥⊥ . Thus B = X⊥⊥ . 2 Assertion (c) is obvious. 20.21 Proposition. Let ι : B → C be an embedding, and let κ : C → D be a quotient morphism such that B ι = ker κ. Then ker ι∗ = (B ι )⊥ , and κ ∗ induces an isomorphism from D ∗ onto ker ι∗ . ∗
Proof. According to 20.16, the morphism κ ∗ is an embedding. From B ι , (D ∗ )κ = ∗ B ικ , D ∗ = 0 we conclude (D ∗ )κ ≤ (B ι )⊥ . Conversely, we infer from χ ∈ (B ι )⊥ that B ι ≤ ker χ . Therefore, there exists a morphism β : D → T such that ∗ χ = κβ = β κ , which is continuous since κ is a quotient map. This means ∗ 2 (B ι )⊥ ≤ (D ∗ )κ .
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20.22 Corollary. For every C ∈ LCA and every closed subgroup B of C we have that (C/B)∗ and B ⊥ are isomorphic as topological groups. An isomorphism is induced by the adjoint of the natural map.
20.23 Remark. It is convenient to express the assertions of 20.21 by means of exact sequences: assume that ι : B → C is an embedding and κ : C → D is a quotient morphism. From 20.13 and 20.16 we know that ι∗ is epic, and that κ ∗ is an embedding. Proposition 20.21 says that if the sequence {0}
/B
ι
/C
κ
2, D
/ {0}
∗ ?_D o
{0}
is exact then the sequence B∗ o
ι∗
C∗ o
κ∗
is exact. If we assume in addition that ι is an open embedding, we infer from 20.15 that ι∗ is a quotient map, and therefore surjective. Now exactness of the sequence {0}
/ B ◦
ι
/C
κ
2, D
/ {0}
∗ ?_D o
{0}
implies exactness of the sequence {0} o
B ∗ lr
ι∗
C∗ o
κ∗
This result will be strengthened in 23.5 below: the additional assumption that ι be open is not necessary. In fact, we will see in 23.2 that every embedding ι in LCA dualizes to a quotient ι∗ , and that every monic η dualizes to an epic η∗ .
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Exercises for Section 20 Exercise 20.1. Show that D 1 contains no nontrivial subgroup of T. 4
Exercise 20.2. Determine the character group of Z(n). Exercise 20.3. Determine the character group of Z. Exercise 20.4. Let D denote the group (Q, +), endowed with the discrete topology. Show that D ∗ is compact and torsion-free. Hint. Use q, nχ = nq, χ . Exercise 20.5. For A ∈ LCA, endow Aut(A) with the modified compact-open topology. Show that ∗ induces a continuous injective anti-homomorphism from Aut(A) to Aut(A∗ ). Exercise 20.6. Find a continuous injective homomorphism from Aut(A) to Aut(A∗ ). Exercise 20.7. Prove that if εA is an isomorphism then Aut(A) and Aut(A∗ ) are isomorphic.
21 Compactly Generated Abelian Lie Groups This section contains the proof of Pontryagin’s Duality Theorem for an important subcategory of LCA. We consider the full subcategory CGAL whose objects are all objects of LCA that are isomorphic to a group of the form Ra × Tb × Zc × F , where a, b, c are nonnegative integers, and F is a finite discrete commutative group. We do not assume that the reader is familiar with Lie Theory, but simply remark (for those who know what that means) that these groups are in fact Abelian Lie groups, and are exactly the compactly generated ones among these. In 21.18 below it will be shown that the class CGAL consists exactly of those compactly generated elements of LCA that ‘have no small subgroups’. The Duality Theorem for CGAL will be derived by proving it first for groups isomorphic to R, T, Z or Z(n) = Z/nZ for some n ∈ N, and then extending it to finite products. The next two lemmas will be needed for this extension. 21.1 Discussion. For A, B ∈ LCA, let α : A × B → A and β : A × B → B be the natural projections, and define α : A → A × B and β : B → A × B by a α = (a, 0) and bβ = (0, b), respectively. Then we have αβ = 0 and βα = 0, and idA×B = αα + ββ. A× O Bo α
? _ A
α
β β
2, ?_B
β∗
/
(A ×O B)∗ s s β ss s s ss α∗ α∗ sssηA,B s s ysss ,2 ? ∗ ∗ A × B∗ A BLR ∗ o _
∗
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For χ ∈ (A × B)∗ , we have αχ ∈ A∗ and βχ ∈ B ∗ , respectively. We obtain a map ηA,B : A × B ∗ → A∗ × B ∗ by putting χ ηA,B = (αχ , βχ). It is easy to see that ηA,B is a morphism in LCA; in fact, it is obtained from the universal property −1 of the product A∗ × B ∗ . An easy computation shows that the inverse ηA,B is given −1
−1 by (ϕ, ψ)ηA,B = αϕ + βψ. Continuity of both ηA,B and ηA,B is secured by 9.8.
Thus we have obtained the following. 21.2 Lemma. For A, B ∈ LCA, we have that ηA,B is an isomorphism from 2 (A × B)∗ onto A∗ × B ∗ . 21.3 Lemma. Assume that A and B are objects of LCA such that εA and εB are isomorphisms. Then εA×B is an isomorphism. Proof. The isomorphism η := ηA,B : (A × B)∗ → A∗ ×B ∗ yields an isomorphism η∗ : (A∗ × B ∗ )∗ → (A × B)∗∗ . By our assumption, mapping (a, b) to (a εA , bεB ) ∗ defines an isomorphism ε from A × B onto A∗∗ × B ∗∗ . Now εA×B = ε ηA−1 ∗,B ∗ η shows that εA×B is an isomorphism, as claimed. 2 Before we proceed to determine the duals of R and T, we determine the possible kernels of morphisms from Rn to Hausdorff topological groups. 21.4 Proposition. Let A be a closed subgroup of Rn . Then the following hold. (a) If A is not discrete then there exists x ∈ R {0} such that Rx ≤ A. (b) If A is subset Z of Rn such that discrete then there is a linear independent c ∼ A = b∈Z Zb. In particular, we have A = Z for c = |Z|. (c) In the general case, a basis B of Rn and disjoint subsets R and Z of B there is such that A = b∈R Rb + b∈Z Zb. Proof. Assume that A is not discrete, and choose a sequence of nonzero elements aν in A converging to 0. Then sν := aaνν accumulates at some vector x, since the sphere {v ∈ Rn | v = 1} is compact. Without loss of generality, we may assume that aν < ν1 and sν − x < ν1 for each ν ∈ N. For r ∈ [0, 1] and ε > 0 pick ν such that ν2 < ε. Then there is an integer z such that z aν ≤ r < (z + 1) aν . Using the triangle inequality, we infer that rx − zaν < ν2 < ε. Thus every neighborhood of rx contains an element of A, and Rx = [0, 1]x ≤ A. This establishes assertion (a). Now let D be a discrete subgroup of Rn . We claim that D ∼ = Zk for some natural number k ≤ n. We proceed by induction on the dimension of the vector subspace W spanned by D. For W = {0}, there is nothing to do. If W = {0}, pick an element m of minimal length in D {0}. The idea is to reduce the problem to
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183
the quotient W/Rm. To this end, we have to check that the image D π under the natural map π : W → W/Rm is a discrete subgroup. Let U be the open ball of radius m
2 around 0 in W . For each d ∈ D∩(Rm+U ), we pick r ∈ R such that d − rm ∈ U . Then we find z ∈ Z such that rm − zm ∈ U and conclude d = zm from d − zm < m and minimality of m. Thus we have D ∩ (Rm + U ) = Zm, and infer that D π is discrete. The induction hypothesis yields D/Zm ∼ = Zk +1 by 6.30. = Zk for some natural number k ≤ n − 1, and D ∼ Returning to the general case, we denote the maximal vector subspace contained in A by M. Then assertion (a) yields that A/M is a discrete subgroup of Rn /M, and we already know A/M ∼ = Zk for some k ≤ n − dim M. As A/M is discrete, the subgroup M is open in A. Applying 4.22, we find a closed subgroup K of A such that M ∩ K = {0} and M + K = A; thus K ∼ = A/M ∼ = Zk and A ∼ = M × K. 2
Duals of Elementary Groups 21.5 Lemma. The set R∗ consists exactly of the maps μv = (r → vr + Z), where v ∈ R is arbitrary. Proof. Let α be a character of R. It suffices to consider the case α = 0. Then Rα is a connected subgroup of T, and contains an interval. This interval generates the connected group T, and we obtain that α is surjective. If r α = Z + 21 then 2r is a nontrivial element of ker α. From
α 21.4 we know that there exists w ∈ R {0} such that ker α = Zw. Then w2 = Z + 21 , and up to replacing w by −w we have α α that w4 = Z + 41 . As w8 has to be contained in the connected component of α T Z, Z + 41 that does not contain Z + 21 , we have w8 = Z + 81 . Inductively, w α = Z + 21n for each natural number n. Thus α coincides we obtain that 2n with μ 1 , because these two morphisms coincide on the dense subgroup generated w 2 by 2wn | n ∈ N . 21.6 Corollary. The set T∗ consist exactly of the maps ρz = (t → zt), where z is an arbitrary integer. Proof. Let π : R → T be the natural map. For every character χ of T we have that πχ is a character of R. Thus we find v ∈ R such that π χ = μv . From 1 ∈ Z = ker π ≤ ker π χ = v1 Z we infer v ∈ Z, and obtain that χ maps Z + r to Z + vr = v(Z + r). 2 Homomorphisms from cyclic groups to arbitrary groups are determined by the image of a generator. As the image of an element of order n has order dividing n, we obtain the following.
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21.7 Lemma. (a) The set Z∗ consists exactly of the maps αt = (z → zt), where t ∈ T is arbitrary.
(b) The set Z(n)∗ consists exactly of the maps νz = nZ + a → Z + za n , where z ∈ {1, . . . , n} is arbitrary. 21.8 Proposition. We have the following isomorphisms. (a) ηR : R → R∗ , mapping v to μv , (b) ηT : Z → T∗ , mapping z to ρz , (c) ηZ : T → Z∗ , mapping t to αt , (d) ηZ(n) : Z(n) → Z(n)∗ , mapping nZ + z to νz . Proof. It is easy to see that the map ηR is a homomorphism. According to 21.5, it is also surjective. If v belongs to ker ηR we have vr ∈ Z for each r ∈ R, and obtain v = 0. Thus ηR is injective. Analogously, one sees that ηT , ηZ and ηZ(n) are bijective homomorphisms. It remains to show that ηA is continuous at 0 and open for each A ∈ {R, T, Z} ∪ {Z(n) | n ∈ N {0}}. Consider the pre-image of $C, Dε % under ηR , where C is a compact subset R, and ε > 0. Then C ⊆ [−a, a] for some a ∈ R, and we see that
of η − aε , aε R ⊆ $C, Dε %. This yields that ηR is continuous at 0. By the Open Mapping Theorem 6.19, the bijective morphism ηR is an isomorphism. The morphisms ηT and ηZ(n) are bijective homomorphisms from discrete groups onto discrete groups, and therefore isomorphisms. In order to see that ηZ is continuous, consider a compact (that is, finite) subset C of Z and ε > 0. For η m = max {|c| | c ∈ C} we obtain D εZ ⊆ $C, Dε %. As T is compact, continuity of m 2 the bijection ηZ implies that it is also open. 21.9 Lemma. For A ∈ {R, T, Z} ∪ {Z(n) | n ∈ N {0}}, the map εA is an isomorphism. ∗
−1 Proof. Evaluating at a ∈ A, one obtains the equalities εR = ηR (ηR ) , εT = ∗ ∗ ∗ −1 −1 −1 ηZ (ηT ) , εZ = ηT (ηZ ) , and εZ(n) = ηZ(n) (ηZ(n) ) . As ηA is an isomorphism ∗
−1 ) is an isomorphism as well, and for each group A in question, we know that (ηA the assertion follows. 2
Combining 21.9 with 21.3, and using the fact that every finitely generated Abelian group is the direct product of finitely many cyclic groups, we obtain: 21.10 Theorem. Pontryagin duality holds in the category CGAL; that is, for every object A ∈ CGAL the map εA is an isomorphism.
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185
21.11 Local homomorphisms. Let A be an Abelian topological group, and let n be a natural number. Assume that there exist neighborhoods U and V of 0 in Rn and A, respectively, and a map ϕ : U → V with the following property: whenever x, y ∈ U are such that x + y ∈ U then x ϕ + y ϕ = (x + y)ϕ . Then the following hold: (a) There is a neighborhood U ⊆ U of 0 in Rn and a homomorphism ψ : Rn → A such that ψ|U = ϕ|U . (b) If ϕ is continuous then ψ is continuous. (c) If ϕ is a homeomorphism then there exists a discrete group D and natural numbers a, b such that a + b = n, and A is isomorphic to Ra × Tb × D. Proof. As U is a neighborhood of 0 in Rn , we find ε > 0 such that the ball U of radius ε around 0 is contained in U (using any euclidean norm on Rn ). For every v ∈ Rn we find some positive integer m such that m1 v lies in U . If an ϕ extension ψ of ϕ|U exists, it has to satisfy v ψ = m m1 v . Although this setting seems to depend on the choice of m, a map ψ : Rn → A is defined by it: if m and k are positive integers with m1 v ∈ U and k1 v ∈ U we take a common multiple l = xm = yk of m and k and observe that 1l v lies in U . Then jl v lies in U for each ϕ ϕ ϕ ϕ natural number j ≤ max{x, y}, and m m1 v = m xl v = l 1l v = k yl v = 1 ϕ k kv . For v, w ∈ Rn we find a positive integer k such that k1 v, k1 w and k1 (v + w) lie in U . This yields that ψ is a homomorphism (here we use that A is Abelian). If ϕ is continuous then the homomorphism ψ is continuous at 0 and therefore continuous, and assertion (a) is established. If ϕ is injective then the kernel ker ψ is discrete, and of the form bi=1 Zvi for linearly independent v1 , . . . , vb , compare 21.4. If U ϕ is a neighborhood in A then B := (Rn )ψ is an open subgroup of A. If ϕ is a homeomorphism, we even obtain that B and A are locally compact, and ψ induces a quotient map from Rn onto B. Thus B is isomorphic to Rn−b × Tb . Applying 4.22, we find a discrete subgroup D of A such that A = B + D and B ∩ D = {0}, and A is isomorphic to Rn−b × Tb × D. 2 21.12 Lemma. Let A ∈ LCA be compactly generated, and assume that there is a discrete subgroup B of A such that A/B belongs to CGAL. Then A belongs to CGAL. Proof. Choose a, b, c and F such that A/B is isomorphic to L = Ra ×Tb ×Zc ×F , and let π : A → L be the induced quotient map.
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Let κ : Ra+b → L0 = Ra × Tb × {(0, 0)} be the natural map. /A | | π || | | _ <| | y /Lo A/B ∼
Ra+b κ
_ L0
=
Aiming at an application of 21.11, we search for neighborhoods U and V in Ra+b and A, respectively, and a map ϕ such that the requirements in 21.11 are satisfied. Pick a neighborhood W of 0 in A such that B ∩ W = {0}, and pick an open neighborhood W of 0 such that W − W − W ⊆ W . Then π |W is injective. As the quotient map π is continuous and open, it induces a homeomorphism from W onto W π . The connected component L0 is open in L. Therefore U := L0 ∩ W π ← is open both in L0 and in W π . We conclude that V := U π ∩ W is open in A, and that the inverse of π |W induces a homeomorphism γ from U onto V . Consider elements x π and y π of U = V π such that x π + y π lies in U . Then π x +y π = (x+y)π = zπ for some z ∈ V , and z−x−y ∈ (W −W −W )∩B = {0} yields x + y = z ∈ V . Replacing V by a still smaller neighborhood if necessary, we can achieve that ← μ := κ|U κ ← induces a homeomorphism from U κ onto U . Now the map ϕ := μγ satisfies the assumptions of 21.11 (c), and we conclude A ∼ = Rk × Tl × D with a discrete group D. As the group A is compactly generated, its quotient D is compactly generated as well. Since D is discrete, this means that D is finitely generated. Thus D and A belong to CGAL. 2 The next result could also be derived from 21.4. We take the opportunity to present a proof that exhibits Pontryagin duality (as far as we have established it yet) at work. 21.13 Proposition. Let n be a natural number, and let A be a closed subgroup of Tn . Then there exist a natural number m and a finite group F such that m ≤ n and A is isomorphic to Tm × F . Proof. Let i : A → B := Tn be the embedding, and let q : B → C := B/A be the natural map. We know that εB is an isomorphism. As A and C are compact, the monomorphisms εA and εC are embeddings. A _
εA
i
q
εB ∼ =
A∗∗
/B
i ∗∗
/ B ∗∗
2, C _
q ∗∗
εC
/ C ∗∗
21. Compactly Generated Abelian Lie Groups
187
As we have not established duality for A, we have to fight against a phantom: we do not know that i ∗∗ is monic. However, we claim that A∗∗ is the direct product of AεA and ker i ∗∗ . In fact, injectivity of iεB = εA i ∗∗ implies AεA ∩ ker i ∗∗ = {0}. Each ∈ A∗∗ is mapped to 0 by i ∗∗ εB −1 qεC = i ∗∗ q ∗∗ = (iq)∗∗ . Since εC ∗∗ −1 is injective, this means i εB ∈ ker q = Ai . Therefore, we find a ∈ A such ∗∗ ∗∗ ∗∗ −1 that i εB = a i , and from i = a iεB = a εA i we obtain that belongs to ε ∗∗ A A + ker i . Thus the claim is established. ∗ The morphism i ∗ induces a surjection p onto D := (B ∗ )i . Let j : D → A∗ be the embedding. The groups A∗ and B ∗ are discrete. Therefore p is a quotient and j is an open embedding. From 20.16 and 20.17 we know that p ∗ is an embedding and that j ∗ is a quotient morphism. { Nn {{ { j {{ {{ { { { { }{ ∗ o A
A _ εA
D [f C CC CC CCp CC CC CC i∗
i
B∗ o
/B
q∗
q
εB ∼ =
/ B ∗∗ A∗∗C ∗∗ = i CC {{ CC {{ CC { CC {{ {{ p∗ j ∗ CC { CC {{ & . { D∗
C∗
2, C _ εC
q ∗∗
/ C ∗∗
As p ∗ is monic, we have ker j ∗ = ker i ∗∗ . In view of our observation that A∗∗ is the direct product of AεA and ker i ∗∗ , we infer that εA j ∗ : A → D ∗ is an isomorphism. The discrete group D is a quotient of B ∗ ∼ = Zn . Therefore, there exists a natural number m ≤ n and a finite group F such that D is isomorphic to Zm × F . Now A∼ 2 = Tm × F . = D∗ ∼ 21.14 Remark. If we had established duality in full strength, we could use a much / Tn be the inclusion map of a closed shorter argument, as follows. Let ι : A subgroup A. Then ι∗ is a quotient map from (Tn )∗ ∼ = Zn onto A∗ . Consequently, we have A∗ ∼ = Zm × F , where F is a finite group of rank at most n − m, and obtain
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F Locally Compact Abelian Groups
A∼ = Tm × F . At the present stage, our problem is that we do not yet know = A∗∗ ∼ that the adjoint of an embedding is a quotient, and we do not know that εA is an isomorphism before we know A ∈ CGAL. We can rephrase the result 21.13 in a nice way, as follows. 21.15 Theorem. A group A ∈ LCA is finitely generated exactly if its dual has an embedding into Tn for some natural number n. In fact, one can choose n as the number of generators for A. Proof. A finitely generated group is at most countable. Therefore, any finitely generated locally compact Hausdorff group is discrete, see 3.13. Thus every finitely generated element A ∈ LCA is a quotient of Zn , where n is the number of generators. Dualizing the quotient map we obtain an embedding as claimed. The reverse statement is 21.13. 2
Approximation of Locally Compact Abelian Hausdorff Groups by Lie Groups 21.16 Theorem. Let A ∈ LCA be compactly generated. Then for each neighborhood U of 0 there exists a subgroup K of A such that K ⊆ U and A/K ∈ CGAL. Proof. Let U be a neighborhood of 0, and choose a neighborhood V of 0 such that V + V + V ⊆ U and V = −V . Without loss, we may assume that U is compact. Pick a compact subset C ⊆ A such that A = C ; without loss, we may assume U ⊆ C. According to 6.31, there exists a discrete subgroup B of A such that B ∩ C = {0} and A/B is compact. Let κ : A → A/B denote the natural map. Then V κ is a neighborhood of 0 in Aκ , and by 19.8 we find a normal subgroup N of Aκ such that N ⊆ V κ and Aκ /N is isomorphic to a closed subgroup of Tn for some natural number n. From 21.13 we then know that there exist a natural number a ≤ n and a finite group F such that Aκ /N ∼ = Tn × F . ← κ Putting M := N , we observe that the continuous bijection M+a → N +a κ is an isomorphism from the (compact) group A/M onto Aκ /N , and B ≤ M ⊆ B +V . We claim that M ∩ V is a subgroup of A. Firstly, we have M ∩ V = −(M ∩ V ). Secondly, for x, y ∈ M ∩ V we obtain from x + y ∈ M + M = M ⊆ B + V that there exists z ∈ V such that x + y − z ∈ (V + V + V ) ∩ B ⊆ U ∩ B = {0}. Thus x + y = z ∈ M ∩ V , and the claim is established. Now M ⊆ B + V implies M = B + (M ∩ V ). Abbreviating K := M ∩ V , we have K ⊆ U and K ∩ B = {0}. Moreover, we obtain that M/K ∼ = B is discrete. As (A/K)/(M/K) ∼ = A/M ∼ = Aκ /N belongs to CGAL, we conclude from 21.12 that A/K belongs to CGAL, as well. 2
21. Compactly Generated Abelian Lie Groups
189
21.17 Corollary. For every A ∈ LCA there are an open subgroup U ≤ A and a compact subgroup K ≤ U such that U/K ∈ CGAL. Recall that a topological group G is said to have no small subgroups if there is a neighborhood U of the neutral element in G such that the trivial group is the only subgroup of G that is contained in U . According to 6.15, this is an extension property. Any discrete group has no small subgroups. Evidently, the groups R and T have no small subgroups. Therefore the class CGAL consists of groups with no small subgroups. Using 21.16, we obtain a reverse statement: 21.18 Theorem. The elements of CGAL are exactly those groups in LCA that are compactly generated and have no small subgroups. Proof. It only remains to show that any compactly generated A ∈ LCA having no small subgroups lies in CGAL. According to 21.16, any neighborhood U of 0 in A contains a subgroup K such that A/K ∈ CGAL. Picking a neighborhood U that contains no subgroup except {0}, we obtain A ∈ CGAL. 2 Taking (Q, +) or ZN with the discrete topology, one sees that the assumption ‘compactly generated’ cannot be dispensed with in 21.18. 21.19 Theorem. The category CGAL is closed with respect to forming closed subgroups or quotients by these, and closed with respect to extensions (inside TA). Proof. We treat extensions first. Assume that A ∈ TA has a subgroup B such that both B and A/B belong to CGAL. Then both B and A/B are locally compact, compactly generated and have no small subgroups. These properties are extension properties by 6.15, and we obtain that A ∈ CGAL by 21.18. Now consider a closed subgroup B of L = Ra × Tb × Zc × F ∈ CGAL. As F is finite, there is an embedding of F into Td for some natural number d. Thus we have an embedding j : L → Ra+c × Tb+d . Let p : Ra+c+b+d → Ra+c × Tb+d be the natural map, and let D be the pre-image of B j under p. Then p induces a quotient map q : D → B. According to 21.4, the group D belongs to CGAL, and is therefore compactly generated. Its quotient B is thus compactly generated. By 6.14, the group B has no small subgroups, and 21.18 yields B ∈ CGAL. Finally, let L be an element of CGAL, and let p : L → Q be a quotient morphism with closed kernel. Then p ∗ : Q∗ → L∗ is an embedding. This yields Q∗ ∈ CGAL and Q∗∗ ∈ CGAL. Since εQ : Q → Q∗∗ is monic, we infer that Q has no small subgroups. Being the quotient of a compactly generated group, the group Q is compactly generated, and an element of CGAL. 2 21.20 Corollary. The class CGAL is the smallest subclass of LCA containing the group R and being closed with respect to the forming of closed subgroups, quotients by closed subgroups, and finite products.
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F Locally Compact Abelian Groups
See 23.14 for another characterization of CGAL inside LCA: the members of CGAL are just those that are compactly generated and have a compactly generated dual. It is natural to ask the question whether closed subgroups of compactly generated locally compact groups are compactly generated. Outside the realm of Abelian groups, this is false. In LCA, however, we have such a theorem. We postpone this discussion until we have the right tools, see 23.13.
Exercises for Section 21 Exercise 21.1. Prove 21.7 in detail, paying special attention to the problem whether the maps in question are well-defined. Exercise 21.2. Determine the duals of R× and C× . Exercise 21.3. Assume that A ∈ LCA has no small subgroups. Show that there is an open subgroup B of A such that B ∈ CGAL. Exercise 21.4. Let A be an Abelian topological group, and assume that there is a subgroup B of A and natural numbers a and b such that B and A/B are isomorphic to Rb and Ra , respectively (as topological groups). Show that A is isomorphic to Ra+b . Hint. Show that A is locally compact and connected, contains no nontrivial compact subgroup, and belongs to CGAL. Exercise 21.5. Show that every group in CGAL contains a unique subgroup that is maximal among the compact subgroups. Exercise 21.6. Verify in detail that each group in CGAL is embedded in some group of the form Ra × Tb . Exercise 21.7. Consider the group Zp with the p-adic topology. For each neighborhood U of 0 in Zp exhibit a subgroup K of Zp such that K ⊆ U and Zp /K ∈ CGAL. Exercise 21.8. Show that Zp does not belong to CGAL. Exercise 21.9. Show that Q, endowed with the discrete topology, does not belong to CGAL. Exercise 21.10. Prove that there is no topology T such that (Q, T ) ∈ CGAL. Exercise 21.11. Exhibit examples of groups in LCA CGAL having no small subgroups, and examples of compactly generated groups in LCA CGAL.
22. Pontryagin’s Duality Theorem
191
22 Pontryagin’s Duality Theorem In this section the proof of the famous Duality Theorem for locally compact Abelian Hausdorff groups is completed. We already know that duality holds for the subcategory CGAL of LCA; that is, the morphism εA is an isomorphism for each A ∈ CGAL. The next step will be to show that εA is an isomorphism for each discrete Abelian group A. To this end, we describe A by a direct limit of finitely generated discrete Abelian groups; that is, a direct limit of elements in CGAL. Let A be a discrete Abelian group. Consider the set J of all finitely generated subgroups of A, ordered by inclusion. Then (J, ≤) is a directed set. For F ∈ J , put DF := F and for E ≤ F ∈ J let dE,F : E → F be the inclusion. Then D is a directed system over DA, and A models the direct limit of D. In fact, the inclusions δF : F → A form the required cone δ. In order to describe the dual of A, define a projective system P ∈ PSJ (CA) as follows: put PF = F ∗ and pE,F = (dE,F )∗ : F ∗ → E ∗ whenever E ≤ F . Then δ ∗ := ((δF )∗ )F ∈J is a cone over P , and the universal property of the projective limit of P yields a morphism ϕ := δ∗ : A∗ → lim P such that ϕπF ∗ = (δF )∗ for each F ∈ J . ←−
22.1 Lemma. The morphism ϕ : A∗ → lim P is an isomorphism. ←−
Proof. We already know that ϕ is a morphism in LCA. In order to see that ϕ is monic, ∗ consider γ ∈ ker ϕ. For each F ∈ J we obtain 0 = γ ϕπF ∗ = γ (δF ) = δF γ = γ |F . From A = J it thus follows that γ = 0. For each F ∈ J , we have that δF is an open embedding. Thus (δF )∗ is a quotient morphism. According to 18.4, this implies that ϕ is epic. As A∗ is compact, we conclude that ϕ is an isomorphism. 2 For every F ∈ J , we have that πF ∗ is a quotient morphism, and infer from 20.16 that αF := εF (πF ∗ )∗ is monic. The family α = (αF )F ∈J forms a cone over the directed system D with vertex (lim P )∗ . The universal property of direct limits ←−
asserts that there is a morphism α : A → (lim P )∗ such that δF α = αF for each ←− F ∈ J. α = a δZ α = a αZ For a ∈ ker α the group Z := a belongs to J , and 0 = a implies a = 0 since αF is monic. Thus α is monic. Let U be a neighborhood of 0 in T that contains no subgroup except the trivial one. For every character χ ∈ (lim P )∗ we find a compact neighborhood V of 0 in lim P such that V χ ⊆ U . ←− ←− According to 18.2, there exists F ∈ J such that K := ker πF ∗ ⊆ V . As the image K χ is a subgroup of T contained in U , we know that K ≤ ker χ. Thus we find a morphism γ : F ∗ → T such that χ = πF ∗ γ . As εF is an isomorphism, we find ∗ ∗ α belongs f ∈ F such that γ = f εF . Now χ = γ (πF ∗ ) = f εF (πF ∗ ) = f αF = f δF
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F Locally Compact Abelian Groups
α , and we have shown that to A α is surjective. As both A and (lim P )∗ are discrete, ←− this means that α is an isomorphism.
22.2 Proposition. For every discrete Abelian group A, the morphism εA is an isomorphism. Proof. We claim that εA = α ϕ ∗ ; as α and ϕ are isomorphisms, this implies that εA is an isomorphism. Pick a ∈ A. Then the group F := a belongs to J , and ∗
∗
αϕ α = ϕa δF = ϕa αF = ϕa εF (πF ∗ ) = ϕπF ∗ a εF = δF ∗ a εF . a
For arbitrary γ ∈ A∗ , this yields 0 / 0 / 0 / / ∗ ∗0 αϕ∗ = γ , (δF )∗ a εF = γ (δF ) , a εF = a, γ (δF ) = a, δF γ = a, γ , γ , a 2
and the claim is established. 22.3 Lemma. For every A ∈ LCA, we have εA∗ (εA )∗ = idA∗ .
/ ∗0 Proof. For arbitrary a ∈ A and arbitrary α ∈ A∗ , we compute a, α εA∗ (εA ) = a, εA α εA∗ = a εA , α εA∗ = α, a εA = a, α . 2
22.4 Proposition. For every A ∈ CA, the morphism εA is an isomorphism. Proof. As A is compact, its dual A∗ is discrete, and εA∗ is an isomorphism by 22.2. Thus (εA )∗ = (εA∗ )−1 is an isomorphism, and εA is an epimorphism by 20.13. As epics from compact groups to Hausdorff groups are quotients, we conclude that εA is an isomorphism. 2 22.5 Proposition. If A ∈ LCA is compactly generated then εA is an isomorphism. Proof. According to 21.16, every neighborhood of 0 in A contains a subgroup K such that A/K ∈ CGAL. As A/K is Hausdorff, the subgroup K is closed in A. Starting with a compact neighborhood, we obtain that K is compact. Now the embedding ι : K → A and the quotient q : A → L := A/K give an exact sequence {0}
/K
i
q
/A
2, L
/ {0}.
As ε is a natural transformation from the identity to the functor commuting diagram K εK
i
K ∗∗
/A
i ∗∗
q
εA
/ A∗∗
2, L
q ∗∗
εL
/ L∗∗
∗∗ ,
we have a
22. Pontryagin’s Duality Theorem
193
We know that εK and εL are isomorphisms, and that εA is monic. This yields that ∗∗ i ∗∗ = εK −1 iεA is monic. From qεL = εA q ∗∗ we infer (AεA )q = L∗∗ , and that ∗∗ the pre-image AεA + ker q ∗∗ of AεA q under q ∗∗ coincides with A∗∗ . According to 20.16, the morphism q ∗ is an embedding. The fact that i ∗∗ is monic implies that i ∗ is epic, see 20.13. Since K ∗ is discrete, this means that i ∗ is a quotient map. Therefore i ∗∗ is an embedding. Applying 20.23 to the exact sequence {0} o
K ∗ lr
i∗
A∗ o
q∗
∗ o ?_L
{0}
we obtain that the sequence {0}
/ K ∗∗
i ∗∗ ∗∗
/ A∗∗
q ∗∗
/ L∗∗
∗∗
is exact, and ker q ∗∗ = (K ∗∗ )i = K εK i = K iεA . Thus A∗∗ = AεA + ker q ∗∗ = AεA , and εA is surjective. As A is compactly generated, the Open Mapping Theorem 6.19 yields that εA is an isomorphism. 2 22.6 Theorem: Pontryagin duality for LCA. The family ε = (εA )A∈LCA is a natural isomorphism from the identity to the functor ∗∗ ; that is, the morphism εA is an isomorphism for each A ∈ LCA. Proof. Pick a compact neighborhood U of 0 in A, and let i : B → A denote the inclusion of the subgroup B generated by U . Then i is an open map, and D := A/B is discrete. Let q : A → D denote the natural map. Applying 20.23 to the exact sequence {0}
/ B ◦
i
/A
q
2, D
A∗ o
q∗
∗ o ?_D
/ {0},
we obtain an exact sequence {0} o
B ∗ lr
i∗
{0}.
Another application of 20.23 yields that the sequence / B ∗∗ / A∗∗ / D ∗∗ {0} i ∗∗ q ∗∗ is exact. As εD is an isomorphism, we have that εA q ∗∗ is surjective, and A∗∗ = ∗∗ AεA +ker q ∗∗ . But εB is an isomorphism as well, and ker q ∗∗ = (B ∗∗ )i = B iεA ≤ AεA yields that εA is surjective. Because D ∗∗ is discrete, the image of B ∗∗ under i ∗∗ is open in A∗∗ , and i ∗∗ is an open embedding. Thus the restriction of εA to the open subgroup B is an open map. Consequently, the morphism εA is open, and we have completed our proof. 2
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F Locally Compact Abelian Groups
23 Applications of the Duality Theorem This section collects bits of information for the sake of easy reference, and completes the picture at places that remained unsatisfactory before the Duality Theorem was established. 23.1 Lemma. Let ϕ : A → B be a morphism in LCA. Let Ass(ϕ) be one of the following assertions about ϕ: ϕ is monic, ϕ is epic, ϕ is open, ϕ is proper, ϕ is an embedding, ϕ is a quotient map, ϕ is an isomorphism. Then Ass(ϕ) holds exactly if Ass(ϕ ∗∗ ) holds. Proof. This follows immediately from ϕ ∗∗ = εA −1 ϕεB and ϕ = εA ϕ ∗∗ εB −1 .
2
23.2 Proposition. Let ϕ : A → B be a morphism in LCA. Then the following hold. (a) The morphism ϕ is monic exactly if ϕ ∗ is epic. (b) The morphism ϕ is epic exactly if ϕ ∗ is monic. (c) The morphism ϕ is proper exactly if ϕ ∗ is proper. (d) The morphism ϕ is an embedding exactly if ϕ ∗ is a quotient morphism. (e) The morphism ϕ is a quotient map exactly if ϕ ∗ is an embedding. Proof. Assertion (b) has been proved in 20.13. Applying (b) to the morphism ϕ ∗ we obtain that ϕ ∗ is epic exactly if ϕ ∗∗ is monic, and assertion (a) follows from 23.1. We know from 20.15 that ϕ ∗ is proper if ϕ is proper. Thus assertion (c) follows from 23.1. If ϕ is a quotient morphism then ϕ ∗ is an embedding by 20.16. Conversely, assume that ϕ : A → B is an embedding. This just means that ϕ is both monic and proper. From assertions (a) and (c) we know that the adjoint ϕ ∗ is both epic and proper, and thus a quotient morphism. This completes the proof of assertion (d). Assertion (e) is now obtained from (d) using 23.1. 2
23. Applications of the Duality Theorem
195
23.3 Corollary. If A is a closed subgroup of B ∈ LCA then every character of A extends to some character of B. Combining 23.2 and 20.23, we obtain: 23.4 Corollary. Assume that the sequence /A
{0}
/B
ι
π
2, C
/ {0}
is exact, where ι is an embedding and π is a quotient morphism. Then ι∗ is a quotient morphism and π ∗ is an embedding, and the dual sequence / A∗ lr
{0}
ι∗
B∗ o
π∗
∗ o ?_C
{0}
is exact, as well. In other words: the dual of an extension in LCA is an extension, as well. We are now in a position to review the preliminary results 20.21 and 20.23. 23.5 Theorem: The annihilator mechanism. Let B be a closed subgroup of ,2 A/B / A be the inclusion, and let q : A A ∈ LCA, let i : B be the ∗ q∗ ∼ ∗ ⊥ ∗ ∗ ∗ ⊥ ∼ natural map. Then B = ker i = ((A/B) ) = (A/B) and B = A /B . It is convenient to visualize this by a correspondence of Hasse diagrams: · {0}
A· ∼ =
o
/
· B⊥
B· ∼ =
o
/
· A∗
{0} · Proof. Applying 23.4 to the sequence {0}
/B
i
/A
q
i∗
A∗ o
q∗
2, A/B
/ {0}
∗ ? _ (A/B) o
{0}
we obtain the extension {0} o
B ∗ lr
and apply 20.21 to see that q ∗ induces an isomorphism from (A/B)∗ onto B ⊥ .
2
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F Locally Compact Abelian Groups
The proof of the next assertion is easy, and left as an exercise. 23.6 Lemma. Let A ∈ LCA. For any subset S ⊆ A, we have (S ⊥ )⊥ = (S ⊥⊥ )εA . 2 23.7 Theorem. Consider an exact sequence / An−1
···
ϕn−1
/ An
ϕn
ϕn+1
/ An+1
/ An+2
/ ···
of arbitrary length in LCA. If each of the morphisms ϕn is proper then each of the adjoints ϕ ∗ is proper, and the dual sequence ··· o
An−1 ∗ o
ϕn−1 ∗
An ∗ o
ϕn ∗
ϕn+1 ∗
An+1 ∗ o
An+2 ∗ o
···
is exact, as well. Proof. We know from 20.15 that the adjoints ϕ ∗ are proper. It suffices to consider ϕ1 / A2 ϕ2 / A3 which is exact at A2 and where ϕ1 and ϕ2 are a sequence A1 proper. We have to show that the dual sequence is exact at A2 ∗ . Factoring the ϕ proper morphisms ϕ1 = π1 ι1 and ϕ2 = π2 ι2 with quotients πi : Ai → Ai i and ϕi embeddings ιi : Ai → Ai+1 we obtain the following commuting diagram: A A1 LL r8 O 3 LLL r r ϕ2 rr LLϕL1 π1 ι2 LLL rrr r ? _ r L r & r ϕ ,2 ϕ2 / A2 A1 1 A2 ι1 π2 Exactness at A2 means exactness of the sequence / Aϕ1 1
{0}
ι1
/ A2
π2
2, ϕ2 A2
/ {0} ∗
∗
and we infer that the dual sequence is exact; that is, we have ker ι1 ∗ = ((A2 2 ) )π2 . ∗ As π1 ∗ is injective and ι2 ∗ is surjective, this means ker ϕ1 ∗ = (A3 ∗ )ϕ2 . 2 ϕ
23.8 Lemma. Assume that A ∈ LCA has a subgroup T isomorphic to Tn for some natural number n. Then the extension splits; that is, there is a subgroup B of A such that T ∩ B = {0} and A = T + B. In fact, one has A = T ⊕ B. Proof. Put Q := A/T . The exact sequence {0}
/T
ι
/A
π
2, Q
/ {0}
dualizes to an exact sequence {0} o
T ∗ lr
ι∗
A∗ o
π∗
∗ ?_Q o
{0} .
23. Applications of the Duality Theorem
197
As T ∗ is isomorphic to Zn , we can apply 6.30 and obtain a morphism κ : A∗ → Q∗ such that π ∗ κ = idQ∗ . In particular, κ is a quotient morphism. Dualizing again, we obtain that the image B of Q under εQ κ ∗ εA −1 has the required properties. 2 Applying 6.23 (b), we infer A = T ⊕ B. 23.9 Theorem. Let (J, ) be a directed set, let D ∈ DSJ (LCA) be a directed system, and assume that λ = (λj )j ∈J is a direct limit of D in LCA, with vertex L. Then λ∗ = (λj ∗ )j ∈J is a projective limit (with vertex L∗ ) of the projective system P ∈ PSJ (LCA) given by Pj = Dj ∗ and pij = dij ∗ . Proof. Let α = (αj )j ∈J be a cone over P , with vertex M. Then γj := εDj αj ∗ defines a cone γ with vertex M ∗ over D, and we obtain that there is a unique morphism γ : L → M ∗ such that λj γ = εDj αj ∗ for each j ∈ J . Now α := ∗ ∗ ∗ γ ) : M → L satisfies α λj = εM (λj γ )∗ = εM (εDj αj ∗ )∗ = εM αj ∗∗ εDj ∗ = εM ( αj εDj ∗ εDj ∗ = αj . It remains to show that α is uniquely determined by this property. If β : M → L∗ is a morphism with βλj ∗ = αj for each j ∈ J , then γj = εDj αj ∗ = εDj λj ∗∗ β ∗ = λj εL β ∗ implies γ = εL β ∗ by the universal property of the direct limit λ. We conclude β = εM β ∗∗ εL∗ −1 = εM β ∗∗ εL ∗ = εM γ∗ = α. 2 Dualizing 23.9, we obtain immediately: 23.10 Corollary. Let (J, ) be a directed set, let P ∈ PSJ (LCA) be a projective system, and assume that λ = (λj )j ∈J is a projective limit of P in LCA, with vertex L. Then λ∗ = (λj ∗ )j ∈J is a direct limit (with vertex L∗ ) of the directed system D ∈ DSJ (LCA) given by Dj = Pj ∗ and dij = pij ∗ . The elements of CGAL have two different decompositions that are very useful. Firstly, there is the maximal compact subgroup of A = Ra × Tb × Zc × F , namely M = {0} × Tb × {0} × F , with complement Ra × {0} × Zc × {0}. Secondly, the maximal vector subgroup A0 = Ra ×{(0, 0, 0)} has complement {0}×Tb ×Zc ×F . In both cases, the complements are not uniquely determined. Via approximation, corresponding decompositions can also be obtained for compactly generated groups in LCA. We consider maximal compact subgroups first.
Maximal Compact Subgroups and Their Complements 23.11 Splitting Theorem. If A ∈ LCA is compactly generated, the following hold. (a) There is a unique maximal compact subgroup M of A. (b) There is a subgroup L of A such that A = M ⊕ L. Moreover, there are natural numbers a and c such that L ∼ = Ra × Zc . As a consequence, we have a c A∼ =M ×R ×Z .
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Proof. Let U be a compact neighborhood of 0 in A. According to 21.16, we find a subgroup K of A such that K ⊆ U and a quotient morphism p : A → G := Ra × Tb × Zc × F with kernel K, where a, b, c are suitable natural numbers, and F is a finite group. As A/K is Hausdorff, the subgroup K is closed and therefore compact. The subgroup C := {0} × Tb × {0} × F of G is compact, and the quotient G/C ∼ = Ra × Zc contains no compact subgroup except the trivial one. Therefore, the subgroup C is maximal among the compact subgroups of G. As the sum of two compact subgroups of an Abelian topological group is a compact subgroup, there is at most one maximal compact subgroup in this case. Because compactness is an extension property 6.15, the pre-image M of C under p is the maximal compact subgroup of A, and assertion (a) is proved. The quotient A/M is isomorphic to Ra × Zc . Let q : A → Ra × Zc be the cor← responding quotient morphism, and put N := (Ra )q . Then A/N ∼ = Zc and 6.30 c yield that there is a subgroup D ∼ = Z of A such that N ∩ D = {0} and A = N + D. We have the following exact sequence: /M
{0}
πN
/N
,2 N/M
/ {0}
Passing to duals, this gives an exact sequence M ∗ lr
N∗ o
πN ∗
∗ ? _ (N/M) o
{0} .
As M ∗ is discrete, the image of (N/M)∗ under πN ∗ is open. From N/M ∼ = Ra we ∗ ∼ a ∗ infer that (N/M) = R is divisible. According to 4.21, the inverse of πN extends to a morphism k : N ∗ → (N/M)∗ . Being open and surjective, the morphism k is in fact a quotient map. Dualizing again, we obtain the embedding k ∗ of (N/M)∗∗ in N ∗∗ . The image of k ∗ εN −1 is the subgroup L that we need in assertion (b). 2 23.12 Example. In general, a locally compact group need not have a maximal compact subgroup. For instance, consider the group Z(2)N , with the discrete topology. We are now in a position to resolve the open problem stated after 21.20: the class of compactly generated locally compact Abelian groups is closed with respect to the forming of closed subgroups. Note, by way of contrast, that this is not true if we drop the assumption of commutativity. In fact, every free non-Abelian group contains subgroups that are not finitely generated (for instance, the commutator subgroup). A more concrete example is the following: let F be a free group with two generators a and b. Then the subgroup generated by S := {a n b | n ∈ N {0}} is not finitely generated; in fact, this subgroup is free over S. Note also that in these (discrete) examples there does exist a maximal compact subgroup, namely, the trivial one. 23.13 Theorem. Let A ∈ LCA be compactly generated. Then every closed subgroup of A is compactly generated, as well.
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Proof. Let B be a closed subgroup of A. By 21.17 there is a compact subgroup K of A such that A/K ∈ CGAL. Let π : A → A/K be the natural map. As K is compact, we have that B π is closed in A/K, compare 6.5. Moreover, the restriction of π to B yields a quotient map onto B π with compact kernel B ∩ K. According to 21.19, the group B π is compactly generated. Since we are dealing 2 with an extension property, this means that B is compactly generated. We are ready to prove another characterization of the members of CGAL. 23.14 Theorem. Let A ∈ LCA. Then A ∈ CGAL if, and only if, both A and A∗ are compactly generated. Proof. If A is compactly generated, we have A ∼ = M ×V ×D, where M is compact, V is a vector group, and D is a finitely generated, discrete Abelian group having no compact subgroups except {0}. This means D ∼ = Zd for some nonnegative ∗ ∗ integer d. The dual A is then isomorphic to M × V × Td . If A∗ is compactly generated then M ∗ is finitely generated, and A∗ ∈ CGAL follows. The converse is obvious. 2
Compactness versus Connectedness 23.15 Definition. A topological group G is called compact-free if it has no compact subgroup except the trivial one. We write Comp(G) := {C ≤ G | C is compact}. 23.16 Remark. In general, the set Comp(G) is not a subgroup of G. For instance,
0 take G = GL(2, C), with the usual topology: the elements a := 01 −1 and 1 1 1 1 b := 0 −1 of Comp(GL(2, C)) have the product ab = 0 1 ∈ / Comp(GL(2, C)). Note also that Comp(G) need not be closed. Again, the group G = GL(2, C) provides an example: pick a sequence of
complex numbers ωn converging to 1 such that ωnn = 1 = ωn , and put an := 01 ω1n . Then each an belongs to Comp(GL(2, C)) but limn→∞ an = 01 11 does not. 23.17 Lemma. Let A ∈ LCA. Then every compact subgroup of A∗ annihilates the connected component A0 , and every connected subgroup of A∗ annihilates Comp(A). Proof. Let ι : → A∗ be the inclusion. The image of A0 under εA ι∗ is a connected subgroup of the discrete group ∗ , and thus trivial. Now we compute A0 , = ∗ ι , (A0 )εA = , (A0 )εA ι = {0}. Now let γ : C → A be the inclusion of a compact subgroup C. As C ∗ is discrete, the connected group is contained in the kernel of γ ∗ : A∗ → C ∗ . However, the morphism γ ∗ does nothing but restrict characters of A to characters of C. 2
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23.18 Proposition. Let A ∈ LCA. Then A is connected exactly if A∗ is compactfree. Proof. If A is connected then every compact subgroup of A∗ is contained in A⊥ = {0} by 23.17. Conversely, if A is not connected we find a proper open subgroup B of A by 6.8. Then the natural map p from A to the discrete group A/B yields an embedding p ∗ of the nontrivial compact group (A/B)∗ into A∗ . 2 23.19 Proposition. For each A ∈ LCA, we have (A0 )⊥ = Comp(A∗ ). Proof. We know from 23.17 that Comp(A∗ ) is contained in (A0 )⊥ . Let π : A → A/A0 be the natural map. For each α ∈ (A0 )⊥ there is a character χ ∈ (A/A0 )∗ such that α = π χ . Let U be a neighborhood of 0 in T containing no subgroups ← except the trivial one. Then U χ is a neighborhood of 0 in A/A0 , and we find an ← open subgroup B of A such that B π ⊆ U χ , see 4.13. We conclude B α = {0} and α ∈ B⊥ ∼ = (A/B)∗ . As A/B is discrete, the group B ⊥ is compact, and α belongs to Comp(A∗ ). 2 In view of the fact that X ⊥ is a closed subgroup of A∗ for each subset X of A, and using duality, we obtain: 23.20 Corollary. For each A ∈ LCA, the set Comp(A) is a closed subgroup of A. If A is totally disconnected then Comp(A) is even an open subgroup (since there exist compact open subgroups, see 4.13). 2 Applying the annihilator mechanism 23.5, we see 23.21 Corollary. For each A ∈ LCA, we have (A/A0 )∗ ∼ = Comp(A∗ ).
2
Divisibility versus Torsion 23.22 Definitions. Let A be an Abelian group, and let n be a natural number. nA Then a homomorphism nA : A → A is given by a = na. The set Tors(A) := n∈N{0} ker nA consists of all elements of finite order in A, and forms a (not necessarily closed) subgroup. We say that A is a torsion group if A = Tors(A), and that A is torsion-free if Tors(A) = {0}. Of course, every compact-free group is torsion-free. The converse is false. We will need to know what the adjoint of nA looks like. 23.23 Lemma. For every A ∈ LCA and every n ∈ N, we have (nA )∗ = nA∗ . ∗
Proof. For a ∈ A and χ ∈ A∗ , we compute a, χ (nA ) = a, nA χ = na, χ = a, nχ . 2
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23.24 Proposition. (a) If A ∈ LCA is divisible then A∗ is torsion-free. (b) The group A is compact and divisible if, and only if, its dual A∗ is discrete and torsion-free. (c) The group A is discrete and divisible if, and only if, its dual A∗ is compact and torsion-free. Proof. Divisibility of A means that nA is surjective for each positive integer n. Thus nA is epic, and nA∗ = (nA )∗ is monic. This means that ker nA∗ = {0} for each positive integer n, and A∗ is torsion-free. This proves assertion (a). Conversely, the assumption that A∗ is torsion-free implies that nA is epic for each positive integer n. In general, this does not mean that nA is surjective. However, if A is compact or discrete, each epic is surjective, and we obtain assertions (b) and (c). 2 23.25 Corollary. An Abelian Hausdorff group A is compact and torsion-free if, and only if, there exists a family d = (dp )p∈P{0} of cardinal numbers such that A∼ = (Q∗ )d0 × Xp∈P (Zp )dp . Proof. From 23.24 (c) we know that A is compact and torsion-free exactly if A∗ is divisible and discrete. Now 4.24 yields that family d = (dj )j ∈P∪{0} " there is∞a (d (d ) ∼ 0 of cardinal numbers such that A = Q × p∈P Z(p ) p ) . Passing to the dual ∗∗ ∗ d 2 A ∼ = A again, we obtain A ∼ = (Q ) 0 × Xp∈P (Zp )dp , as required. 23.26 Lemma. Let A ∈ CA. Then the group A is divisible if, and only if, it is connected. Proof. If A is compact and connected then the dual A∗ is discrete and compact-free, and therefore torsion-free. Now we can apply 23.24, and infer that A is divisible. Conversely, assume that A is compact and divisible. Then A∗ is discrete and torsionfree, and Comp(A∗ ) = {0}. According to 23.18, this means that A is connected. 2 23.27 Corollary. Each connected group in LCA is divisible. Proof. As A is connected, it is compactly generated. According to 23.11, we have A ∼ = M × Ra × Zc for suitable natural numbers a, c and a compact group M. Since A is connected, we have that M is connected and c = 0. Now M is divisible by 23.26, and we obtain that A ∼ 2 = M × Ra is divisible. The discrete group Q shows that divisible groups in LCA need not be connected. Applying 4.22, we obtain the following corollary to 23.27.
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23.28 Proposition. If the connected component of A ∈ LCA is open in A then A∼ = A0 × D, where D is a discrete Abelian group. 23.29 Proposition. If A ∈ LCA is a torsion group then both A and A∗ are totally disconnected. Proof. In a torsion group, every element is contained in a (finite cyclic) compact subgroup. Thus 23.17 yields that the connected component of A∗ is trivial. If A0 is nontrivial, we find a neighborhood U = A0 of 0 in A0 and a subgroup K of A0 such that K ⊆ U and A0 /K ∈ CGAL, see 21.16. Then A0 /K is connected, and isomorphic to Ra × Tb for some a, b ∈ N. As A0 /K is not trivial, we have that A0 has a quotient which is not a torsion group, contradicting our assumption on A. 2 Note that a torsion-free group can have a totally disconnected dual; for example, take the torsion-free compact group Zp . 23.30 Proposition. A group A ∈ LCA is compact and totally disconnected exactly if its dual A∗ is a discrete torsion group. Proof. If A is compact and totally disconnected then it models a projective limit of finite groups, see 19.9. Thus A∗ models a direct limit of finite groups, and we obtain Tors(A∗ ) = A∗ . Conversely, if A∗ is discrete then A is compact, and Tors(A∗ ) = A∗ implies A∗ ≤ (A0 )⊥ , which means that A0 is trivial. 2
Exercises for Section 23 Exercise 23.1. Prove 23.10. Exercise 23.2. Show that Zp ∗ is isomorphic to Z(p ∞ ). Exercise 23.3. Show that the (discrete) Prüfer group Z(p∞ ) is not finitely generated (and thus not compactly generated), but has a compact group of automorphisms: in fact, one has Aut(Z(p∞ )) = Z× p = Zp pZp . Exercise 23.4. Prove 23.6. Exercise 23.5. Exhibit examples of torsion-free compact Abelian groups that are not connected, or not totally disconnected. Exercise 23.6. Determine Aut(T)n . Exercise 23.7. Let Q be the group (Q, +), endowed with the discrete topology. Show that Q∗ models a projective limit of circle groups. Prove also that Q∗ is compact, connected, and torsion-free.
24. Maximal Compact Subgroups and Vector Subgroups
203
24 Maximal Compact Subgroups and Vector Subgroups We have seen in 23.11 that every connected group A ∈ LCA splits as A = M ⊕ V , where M is the maximal compact subgroup, and V is isomorphic to Rn , for some natural number n. In the present chapter, we study subgroups like V in arbitrary groups A ∈ LCA, and conclude with a generalization of the splitting result 23.11 under the assumption that A possesses a maximal compact subgroup. 24.1 Definition. A topological group V is called a vector group if there is a natural number n such that V is isomorphic to Rn (as topological group). A subgroup V of a topological group G is called a vector subgroup of G if V is a vector group (with the induced topology). A maximal vector subgroup of G is a vector subgroup V of G such that no vector subgroup of G properly contains V . Of course, every vector group belongs to LCA. Being locally compact, a vector subgroup of a Hausdorff group is always closed, compare 4.8. As vector groups are connected, the vector subgroups of a topological group are exactly those of its connected component. If A0 is the connected component of a locally compact Abelian Hausdorff group A then A0 is the direct product of its maximal compact subgroup and some vector subgroup, compare 23.11. A compact Hausdorff group contains no vector subgroup except the trivial one; in fact, a vector subgroup of such a group is closed, therefore compact and trivial. Therefore, the complement to the maximal compact subgroup of the connected component is a maximal vector subgroup, and we have the following. 24.2 Lemma. Every group A ∈ LCA possesses a maximal vector subgroup.
2
24.3 Lemma. Let A be an Abelian topological group. Then a vector subgroup V of A is a maximal vector subgroup of A exactly if A/V contains no vector subgroup except the trivial one. Proof. Let V be a vector subgroup of A. If V is not a maximal vector subgroup then there is a vector subgroup W of A such that V is properly contained in W , and W/V is a nontrivial vector subgroup of A/V . Conversely, assume that A/V has a nontrivial vector subgroup X. Then the pre-image W of X under the natural map from A to A/V is a connected and compact-free member of CGAL, since all these properties are preserved under extensions, see 6.15 and 21.19. Every connected group in CGAL is of the form Ra × Tb for natural numbers a, b. Thus W is a vector 2 subgroup of A, and V is not maximal. The main step in the previous proof is worth remembering; it says that the class of vector groups is closed with respect to extensions (inside TA):
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24.4 Corollary. If A is an Abelian topological group with a subgroup V such that both V and A/V are vector groups then A is a vector group. 24.5 Lemma. Consider A ∈ LCA, and let V be a maximal vector subgroup of A. Then the connected component of A/V is compact. Proof. We know from 24.3 that every vector subgroup of A/V is trivial. Using the fact that the connected component of A/V is a direct product of a compact subgroup and a vector subgroup, see 23.11, we obtain the assertion. 2 As RN shows, a topological Abelian group need not have maximal vector subgroups in general. In the sequel, we are going to give an explicit description of all vector subgroups of a given group A ∈ LCA. 24.6 Theorem: Morphisms of vector groups. Let V be a vector group, and let W be the additive group of some vector space over R; not necessarily of finite dimension. Moreover, assume that W carries a Hausdorff topology such that W is a topological group and that multiplication by scalars is a continuous map from R × W to W . Then every continuous homomorphism from V to W is an R-linear map. Proof. Every homomorphism between Abelian groups is Z-linear, where the multiplication by positive integers is of course defined as repeated addition, and multiplication by −1 is inversion. Since V is uniquely divisible, every homomorphism from V to W is a Q-linear map. It remains to show that every continuous Qlinear map ϕ : V → W is R-linear. Fix v ∈ V . For every r ∈ R we can pick a sequence (qn )n∈N of rational numbers qn such that r = limn→∞ qn . By continuity of ϕ and the multiplication by scalars, we obtain (rv)ϕ = (limn→∞ qn v)ϕ = 2 limn→∞ qn v ϕ = rv ϕ . 24.7 Corollary. The topological ring Mor(Rn , Rn ) is isomorphic to the ring Rn×n of all n × n matrices over R. 2 24.8 Lemma. Assume that G is a topological group, and that V is a vector group. Let W be a vector subgroup of V , and let γ : W → G be a continuous homomorphism. Then γ := {(w, wγ ) | w ∈ W } is a vector subgroup of V × G; in fact, it is isomorphic to W . If V × {1} is a maximal vector subgroup of V × G then every vector subgroup of V × G arises in this way, and the maximal ones are obtained for W = V . In the latter case, we have γ ∩ ({0} × G) = {(0, 1)} and V × G = γ ({0} × G). Proof. Let W be a vector subgroup of V , and let γ : W → G be a continuous homomorphism. Then the map ϕ : W → γ defined by w ϕ = (w, wγ ) is a −1 continuous homomorphism, with continuous inverse (given by (w, wγ )ϕ = w). Thus γ is a vector subgroup of V × G.
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Now let X be any vector subgroup of V × G. The canonical projection from V × G onto V induces a continuous homomorphism π from X to V . By 24.6, this homomorphism is an R-linear map, with kernel X ∩ ({0} × G). If V × {1} is a maximal vector subgroup of V × G then G contains no vector group except the trivial one, and we infer that π is injective. Thus there is an R-linear (and thus continuous) map ψ : V → X such that π ψ = idX . Composing ψ|Xπ with the canonical projection from V × G to G, we obtain a continuous homomorphism γ from W := Xπ to G such that γ = X. The rest of the assertion is now obvious. 2 24.9 Corollary. Assume that A ∈ LCA is connected, and let M be a maximal compact subgroup of A. Then A = M ⊕ V for each maximal vector subgroup V . 24.10 Theorem. Let A ∈ LCA, and assume that V and are maximal vector −1 subgroups of A and A∗ , respectively. Then A∗ = ⊕ V ⊥ , and A = V ⊕ ( ⊥ )εA . Moreover, the groups V and are isomorphic. Proof. According to 23.11, the connected component of A∗ splits as = ⊕ , where is the maximal compact subgroup. From ≤ Comp(A∗ ) = (A0 )⊥ ≤ V ⊥ we conclude + V ⊥ = + + V ⊥ = + V ⊥ . The embedding ι : V → A gives a quotient morphism ι∗ : A∗ → V ∗ ∼ = V with kernel V ⊥ . By 24.6, the ∗ ∗ restriction of ι to is an R-linear map, and ι is closed in V ∗ . Thus + V ⊥ is closed in A∗ . As a, + V ⊥ = {0} implies a ∈ Comp(A) ∩ V = {0}, we have A∗ = {0}⊥ = + V ⊥ = + V ⊥ . Dually, we have A∗∗ = V εA + ⊥ . Now V εA ∩ ⊥ = (V ⊥ )⊥ ∩ ⊥ = (V ⊥ + )⊥ = (A∗ )⊥ = {0}. As ι∗ induces an injective R-linear map from to V ∗ ∼ = V , we know that dim ≤ dim V . Duality yields the reverse inequality, and we obtain ∼ = V. 2 24.11 Theorem: Splitting of vector subgroups. For each A ∈ LCA, we have: (a) Every vector subgroup of A is contained in a maximal vector subgroup of A. (b) Every vector subgroup V of A has a complement; that is, there exists a closed subgroup B of A such that A = B ⊕ V . (c) For every pair (V , W ) of maximal vector subgroups there is an automorphism α of A such that V α = W . Proof. Let V be a maximal vector subgroup of A. Without loss of generality, we may assume that A = V × G, where G ∼ = ⊥ , compare 24.10. Assertion (a) follows from 24.8, and assertion (b) is clear from 24.10. If W is another maximal vector subgroup of A, we find a continuous homomorphism γ : V → G such that W = γ . The map α : A → A defined by (v, g)α = (v, g + v γ ) is a continuous −1 homomorphism with continuous inverse (given by (v, g)α = (v, g − v γ )). This proves assertion (c). 2
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If a locally compact Abelian group possesses a maximal compact subgroup, it behaves as if it were compactly generated, compare 23.11: 24.12 Theorem: Splitting of maximal compact subgroups. Assume that A ∈ LCA has a maximal compact subgroup M. Then the extension A/M splits; in fact, there are a vector subgroup V ∼ = Ra and a discrete torsion-free subgroup D of A such that A = M ⊕ V ⊕ D. Proof. Pick maximal vector subgroups V of A, and of A∗ . According to 24.10, −1 we have A = B ⊕ V , where B = ( ⊥ )εA . By 23.17, the maximal compact subgroup M of A satisfies M, = 0, and using 23.6 we obtain M ≤ B. It remains to find some discrete torsion-free subgroup D such that B = M ⊕ D. To this end, we dualize the exact sequence {0}
/M
ι
/B
π
,2 B/M
/ {0} .
As ι is an embedding and π is a quotient, we have that ι∗ is a quotient, and π ∗ is an embedding; compare 23.4. We obtain an exact sequence {0} o
M ∗ lr
ι∗
B∗ o
π∗
∗ ? _ (B/M) o
{0} .
Now B/M contains no compact subgroup apart from {0}, and we infer from 23.18 that := (B/M)∗ is connected. As the quotient M ∗ is discrete, the kernel of ι∗ is open. According to 4.22, there is a discrete subgroup of B ∗ such that ∗ B ∗ = ⊕ π . The restriction ι∗ | : → M ∗ is an isomorphism of topological (in fact, discrete) groups. Therefore, we find a morphism σ : M ∗ → B ∗ such that σ ι∗ = idM ∗ . Thus the morphism κ = εB σ ∗ εM −1 : B → M satisfies ικ = (εM ι∗∗ εB −1 )(εB σ ∗ εM −1 ) = εM (σ ι∗ )∗ εM −1 = idM . The kernel D of κ is the subgroup we are searching for. In fact, we have B = M + D and M ∩ D = {0}. The connected component of B is compact since V is a maximal vector subgroup of A; see 24.5. Therefore, the quotient B/M is totally disconnected. Being compact-free, this quotient is discrete; compare 4.13. In particular, the group M is open in B, and D is discrete. Finally, we have B = M ⊕ D by 6.23. 2 24.13 Remark. The additive group of a vector space of infinite dimension over a finite field, taken with the discrete topology, shows that an Abelian locally compact group need not possess any maximal compact subgroup. A maximal compact subgroup of an Abelian topological group is unique (if it exists at all). The complement V ⊕ D constructed in 24.12, however, is unique only in rare instances. In fact, we may choose any maximal vector subgroup to play the role of V . The groups R × Z or T × Z may serve as examples that show that D is not uniquely determined.
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Exercises for Section 24 Exercise 24.1. Find maximal vector subgroups in the multiplicative groups R× = R {0} and C× = C {0}. What can be said about uniqueness? Exercise 24.2. Show that the groups R× and C× possess maximal compact subgroups. Exercise 24.3. Find decompositions of R× and C× as in 24.12. Exercise 24.4. Show that the additive group Qp of p-adic numbers does not possess maximal compact subgroups, and that {0} is a maximal vector subgroup. Exercise 24.5. Let V be a topological vector space over R, and assume that V is a Hausdorff space. Show that V does not contain maximal vector subgroups unless its dimension is finite. Hint. Use that every finite-dimensional subspace of V is a vector subgroup, cf. 26.40. Exercise 24.6. Use Exercise 24.5 to prove that every locally compact Hausdorff vector space over R has finite dimension. (See 26.40 for a stronger result.) Exercise 24.7. Give explicit examples of locally compact Abelian torsion groups possessing no maximal compact subgroups. Exercise 24.8. Give explicit examples of locally compact Abelian torsion-free groups possessing no maximal compact subgroups.
25 Automorphism Groups of Locally Compact Abelian Groups In this section, we show that dualizing implements an isomorphism of topological groups between Mor(A, B) and Mor(B ∗ , A∗ ) for each pair of elements A, B ∈ LCA. This will have several useful applications.
Adjoints of Groups of Homomorphisms or of Automorphisms 25.1 Theorem. Let A, B ∈ LCA, and endow the Abelian groups Mor(A, B) and Mor(B ∗ , A∗ ) with the topologies induced by the compact-open topologies on C(A, B) and C(B ∗ , A∗ ), respectively. Then mapping ϕ to ϕ ∗ defines an isomorphism of topological groups from Mor(A, B) onto Mor(B ∗ , A∗ ). Proof. We know that mapping ϕ to ϕ ∗ defines a homomorphism from Mor(A, B) to Mor(B ∗ , A∗ ). Mapping ψ to εA ψ ∗ εB −1 gives the inverse of this homomorphism. Both homomorphisms are continuous; see 9.9 and 9.4. 2
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25.2 Corollary. For every A ∈ LCA, mapping ϕ to ϕ ∗ is an anti-isomorphism of topological rings from Mor(A, A) onto Mor(A∗ , A∗ ). Using 9.13 we have another corollary. 25.3 Theorem. Let A ∈ LCA, and endow Aut(A) and Aut(A∗ ) with the modified compact-open topologies. Then mapping α to (α ∗ )−1 is an isomorphism of topological groups from Aut(A) onto Aut(A∗ ). As the set of all continuous maps from any space to a totally disconnected one has to be totally disconnected, we obtain an important application. 25.4 Corollary. For all A, B ∈ LCA we have the following. (a) If B is totally disconnected then the groups Mor(A, B) and Mor(B ∗ , A∗ ) are totally disconnected. (b) If A = Comp(A), in particular if A is compact or A = Tors(A), then the groups Mor(B, A∗ ) and Mor(A, B) are totally disconnected. (c) If A is either totally disconnected or A = Comp(A) then the ring Mor(A, A) and the group Aut(A) are totally disconnected.
A Characterization of Vector Groups 25.5 Theorem. Let A ∈ LCA, and assume that is a connected subset of Mor(A, A) with the following property: whenever a γ = a for a pair (a, γ ) ∈ A× then either a = 0 or γ = idA . If contains {idA } as a proper subset then there is a natural number n such that A ∼ = Rn . Proof. Let μ : M → A be the inclusion map, where M is the maximal compact subgroup of the connected component A0 . Mapping ϕ to μϕ defines a continuous homomorphism from Mor(A, A) to Mor(M, A). Applying 25.4, we obtain that Mor(M, A) is totally disconnected. Therefore, the set μ = {μγ | γ ∈ } consists of a single element, and we have μ = {μ idA } = {μ}. For γ ∈ and m ∈ M we observe mγ = mμγ = mμ = m. Picking γ ∈ {idA }, our assumption now yields M = {0}, and A0 ∼ = Rn by 21.16. Now consider the natural map π : A → Q := A/A0 . As Q is totally disconnected, we know that Mor(A, Q) is totally disconnected. Reasoning as above, we see that the set π consist of the single element π = idA π . Our assumption implies that the endomorphism idA −γ ∈ Mor(A, A) is monic for each γ ∈ {idA }. The restriction of such an endomorphism to A0 ∼ = Rn is an isomorphism, and we 2 conclude A = A0 .
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As a corollary, we obtain a well-known result from the theory of topological vector spaces. Note that we use only a part of the axioms of a topological vector space. 25.6 Theorem. Let V be the additive group of a vector space over R. Assume that V carries a topology such that V is a topological group, and that multiplication by positive scalars is a continuous map from V × {r ∈ R | r > 0} to V . If V is locally compact Hausdorff then its dimension over R is finite. Proof. Let ω : V × {r ∈ R | r > 0} → V be the multiplication by positive scalars. Then mapping r to r idV is a homomorphism from the group of positive real numbers to Aut(V ). According to 9.7, this homomorphism is continuous since ω is continuous. If V is locally compact Hausdorff, we can apply 25.5 and conclude that there is a natural number n such that V ∼ = Rn . According to 24.6, any isomorphism is an 2 R-linear map, and the assertion follows.
The Structure of Rings of Endomorphisms and of Groups of Automorphisms In the sequel, we are going to apply the description of homomorphisms between direct products of Abelian groups, as discussed in 11.15. As a vector group V contains no compact subgroup except the trivial one, we have Mor(A, V ) = {0} for each topological Abelian group A with Comp(A) = A. This has the following immediate consequence. 25.7 Proposition. Let V ∼ = Rn be a vector group, and assume that A ∈ LCA satisfies Comp(A) = A. Then the topological ring Mor(V × A, V × A) is isomorphic to the ring of matrices λ μ λ ∈ Mor(V , V ), μ ∈ Mor(V , A), V V = , × A, V × A)] [Mor(V A A γ γ ∈ Mor(A, A). endowed with the product of the respective compact-open topologies. We have automorphisms Aut(V ) ∼ = GL(n, R), and Aut(A) ∼ = Aut(A∗ ). Trans∗ ∗ lating Mor(V , A) into Mor(A , V ) via adjunction may further help understanding the structure of Aut(V × A). Firstly, the isomorphism V ∗ ∼ = Rn induces the struc∗ ∗ ture of a topological vector space over R on Mor(A , V ); compare 11.13. Note that the dimension of this vector space is infinite, in general. If A is compact, then A∗ is discrete. In this case, the compact-open ∗topology on ∗ Mor(A∗ , V ∗ ) is the topology induced by the product topology on V ∗ A ∼ = (Rn )A .
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Let us view the situation discussed above in a more general setting. Consider the product A × B for A, B ∈ LCA. Then the factor A × {0} is characteristic in A × B exactly if Mor(A, B) = {0}. Examples are the pairs (A, B) where A = Comp(A) and Comp(B) = {0}, and pairs where A is connected and B is totally disconnected. Specifically, we observe Mor( Ta × Rb × F, Zc ) = {0}, Mor( Ta × Rb , F ) = {0}, Mor( Ta , Rb ) = {0}, and Mor( F, Rb ) = {0} for all natural numbers a, b, c and every finite group F . This leads to the following. 25.8 Theorem. Let a, b, c be natural numbers, let F be a finite (discrete) group, and put A = Ta × Rb × F × Zc . Then the following hold. (a) The topological ring Mor(A, A) of all endomorphisms of the group A is isomorphic to the ring of matrices ⎧ ⎫ ⎛ ⎞ t ∈ Mor(Ta , Ta ), ⎪ ⎪ ⎪ ⎪ t 0 0 0 ⎪ ⎪ ⎪ m ∈ Mor(Rb , Ta ), r ∈ Mor(Rb , Rb ),⎪ ⎨⎜ ⎬ ⎟ m r 0 0 a ⎜ ⎟ n ∈ Mor(F, T ), f ∈ Mor(F, F ), ⎝ n 0 f 0⎠ ⎪ ⎪ u ∈ Mor(Zc , Ta ), v ∈ Mor(Zc , Rb ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ u v w z ⎭ w ∈ Mor(Zc , F ), z ∈ Mor(Zc , Zc ), endowed with the product of the respective compact-open topologies. (b) The group Aut(A) is isomorphic to a semidirect product of
Mor(Rb × F × Zc , Ta ) Mor(Zc , Rb × F ) with
Aut(Ta ) × Aut(Rb ) × Aut(F ) × Aut(Zc ) ;
the homomorphisms involved can be read off from the matrix description ⎧ ⎫ ⎛ ⎞ t ∈ Aut(Ta ), ⎪ ⎪ ⎪ ⎪ t 0 0 0 ⎪ ⎪ b a b ⎪ ⎪ ⎨⎜ ⎬ m ∈ Mor(R , T ), r ∈ Aut(R ), ⎟ m r 0 0 a ⎜ ⎟ n ∈ Mor(F, T ), f ∈ Aut(F ), . ⎝ n 0 f 0⎠ ⎪ ⎪ u ∈ Mor(Zc , Ta ), v ∈ Mor(Zc , Rb ),⎪ ⎪ ⎪ ⎪ ⎪ u v w z ⎪ ⎩ ⎭ w ∈ Mor(Zc , F ), z ∈ Aut(Zc ),
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Thus Aut(A) is isomorphic to the semidirect product ab
R × F a × Tac Rbc × F c (GL(a, Z) × GL(b, R) × Aut(F ) × GL(c, Z)) .
(c) The connected component of 0 in Mor(A, A) is given as ⎧⎛ ⎫ ⎞ 0 0 0 0 ⎪ b a b b ⎪ ⎪ ⎪ ⎨⎜ ⎟ m ∈ Mor(R , T ), r ∈ Mor(R , R ),⎬ ⎜m r 0 0⎟ n ∈ Mor(F, Ta ), . ⎝ n 0 0 0⎠ ⎪ ⎪ u ∈ Mor(Zc , Ta ), v ∈ Mor(Zc , Rb ) ⎪ ⎪ ⎩ ⎭ u v 0 0 (d) The connected component of 1 in Aut(A) is isomorphic to a semidirect product
ab (R × Tac ) Rbc GL+ (b, R), where GL+ (b, R) = {M ∈ GL(b, R) | det M > 0} is the connected component of GL(b, R). In the matrix description, the connected component of 1 in Aut(A) is ⎧⎛ ⎫ ⎞ 1 0 0 0 ⎪ ⎪ b a b ⎪ ⎪ ⎨⎜ ⎟ m ∈ Mor(R , Ta ), r ∈ Aut(R )1 , ⎬ m r 0 0 ⎜ ⎟ n ∈ Mor(F, T ), . ⎝ n 0 1 0⎠ ⎪ ⎪ u ∈ Mor(Zc , Ta ), v ∈ Mor(Zc , Rb )⎪ ⎪ ⎩ ⎭ u v 0 1 We have obtained quite good information about the automorphism group of any element of the category CGAL. We state a consequence explicitly. 25.9 Theorem. For each element A ∈ CGAL, the automorphism group Aut(A) is a compactly generated locally compact group. 2 25.10 Remarks. Even if the group A is compact (but not in CGAL), its automorphism group need not be compactly generated. For instance, let Q be the additive group of rational numbers, endowed with the discrete topology. Then its compact dual Q∗ has the same group of automorphisms, namely, the multiplicative group of nonzero rational numbers. This follows immediately from the fact that an automorphism of Q is determined by the image of 1. The same fact also yields that Aut(Q) is discrete. Now Aut(Q) is not compactly generated; in fact, every finitely generated subgroup of Aut(Q) is cyclic. We have already seen in 11.4 that there exists a discrete Abelian group G such that Aut(G) is not locally compact. Dualizing, we find a compact Abelian group G∗ whose automorphism group Aut(G∗ ) is not locally compact.
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26 Locally Compact Rings and Fields We have already seen in 11.12 that every topological ring R is embedded in the ring Mor(A, A) of all endomorphisms of its underlying additive topological group A. In this section, we draw some interesting consequences from this observation. 26.1 Lemma. Let R be a topological ring. Then the connected component R0 is an ideal. The closure of any ideal is an ideal, as well. Proof. For each r ∈ R, the sets R0 r and rR0 are connected, and contain 0. Thus we have R0 r ⊆ R0 ⊇ rR0 , and R0 is an ideal. Now let J be an ideal of R. Then R × J and J × R are contained in the pre-image of J under multiplication. Therefore, the closures R × J and J × R are contained in the pre-image of J under multiplication; that is, the closure of any ideal is an ideal, as well. 2 In view of 26.1, it is no great loss to concentrate on Hausdorff rings: one can always pass to the quotient by the closure of the ideal {0}, and obtain a Hausdorff ring. As a field F has no ideals except {0} and F , we have an immediate corollary: 26.2 Lemma. If F is a topological field then either F is connected or F is totally 2 disconnected. If F is not Hausdorff then the topology is the indiscrete one. Every discrete field is of course locally compact and totally disconnected. Apart from these trivial examples, there are two fundamental constructions for locally compact totally disconnected fields. 26.3 Examples. Let F be a commutative field. On the set F N we use componentwise addition, and define a multiplication as follows: the product of x = (xn )n∈N and y = (yn )n∈N is xy = (zn )n∈N , where zn = nj=0 xj yn−j . This multiplication may be interpreted nicely if we think of the elements of F N as “formal power series”, identifying (xn )n∈N with n∈N xn t n . The set F N with these operations is called the ring of formal power series over F , and denoted by F [[t]]. (Here t is just a name for an “indeterminate” over F .) We leave it as an exercise to verify that the ring F [[t]] of formal power series over a commutative field is a commutative ring, and that xy = 0 implies 0 ∈ {x, y} in F [[t]]. The field of quotients of F [[t]] can be obtained as follows. For every element x = (xn )n∈Z ∈ F Z we define Nx := inf {m ∈ Z | xm = 0}. Let K be the set of those elements of F Z whose support is bounded below; that is, the set {(xn )n∈Z | Nx > −∞}. The product of x = (xn )n∈Z and y = (yn )n∈Z is given by xy = (zn )n∈Z , where ! 0 if n < Nx + Ny , zn = n−Ny if n ≥ Nx + Ny . j =Nx xj yn−j
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With component-wise addition and this multiplication, the set K forms a commutative field, denoted by F ((t)). This field is also known as the field of formal Laurent series over F . If F is a finite field, we give it the discrete topology, and take the ring F [[t]] of formal power series with the product topology. This turns F [[t]] into a compact totally disconnected ring. The field F ((t)) is given the (unique) topology containing the product topology on F [[t]] such that the additive group K of F ((t)) becomes a topological group, compare 3.23. We leave it as an exercise to show that this topology turns F ((t)) into a locally compact totally disconnected field. 26.4 Examples. We generalize the construction of the field of formal Laurent series, as follows. Let F be a commutative field, and let σ be an automorphism of F . Again, let K be the set of those (xn )n∈Z ∈ F Z whose support is bounded below, and endow K with component-wise addition. However, we use σ to distort the multiplication that we used in 26.3, and put xy = (zn )n∈Z , where ! 0 if n < Nx + Ny , zn = n−Ny j σ if n ≥ Nx + Ny , j =Nx xj yn−j where Nx := inf {m ∈ Z | xm = 0}, as above. With these operations, the set K forms a field that will be denoted by Fσ ((t%), and called the field of skew Laurent series over F , with respect to σ . The distorted multiplication may be interpreted as a multiplication of “formal Laurent series” x = n≥Nx xn t n and y = n≥Ny yn t n , where xy =
*
n *
n≥Nx +Ny j =Nx
xj t n yn−j t n−j =
*
n *
j
σ xj yn−j t n,
n≥Nx +Ny j =Nx
in other words, using the rule tf = f σ t. We leave the details for an exercise, see Exercise 26.7. The field of formal Laurent series over a finite field has the same characteristic (cf. 26.13 below) as the finite field one starts with; that is, there is a prime number p such that px = 0 for each element x of the field. There are also examples of locally compact totally disconnected fields of characteristic 0; that is, fields that contain a copy of the field of rational numbers (with an unusual topology, though). 26.5 Examples. The compact ring Zp of p-adic integers (see 17.2, Exercise 17.8, or 8.49, 8.58) is an integral domain; that is, a commutative ring such that xy = 0 implies 0 ∈ {x, y} for x, y ∈ Zp . Let Qp be a quotient field of Zp , and give it the unique topology containing the topology of Zp such that the additive group is a topological group, compare 3.23. Since Zp is a neighborhood of 1, it is easy to
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show that this topology turns Qp into a locally compact, totally disconnected field, called the field of p-adic numbers. Again, the details are left for an exercise, see Exercise 26.3. 26.6 More examples. Let C be a commutative topological Hausdorff field, and let F be a finite central extension of C; that is, a field containing C as a subfield of its center such that the dimension d of F as a (left) vector space over C is finite. Via multiplication from the right, we identify F with a subring of the ring C d×d of d × d matrices over C. The product topology turns C d×d into a Hausdorff ring, and F becomes a topological Hausdorff field, compare 3.7. If C is locally compact then this construction yields that F is locally compact, as well. The totally disconnected fields Fp ((t)) and Qp are far from being algebraically closed. Thus there are lots of (commutative) finite extensions (obtained by adjoining a root of an irreducible polynomial). We are going to show that every non-discrete locally compact totally disconnected field is a finite central extension of one of the fields constructed in 26.3 and 26.5. The commutative fields of positive characteristic among these are in fact exactly the fields of formal Laurent series over finite fields. Note that a topological ring may well be neither connected nor totally disconnected: for instance, form the product of a connected ring (like Rn×n ) and a totally disconnected one (like Zp ). In this example, the subsets C := Rn×n × {0} and D := {0} × Zp satisfy CD = {0}. This is a special case of a very general phenomenon. 26.7 Lemma. Let R be a locally compact Hausdorff ring, and let A denote the underlying additive topological group. Then xz = 0 = zx holds for each pair (x, z) ∈ Comp(A) × R0 , and xy = x = yx for each pair (x, y) ∈ Comp(A) × R1 . Proof. For the sake of readability, we abbreviate B = Comp(A) and C = R0 . Let β : B → A denote the inclusion. Now C ρ is a connected subset of Mor(A, A), and βC ρ is connected in Mor(B, A). As Mor(B, A) is totally disconnected by 25.4, we conclude that βC ρ consists of a single element. This element is β0ρ , and xz = 0 = zx is established for each pair (x, z) ∈ Comp(A) × R0 . The rest follows 2 from the observation R1 = 1 + R0 .
Connected Locally Compact Rings 26.8 Theorem. Let R be a connected locally compact Hausdorff ring. Then there is a natural number n such that the additive group of R is isomorphic to Rn , and R is isomorphic to a closed subring of Rn×n . Moreover, this subring contains the center R idRn ; that is, the given ring R is an R-algebra of finite dimension.
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Proof. The underlying additive topological group A is a locally compact connected Abelian Hausdorff group, and Comp(A) = 0 by 26.7. Therefore, there is a natural number n such that A ∼ = Rn . According to 11.12, we have an embedding ρ of R into the topological ring Mor(A, A), which is isomorphic to Rn×n . It remains to show that R ρ contains R idRn . But the additive group of R contains the rational span of 1 (since A is uniquely divisible), and thus R ρ contains the closure R idRn of Q idRn . 2 There are two connected locally compact fields that are well known: the fields of real numbers, and of complex numbers, respectively. In Exercise 3.2, we have introduced a third example: Hamilton’s quaternions H. As H is not commutative, we have thus three non-isomorphic connected locally compact fields. In fact, these three represent all possibilities: 26.9 Theorem. Let F be a connected locally compact Hausdorff field. Then F is isomorphic to one of the three fields R, C, and H. Proof. According to a result of Frobenius, compare [32], Section 7.7, every division algebra of finite dimension over R is isomorphic to one of the three fields R, C, H. 2
Compact Rings The compact ring Zp was introduced in Exercise 18.6 and Exercise 18.7 by a projective limit over a system of finite rings. The next result shows that every compact Hausdorff ring can be described by such a construction. 26.10 Theorem. Every compact Hausdorff ring is totally disconnected. Moreover, the open ideals form a neighborhood basis at 0. Proof. From 26.7 we infer that every compact Hausdorff ring is totally disconnected. Therefore, the additive group A has a neighborhood basis consisting of open subgroups, see 4.13. Let B be an open subgroup of A. For each pair p = (r, s) ∈ R 2 , continuity of multiplication yields the existence of open neighborhoods Up , Vp , and Wp of r, 0, and s, respectively, such B. As R is that Up Vp Wp ⊆ 2 such that U = R = compact, there exists a finite subset F ⊆ R f ∈F f f ∈F Wf . Then V := f ∈F Vf is a neighborhood of 0 satisfying RV R ⊆ B. Let J be the subgroup of B generated by RV R. From RV R = RRV RR we infer that J = RJ R is an ideal of R. This ideal is contained in B, and is open since it contains V . 2 26.11 Corollary. Every compact Hausdorff ring models the projective limit of a 2 projective system of finite rings in TR.
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26.12 Corollary. Every compact Hausdorff field is finite. Proof. Theorem 26.10 means that the only proper ideal (namely, {0}) of a compact Hausdorff field is open. Thus every compact Hausdorff field is discrete, and therefore finite. 2
Compact (and Thus Finite) Fields We know from 26.12 that every compact Hausdorff field is finite. For a deeper understanding of the structure of finite fields and their automorphisms, we need Wedderburn’s result that every finite field is commutative (see 26.17 below), and some information about roots of unity. This information will also be used to derive information about totally disconnected locally compact fields. Let us first recall some basic notions: 26.13 Definitions. For each ring R, there is a unique ring homomorphism ζ : Z → R, mapping z ∈ Z to z·1 ∈ R. Clearly, the image Zζ =: 1 R is the smallest subring of R. Like every ideal in Z, the kernel of ζ is a principal ideal: we find j ∈ N such that ker ζ = j Z. We call j the characteristic of R, and write char R = j . If F is a field of characteristic p > 0 then 1 F ∼ = Z(p) := Z/pZ is called the prime field of F . For fields of characteristic 0, the subring 1 ∼ = Z is not a field, but {a ζ (bζ )−1 | a, b ∈ Z, b > 0} forms a subfield isomorphic to Q, which we call the prime field in that case. In a topological field F , the closure of the prime field is called the topological prime field of F . If F is Hausdorff, the topological prime field is contained in the center of F . Each topological field is a topological vector space over its topological prime field. 26.14 Lemma. Every locally compact Hausdorff ring of positive characteristic is totally disconnected. Proof. The additive group A of a ring of characteristic j > 0 satisfies j A = {0}, and is a torsion group. If A is locally compact, this yields that A is totally disconnected, 2 see 23.29. 26.15 Lemma. For every field F , we have char F ∈ P∪{0}. For every finite field F , there is a positive integer n such that |F | = (char F )n . Proof. The first assertion follows from the observation that Z(char F ) ∼ = Zζ does not contain divisors of zero. The second assertion is an immediate consequence of the fact that F is a vector space over its prime field. 2
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26.16 Lemma. Let p be a prime, and let R be a commutative ring of characteristic p. Then frobp : R → R : x → x p is a ring homomorphism, called the Frobenius endomorphism of R. Proof. Commutativity of R immediately yields that frobp is multiplicative. The binomial expansion (x + y)p = k≤p pk x k y p−k reduces to (x + y)p = x p + y p because the prime p divides pk whenever 1 ≤ k ≤ p − 1. Thus frobp is additive, as well. 2 We are ready to give Witt’s beautiful proof of Wedderburn’s theorem: 26.17 Wedderburn’s Theorem. Every finite field is commutative. Proof. Let F be a finite field. For x ∈ F , put Zx := {y ∈ F | xy = yx} and Cx := Zx ∩ F × . Then Zx is a subfield of F , and so is the center Z := x∈F Zx . Putting q := |Z|, dx := dimZ Zx and d := dimZ F , we obtain |F | = q d and |Zx | = q dx . For each x ∈ F × , the conjugacy class x Fx has |F × |/|Cx | = (q d − 1)/(q dx − 1) elements. Choosing a set R of representatives for the conjugacy classes in F Z, we obtain * qd − 1 ) × q d − 1 = |F × | = Z × ∪ xF = q − 1 + . q dx − 1 x∈R
x∈R
Aiming at a contradiction, we assume Z = F , then d > 1. Using the set d := {a ∈ C× | ord(a) = d}of primitive d-th roots of unity, we define the cyclotomic polynomial d (X) := ω∈d (X − ω). It is well known that this polynomial has
−1 coefficients in Z. Quite obviously, it divides X X e −1 whenever e is a proper divisor of d. For each x ∈ R, we infer that d (q) divides |x Fx |. Since d (q) also divides q d − 1, we conclude that q − 1 is a multiple of d (q), as well. Now the bound 2 |q − ω| > q − 1 yields the contradiction | d (q)| > q − 1. d
We use Wedderburn’s Theorem to refine 26.12: 26.18 Theorem. Every compact Hausdorff field is finite and commutative.
2
In order to show that finite fields exist for each possible order (that is, for each power of any prime, cf. 26.15), and that they are uniquely determined by their order, we use the fact that every commutative field F possesses an algebraic closure AF : that is, a commutative field AF containing F such that every element of AF is the root of some nontrivial polynomial over F , and every polynomial over AF splits as a product of factors of degree 1 in AF [X]. Recall that the algebraic closure is determined up to F -linear isomorphism. For details and proofs, consult any serious introduction to algebra (e.g., see [33], 8.1).
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26.19 Construction. For p ∈ P and n ∈ N {0}, let AFp denote any algebraic n closure of Z(p), and put Fpn := {a ∈ AFp | a p = a}. Then Fpn is a (commutative) field of order pn , its multiplicative group is cyclic, and every automorphism of Fpn is of the form x → x s with s ∈ N. Proof. We have Fpn = Fix(ϕ n ), where ϕ := frobp : AFp → AFp is the Frobenius endomorphism. This shows that Fpn is a subfield of AFp . Each element of Fpn is a n root of the polynomial Xp − X. Over AFp , this polynomial splits as the product of n n pn linear factors. Looking at the formal derivative (X p − X) = pn Xp −1 − 1 = n n −1 ∈ AFp [X], we see that X p − X has no multiple roots. This means X p − X = n a∈Fpn (X − a), and |Fp n | = p follows. The multiplicative group of any finite (commutative) field is cyclic by 6.33. In particular, any automorphism α of Fpn is α n α ns = x s determined by the image a s of a generator a of F× p n , and x = (a ) = a follows. 2 26.20 Corollary. Every finite field F is isomorphic to F|F | . Proof. Since F is commutative, we may assume F ≤ AFp , where p := char F . × Every element a ∈ F × satisfies a |F | = 1. Therefore, each element of F is a root × of the polynomial X |F | − X = X(X|F | − 1), and F ⊆ Fpn follows. 2 26.21 Remark. More precisely, one can show that Aut(Fpn ) is the cyclic group generated by the Frobenius automorphism frobp .
Totally Disconnected Locally Compact Fields 26.22 Lemma. Let R be a locally compact ring. (a) For every compact open subgroup B of the additive group of R, the subset RB := {r ∈ B | Br ≤ B} is an open subring of R. (Note that nothing is said about the existence of compact open subgroups !) (b) The ring R contains a compact open subring if, and only if, it is totally disconnected and there is an invertible element b which is contained in a compact subgroup of the additive group of R. Proof. The set RB is the pre-image of the open subset $B, B% of C(R, R) under the continuous map ρ, and therefore open; compare 11.12. It is closed under multiplication since that operation is associative, and closed under addition because B is an additive subgroup and the distributive law holds. If C is a compact open subring of R, then the invertible element 1 is contained in the compact additive group of C. Conversely, assume that b ∈ R is invertible
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and contained in a compact subgroup D of the additive group of R. We pick a compact open subgroup E of the additive group of R. Then the group B := D + E is compact and open. From bRB ⊆ B we infer RB ⊆ Bb−1 . Therefore, the open 2 (and thus closed) subring RB is compact. Assertion (b) of Lemma 26.22 applies to every non-discrete totally disconnected locally compact field, and to every locally compact ring of positive characteristic (note that such a ring is totally disconnected by 26.14). However, the discrete rings Z and Q show that the additional assumption about invertible elements cannot be dispensed with. Recall also that Weil’s Lemma 6.26 shows that the closure of a cyclic subgroup in a locally compact group is either compact or isomorphic to the discrete group Z. We are going to find a maximal compact open subring in every non-discrete totally disconnected locally compact field. Convergence of powers of elements will be encoded in the following construction (which also leads to very suitable neighborhood bases in certain compact rings). Let R be a ring, let J = R be a left ideal in R, and let n be a positive integer. Inductively, we put J0 := R and Jk+1 := {xy | x ∈ Jk , y ∈ J }. Thus we obtain Jn+1 ⊆ Jn , and {Jk | k ∈ N} forms a filterbasis. 26.23 Lemma. Let R be a compact open subring of a topological field. If J = R is a compact open left ideal of R then k∈N Jk = {0}. In other words: the filterbasis {Jk | k ∈ N} converges to 0, see 1.23. Proof. For each a ∈ R {0}, we have mod ρa = |R/Ra|−1 , see 13.16. If a belongs to J then Ra ⊆ J yields |R/Ra| ≥ |R/J | =: α > 1, and mod ρa ≤ α −1 . −m holds for each x ∈ J . For any Inductively, we conclude m that mod ρx ≤ α nonzero element s in k∈N Jk we would obtain mod ρs ≤ inf{α −m | m ∈ N} = 0. 2 This is impossible. We are going to show (in 26.34 below) that the topology on a disconnected locally compact field is described by an absolute value. This absolute value is constructed from the module function, as follows. 26.24 Definition. Let F be a (totally) disconnected locally compact field. (a) For each a ∈ F {0}, we put |a| := mod ρa . Putting |0| := 0 we extend this to a map ! mod ρa | · | : F → [0, ∞[ : a → 0 called the absolute value on F .
for a = 0, for a = 0,
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(b) For ε ∈ R, we write Cε := {a ∈ F | |a| ≤ ε}. For the following discussion, we consider a fixed (totally) disconnected locally compact field F , and assume that F is not discrete. We are going to show that the map that we call the absolute value on F in fact has the usual properties of an absolute value, cf 26.34 below. 26.25 Lemma. The absolute value | · | : F → [0, ∞[ is continuous. Proof. From 13.4 and 11.11 we know that the restriction of the absolute value to the multiplicative group F × is continuous. It remains to show that it is continuous at 0. Pick a compact open subgroup A0 of the additive group of F , and some a ∈ A0 {0}. According to 4.13, we find an open subgroup A1 ≤ A0 such that a∈ / A1 , and |A0 /A1 | ≥ 2. Proceeding inductively, we find a sequence (An )n∈N of subgroups of A0 satisfying |A0 /An | ≥ 2n . For ε > 0, we pick n ∈ N such that 2−n < ε. As multiplication in F is continuous at 0 and A0 is compact, there is a neighborhood U of 0 such that A0 U ⊆ An . For each u ∈ U , we have |u| = mod ρu = |A0 /A0 u|−1 ≤ |A0 /An |−1 = 2−n < ε. This proves continuity at 0 for the absolute value. 2 26.26 Lemma. Let R be a compact open subring of a Hausdorff topological field. Then each left ideal J = {0} in R is open (and thus closed, and compact) in R. Proof. For a ∈ J {0}, the map ρa is an open map (clearly, the inverse ρa−1 = ρa −1 is continuous). Thus the left ideal J contains the open set Ra, and a belongs to the interior J ◦ of J in R. Therefore, the additive subgroup J is open in R. 2 26.27 Lemma. Let R be an open compact subring of a Hausdorff topological field. Then R is a local ring, that is, the set P of non-invertible elements of R forms an open compact ideal in R. Proof. Clearly, we have RP ∪ P R ⊆ P , in particular −P ⊆ P . It remains to show P + P ⊆ P . Aiming at a contradiction, we assume that there are a, b ∈ P such that a + b ∈ R P . Then c := a(a + b)−1 and d := b(a + b)−1 are elements of P satisfying c + d = 1. Now J := Rc ⊆ P is an open left ideal in R, and the filterbasis (Jn )n∈N converges to 0, see 26.23. For n ∈ N, put sn := nk=0 ck ∈ R. The sequence (dsn )n∈N converges to 1 because dsn = (1−c)sn = 1−cn+1 belongs to 1 + Jn+1 . But this means that the sequence (sn )n∈N converges to d −1 , and we arrive at the contradiction d −1 ∈ J ⊆ P . Now that we know that P is additively closed, it suffices to show that P is open, then P is also closed in R, and thus compact. If P = {0} then R is a compact Hausdorff field, and thus finite and discrete by 26.18. If P contains a nonzero element, we use 26.26 to see that P is open. 2
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26.28 Remark. Quite obviously, the ideal P is the maximal ideal in R. Note that, historically, the ring of p-adic integers was introduced as a localization of the ring Z at the prime p: that is, with the intention to make invertible all elements of Z pZ, ending up with a local ring in the sense of 26.27. 26.29 Lemma. For each ε > 0, the set Cε is a compact neighborhood of 0. Proof. Continuity of the absolute value yields that Cε is closed in F . Choose an open compact subring R of F , and let P denote the set of non-invertible elements in R. We claim that there is a finite set E ⊆ F such that Cε ⊆ P ∪ (P P x)E: this will exhibit Cε as a closed subset of a compact set, and thus as a compact set itself. According to 26.27, the compact ideal P is open in R, and open in F , as well. Consequently, the set P x is open, and P P x is compact. The continuous image K := {|r| | r ∈ P P x} ⊆ ]0, ∞[ of that compact set is compact, as well, and μ := min K is a positive integer. Pick x ∈ P {0}, then α := |x| < 1 holds because of |R/Rx| ≥ |R/P |. For a ∈ Cε P , put m := ma := min {n ∈ N | ax n ∈ P }. Then m ≥ 1, / P implies ax m ∈ / P x since x has an inverse in F . In other words, and ax m−1 ∈ we have a ∈ (P P x)x −m . Now 0 < μ ≤ |ax m | = |a| · |x|m ≤ εα m yields ε ln μ ≤ ln ε + m ln α, and α < 1 leads to ma = m ≤ ln μ−ln bound does ln α . This −k ln μ−ln ε not longer depend on a, and putting E := x | k ∈ N, k ≤ ln α we obtain Cε ⊆ P ∪ (P P x)E, as claimed. 2
Non-Archimedean Valuations 26.30 Definition. Let F be any field. A map ν : F → R ∪ {∞} is called a nonarchimedean valuation on F if the following are satisfied for a, b ∈ F : (a) ν(a) = ∞ ⇐⇒ a = 0, (b) ν(ab) = ν(a) + ν(b), (c) ν(a + b) ≥ min{ν(a), ν(b)}. A map | · | : F → [0, ∞[ is called a non-archimedean absolute value on F if it satisfies (a) |a| = 0 ⇐⇒ a = 0, (b) |ab| = |a| · |b|, (c) |a + b| ≤ max{|a|, |b|}
(the ultrametric inequality).
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Note that ν(1) = 0 = ν(−1) and ν(−a) = ν(a) follow immediately from these properties; similarly, we have |1| = 1 = | − 1| and | − a| = |a|. Moreover, we find ν(n · a) ≥ ν(a) and |n · a| ≤ |a|, for each positive integer n. This strange behavior, in marked contrast to Archimedes’ Axiom which holds for the usual absolute value on the real numbers, is the reason for the term “non-archimedean”. It is easy to see that Rν := {x ∈ F | ν(x) ≥ 0} forms a subring of F , called the valuation ring corresponding to ν. Moreover, the set Pν := {x ∈ F | ν(x) > 0} is an ideal in Rν , called the valuation ideal corresponding to ν. Non-archimedean absolute values are absolute values in the usual sense, the triangle inequality is replaced by the even stronger ultrametric inequality. Clearly, every non-archimedean valuation yields a non-archimedean absolute value: choose α ∈ R with α > 1, and put |a| := |a|ν := α −ν(a) . The freedom of choice for α reflects the fact that we may replace ν by any multiple ρν, where ρ is a positive real number. Quite obviously, we have Rν = Rρν and Pν = Pρν . Non-archimedean valuations (and the corresponding absolute values) play an important role in number theory. This is due to the fact that non-archimedean valuations on the field Q encode properties of prime numbers: 26.31 Example. Each positive rational number q has a unique prime factorization q = p∈P pνp (q) , where νp (q) ∈ Z. For fixed p ∈ P, we extend νp to a nonarchimedean valuation by putting νp (−q) := νp (q) and ν(0) := ∞, of course. This valuation is called the p-adic valuation. In the valuation ring Rνp corresponding to the p-adic valuation of Q, every prime except p is invertible: in fact, for t ∈ P {p} we have νp (t) = 0 = νp (t −1 ), and t −1 ∈ Rνp . The valuation ideal consists of those elements of Q that are properly divisible by p. One may think of Rνp as the ring obtained from Z by “adding inverses” for all elements of Z pZ: in algebra, this process is known as “localization at the prime ideal pZ”. This valuation simply measures how often q is divided by p, whence its importance in number theory. Moreover, these are the only interesting examples of non-archimedean valuations on Q: 26.32 Lemma. For each non-archimedean valuation ν : Q → R with ν(Q× ) = {0}, there are a prime p and a positive real number ρ such that ν = ρνp . Proof. For z ∈ Z, we have ν(z) ≥ 0. As ν(Z {0}) = {0} would imply ν(Q× ) = {0}, the intersection Pν ∩ Z is different from {0}, and there is a positive integer p with Pν ∩ Z = pZ. For a, b ∈ Z with ab = p we find ν(a) ≥ 0 ≤ ν(b) but ν(a) + ν(b) = ν(ab) = ν(p) > 0. Thus one of a, b is contained in Pν ∩ Z = pZ, and p ∈ P has been proved. Putting ρ := ν(p) and using 26.30 (b), we obtain ν = ρνp , as claimed. 2
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We return to the general case of totally disconnected locally compact fields. 26.33 Theorem. Let F be a totally disconnected locally compact field. Then the following hold. (a) There exists an open compact subring R such that R = RP , where P is the maximal ideal of R, cf. 26.22 and 26.27. (b) This property determines R uniquely: in fact, we have R = D1 . (c) Every element of F R belongs to {x −1 | x ∈ R {0}}. Proof. According to 26.22, there exists a compact open subring R0 in F . Every such ring is a local ring by 26.27; we denote the maximal ideal of R0 by P0 . Inductively, maximal ideal of Rn+1 . We we now define Rn+1 := RPn , and let Pn+1 denote the obtain an ascending chain of subrings, its union R := n≥0 Rn is an open subring, again. Since each of the rings Rn is contained in the compact set C1 (cf. 26.29), the open (and thus closed) subring R is in fact compact. We denote its maximal ideal by P . Now R0 is an open subgroup of the compact additive group R, and the quotient R/R0 is finite. This means that the chain R0 ≤ R1 ≤ R2 ≤ · · · becomes stationary: we have Rm = Rm+1 for some m, and R = Rm = Rm+1 = RPm = RP follows. This proves assertion (a). In order to verify assertion (c), consider a ∈ F R. We know from 26.23 that the filter basis {Pj | j ∈ N} converges to 0. Continuity of multiplication at (a, 0) yields the existence of j ∈ N with aPj ⊆ R. The minimal choice for j gives aPj −1 ⊆ R, and aPj = aPj −1 P ⊆ P follows from assertion (a). Thus there is x ∈ Pj with ax ∈ R P , which means (ax)−1 ∈ R and a −1 = x(ax)−1 ∈ R. It remains to prove assertion (b). For a ∈ R, we have mod ρa = |R/Ra|−1 ≤ 1, and R ⊆ D1 follows. For a ∈ F R, we have a −1 ∈ R by c, and 1 ∈ / Ra −1 −1 −1 −1 implies 1 < |R/Ra | = |a | = |a|. This means F R ⊆ F D1 . 2 26.34 Theorem. Let F be a non-discrete locally compact field. If F is totally disconnected, then the absolute value introduced in 26.24 is a non-archimedean absolute value, in the sense of 26.30. More precisely, let R = D1 be the local ring found in 26.33, let P denote its maximal ideal, and write α := |R/P |. Then the map νF : F → Z ∪ {∞} : a → sup{n ∈ Z | |a| ≤ α −n } is a non-archimedean valuation, and one of the corresponding absolute values is ! mod ρa = |R/P |−νF (a) if a = 0, |a| = 0 if a = 0.
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Proof. In the proof of 26.33 (b), we have seen that P = {x ∈ R | |x| < 1}. Since P is compact (cf. 26.27), there exists z ∈ P such that |z| = sup {|x| | x ∈ P }. Putting α := |z|−1 , we obtain P = Cα −1 = zR = Rz and |R/P | = α. Since mod is multiplicative, we infer Pn = Cα −n for each positive integer n. For a ∈ Pn Pn+1 = Cα −n Cα −n−1 , we thus have |a| = α n and a −n ∈ C1 . This gives νF (a) = −n, and |a| = α −νF (a) , as claimed. For a ∈ F R, the inverse a −1 belongs to R by 26.33 (c). Now |a −1 | = α −n gives |a| = α n , and νF (a) = −n follows. We have thus seen that |a| = α −νF (a) holds for each a ∈ F , with the usual convention for |0| = 0 = α −∞ . Multiplicativity of mod (cf. 13.4) immediately yields property 26.30 (b) for νF . It remains to show νF (a + b) ≥ min{ν(a), ν(b)}: this follows immediately from the observation that each of the balls Cα −k = zk C1 = zk R is closed under addition. 2 26.35 Corollary. The topology induced by the metric d(a, b) := |a − b| coincides with the original topology on F . Proof. A basis for the topology induced by the metric d is given by the ε-balls Bε (x) := {y ∈ Z | d(x, y) < ε}, see 1.2. Since the absolute value on F takes its values in {α n | n ∈ Z} ∪ {0}, every one of these balls coincides with one of the sets x + Cα k , with k ∈ Z. Thus the topology induced by the metric coincides with the group topology defined by the filterbasis {Cα k | k ∈ Z} on the additive group of F , cf. 3.22. This filterbasis of open subgroups converges to 0 by 26.23, and thus generates the filter of neighborhoods of 0, with respect to the original topology. Therefore, the topologies coincide. 2
Finding Familiar Subfields Our next aim is to show that every non-discrete totally disconnected locally compact field contains a subfield isomorphic to a p-adic number field Qp or a field Fp ((t)) of Laurent series. For fields of characteristic 0, these subfields will be obtained as the closure of the prime subfield (that is, the smallest subfield, generated by 1 by repeated addition and division). In particular, it will turn out that these subfields are contained in the center. In fields of positive characteristic, we have to work harder to see that the center is not discrete. 26.36 Theorem. Let F be a nondiscrete locally compact field of positive characteristic, let R = D1 be the valuation ring (see 26.33), with maximal ideal P . Then F contains a subfield isomorphic to R/P (and thus isomorphic to F|R/P | ).
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Proof. The order q := |R/P | is a power of the characteristic of R/P . Since F has positive characteristic, we have char F = char R = char R/P , and x → x q is a power of the Frobenius endomorphism of F . We know from 26.17 and 6.33 that (R/P )× is cyclic. Choose a ∈ R such that P + a is a generator of that group. Then n P + a q = (P + a)q = P + a yields a q ∈ P + a, and the sequence (a q )n∈N is contained in the compact set P + a, where it accumulates at some element b = 0. For m > n we observe n m n m n n m−n n n m−n q n a q − a q = a q 1 − a q −q = a q 1 − a (q −1)q = a q 1 − a q . n
m
m−n
∈ P , and Together with a ∈ R P and a q − a q ∈ P , this yields 1 − a q n m q q n a −a ∈ Pq follows. According to 26.23, the filterbasis {Pk | k ∈ N} converges to 0, and our sequence has at most one accumulation point. This shows that the sequence converges to b. n As the Frobenius endomorphism is continuous, we have bq = limn∈N (a q )q = n+1 limn∈N a q = b. Now bq−1 = 1, and P + b = P + a has order q − 1: this means that b has order q − 1. The smallest subfield B of F containing b is commutative. Surely B contains {bk | k ∈ N} = {bk | k ∈ Z, k < q} ⊆ {x ∈ B | x q = x}. This is the set of fixed points of an endomorphism of B, and thus a subfield of B. Minimality of B yields that B = {bk | k ∈ Z, k < q} is a field with q elements. From 26.20 we know that every field with |R/P | elements is isomorphic to F|R/P | . 2 26.37 Lemma. Let F be a nondiscrete locally compact Hausdorff field of positive characteristic with valuation ring R and valuation ideal P , let p ∈ P P2 , and let B be a finite subfield isomorphic to R/P , as in 26.36. Then every element of R is the limit of a series n≥0 bn pn with bn ∈ B. Proof. The subfield B is a set of representatives for R/P because different elements a, b ∈ B have a difference a − b ∈ R P . For x ∈ R, let b0 be the representative j for x, and choose bj inductively as the representative for (x − n=0 bn pn )p−j . j Then x − n=0 bn p n ∈ Pn and the fact that the filterbasis (Pn )n∈N converges to 0 j 2 yield x = limn∈N n=0 bn pn . 26.38 Remark. While addition of elements of R can easily be expressed in terms of power series as in 26.37, multiplication is more delicate because p need not commute with every element of B. 26.39 Theorem. Every nondiscrete locally compact Hausdorff field F of positive characteristic p contains a central subfield isomorphic to Fp ((t)). Proof. First of all, we note that every topological automorphism of F leaves invariant both the valuation ring R and its maximal ideal P because both are uniquely
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determined, see 26.33. In particular, this is true for each inner automorphism ia : x → a −1 xa with a ∈ F × . Thus ia induces an automorphism of R/P ∼ = F|R/P | , and by 26.19 there is some s ∈ N such that P + a −1 xa = P + x s holds for each x ∈ R. For x, y ∈ P P2 , the inner automorphisms ix and iy induce the same automorphism of R/P : in fact, we have x −1 y ∈ R (look at the absolute value!), and ix −1 y = ix−1 iy induces the identity on R/P because that finite field is commutative. According to 26.36, there is a subfield B < F with B ∼ = R/P . We are going to construct p ∈ P P2 such that p −1 bp = bs holds for each b ∈ B (and not only modulo P ). Pick u ∈ P P2 , then u−1 bu ∈ P + bs yields u−1 bub−s ∈ P + 1, and we infer bub−s ∈ P u + u ⊆ P2 + u, for each b ∈ B × . As B × is commutative, we check easily that p := b∈B × bub−s satisfies apa −s = p, for each a ∈ B × . This means that p−1 ap = a s holds for each a ∈ B, and it remains to verify x ∈P P2 . Using the fact that the characteristic of F divides |B|, we compute p = b∈B × bub−s ∈ P2 + b∈B × bub−s = P2 + |B × | · u = P2 + (|B| − 1) · u = P2 − u ⊆ P P2 . Let r be the order of the automorphism induced by ip on R/P . Then z := p r commutes with each element of B, and the subfield generated by B and z centralizes B ∪ {p}. According to 26.37, every element of F lies in the closure of the subring generated by B ∪ {p}. This shows that z belongs to the center of F . It remains to show that the subfield C generated by {z} has a closure isomorphic to Fp ((t)) in F . To this end, we first remark that C is a non-algebraic extension of Fp because z ∈ Pr implies νF (z) = r > 0. This means that z → t extends to an isomorphism of fields from C onto the quotient field Fp (t) of the ring Fp [t] of polynomials. This isomorphism maps the valuation induced on C to a valuation equivalent to the valuation induced on Fp (t) by the one on Fp ((t)). Therefore, we have an isomorphism of topological fields, which extends to an isomorphism of completions from C onto Fp ((t)), cf. 8.25 and 8.57. 2
Locally Compact Vector Spaces In order to show that every (non-discrete) locally compact field is a finite extension of one of the examples that we already know, we take a more general point of view. 26.40 Theorem. Let F be a non-discrete locally compact field, and let V be a topological left vector space over F . If V is Hausdorff then the following hold: (a) For every v ∈ V {0}, the map μv : F → F v : f → f v is a homeomorphism. (b) Every finite-dimensional subspace of V is closed.
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(c) If V is locally compact then F has finite dimension. Proof. Clearly, the map μv is a continuous homomorphism of (additive) groups. Therefore, it suffices to show that μ−1 v is continuous at 0; we will show that, for each ε > 0, the pre-image Cε v of Cε under μ−1 v contains a neighborhood U of 0 in F v. Since F is not discrete, we find f ∈ Cε {0}. As F v is Hausdorff, there is a neighborhood N of 0 in F v such that f v ∈ / N. Multiplication by scalars is a continuous map from F × V to V . Using continuity at (0, 0), we find η > 0 and a neighborhood M of 0 in F v such that Cη M ⊆ N. Picking t ∈ Cη {0} and putting U := C1 tM, we obtain U = C1 U ⊆ Cη M ⊆ N. This is the neighborhood we are looking for: for any sv ∈ U Cε v, we would have |s| > ε and |f s −1 | = |f | · |s|−1 < 1, and f s −1 ∈ C1 would then imply f v = (f s −1 )sv ∈ C1 U = U ⊆ N, contradicting our choice of N. Thus we have proved assertion (a). Assertion (b) is now proved by induction on the dimension of the subspace: one-dimensional subspaces are locally compact by assertion (a), and thus closed in the Hausdorff group V by 4.7. In any subspace W of finite dimension d > 1, we pick a subspace F w of dimension 1. The quotient V /F w is Hausdorff by 6.6, and our induction hypothesis applies to W/F w. The pre-image W of W/F w is closed in V . Finally, assume that V is locally compact. Pick a compact neighborhood U of 0 in V such that C1 U = U , and a sequence of scalars sn ∈ C 1 {0}. Continuity n of multiplication by scalars yields that {sn U | n ≥ 1} forms a neighborhood basis at 0 in V . Pick s ∈ C 1 , then the compact set U is covered by translates of the open 2 set U ◦ , and there is a finite subset E of U such that U ⊆ E + sU ◦ ⊆ E + sU . We claim that the subspace W generated by E in V coincides with V . Suppose otherwise, and pick v ∈ V W . Since V /W is Hausdorff by assertion (b), we find a positive integer n such that (v + sn U ) ∩ W = ∅. Now sk U ⊇ sk+1 U yields that δ := inf {|x| | x ∈ F, (v + xU ) ∩ W = ∅} is positive, and we find y ∈ C 3 δ with |y| > δ. We choose u ∈ U such that v + yu ∈ 2 W , then there exists e ∈ E with v + yu ∈ e + sU . From v + yu ∈ v + yu + ysU we infer v + yu − ye ∈ (v + ysU ) ∩ M = ∅, a contradiction. This shows W = V , 2 and assertion (c) is proved. 26.41 Remarks. Of course, the assertions of 26.40 hold, mutatis mutandis, for right vector spaces, as well. The results are true in even greater generality: what one actually needs is that the topology on the field is induced by the metric defined by an absolute value, and that it is complete with respect to this metric.
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A very special type of vector space is obtained from a field F with a subfield S by restricting the multiplication in F to maps from S × F or F × S to F . This turns F into a left or right vector space over S, respectively. Note that the dimensions may well differ for these two vector spaces ([7], see [8], 11.5 for an overview and [58] for exhaustive results). 26.42 Corollary. Let F be a locally compact Hausdorff field, and let S be a closed, non-discrete subfield. Then the dimensions of F as left or right vector space over S are both finite. 2 26.43 Theorem. Every non-discrete locally compact Hausdorff field has finite dimension over its center. Proof. Let F be a non-discrete locally compact Hausdorff field. If F is not totally disconnected then F is connected by 26.2, and forms an algebra of finite dimension over R, cf. 26.8. In fact, there remain three possibilities, namely R, C, and H, see 26.9. Now consider the case where F is totally disconnected. In the light of 26.42, it suffices to show that the center is not discrete. This has been proved in 26.39 for the case of positive characteristic, and it remains to discuss the case where F is a totally disconnected field containing Q. According to 26.22 (b), there exists a compact open subring R in F . Clearly, the subring Z is contained in R. Now the closure Q contains the infinite compact subring Z, and is not discrete. As the prime 2 field Q is contained in the center of F , the center is not discrete, as claimed. 26.44 Theorem. Assume that F is a locally compact field containing Q as a dense subfield. Then either F = Q is discrete, or F ∼ = R, or there exists a prime p such that F ∼ = Qp . Proof. If F is not totally disconnected then F is connected by 26.2, and forms an algebra of finite dimension over R, cf. 26.8. In fact, there remain three possibilities, namely R, C, and H, see 26.9, and our assumption that Q is dense leaves only the case F ∼ = R. Now consider the case where F is totally disconnected. We have seen in 26.34 that there is a non-archimedean valuation νF on F , and this valuation induces the original topology on F , see 26.35. If Q is discrete (with respect to the topology induced by the restriction νF |Q ) then it is closed in F by 4.6, and F = Q follows. If Q is not discrete then νF (Q× ) = {0} yields that νF |Q is equivalent to a p-adic valuation νp , see 26.32, and induces the p-adic topology Tp . The locally compact closure F of Q is complete by 8.25, and thus is (isomorphic to) the completion Qp of (Q, Tp ), cf. 8.58 and 8.57. 2
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Exercises for Section 26 Exercise 26.1. Show that the ring F [[t]] of formal power series over a finite field F is an integral domain (that is, a commutative ring without zero divisors: xy = 0 implies 0 ∈ {x, y}), and that F ((t)) is a field of quotients for F [[t]]. Exercise 26.2. Show that the product topology on F N makes the ring F [[t]] of formal power series over a finite field a compact totally disconnected ring. Exercise 26.3. Verify that Qp is a topological field. Exercise 26.4. Prove that no commutative finite extension of Qp is algebraically closed. Hint. Investigate norms of n-th roots of p. Exercise 26.5. Show that no commutative finite extension of Fp ((t)) is algebraically closed. Exercise 26.6. Why is C the only algebraically closed locally compact commutative field? Exercise 26.7. Let F be a commutative topological field, and let σ be an automorphism of F . Verify that the set Fσ ((t%) of skew Laurent polynomials over F is a topological field, and determine its center. Exercise 26.8. Let F be a topological commutative field, and assume that σ : x → x¯ is a continuous automorphism of F such that σ 2 = idF . Moreover, assume that there exists ¯ s ∈ F such that a a¯ + bs b = 0 implies (a, b) = (0, 0). Show that the set HF,σ,s := a b a, b ∈ F forms a topological field (with noncommutative multiplication), and ¯ −s b a¯
determine the center. This construction generalizes the field H of Hamilton’s quaternions, see Exercise 3.2. Exercise 26.9. Let F be a finite (commutative) field, with q elements. Verify that |a| = q −Na holds for each a ∈ F ((t)), where Na := inf {n ∈ N | an = 0}. Hint. Consider a ∈ F [[t]] first, and show F [[t]]ρa = {x ∈ F [[t]] | Nx ≥ Na }. Exercise 26.10. Let p be a prime. Show that Xp − t is irreducible over F ((t)). What are the roots of that polynomial, in an algebraic closure of F ((t)) ? Hint. Eisenstein’s criterion for irreducibility may be helpful: consider the ring F [[t]]. Exercise 26.11. Show that the following polynomials are irreducible over F2 : X2 + X + 1,
X 3 + X + 1,
X 4 + X + 1.
What about X 5 + X + 1 ? Exercise 26.12. Let F8 be a field with 8 elements (e.g. the one obtained by adjoining an element a to F2 with a 3 = a + 1, cf. Exercise 26.11). Show that X 2 + X + 1 is irreducible over F8 . Hint. One possibility is to show first that every root of X 2 + X + 1 also is a root of X 3 − 1. Then argue that the multiplicative group of F8 does not contain elements of order 3. Exercise 26.13. Construct fields of order 9, 25, and 49. Hint. Start from fields of order 3, 5 and 7, respectively.
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27 Homogeneous Locally Compact Groups In this chapter, we determine all locally compact Abelian groups with the property that the group of all topological automorphisms acts transitively on the set of nontrivial elements. This will lead to characterizations of certain topological vector spaces. 27.1 Definition. A topological group H is called homogeneous, if the group of all topological automorphisms of H acts transitively on the complement of the neutral element in H . The first examples that come to mind are additive groups of topological (skew) fields, or of vector spaces of finite dimension over topological fields. Our investigation is guided by the question how far a general homogeneous group can deviate from these examples. Of course, the crucial point is that we admit only topological automorphisms. If the topology is ignored then the Abelian case is almost trivial, while the non-Abelian case appears to be very hard. For instance, there are homogeneous non-Abelian discrete torsion-free groups; see [17]. We only consider Hausdorff groups, except in Lemma 27.2. Assertion (a) of that lemma shows that this is no substantial loss of generality.
Homogeneous Topological Groups In a homogeneous group H , there are no characteristic subsets apart from {1} and H {1}. This is the key argument in the proof of the following. 27.2 Lemma. Let H be a homogeneous group. Then the following hold. (a) Either H is a Hausdorff group, or H has the indiscrete topology. (b) Either H is torsion-free and divisible, or H has exponent p (that is, there exists a prime p such that every element of H {1} has order p). (c) Either H is Abelian, or H = {g −1 h−1 gh | g, h ∈ H }, and the center of H is trivial. (d) Either H is covered by compact subgroups (that is, we have Comp(H ) = H ), or H is compact-free (that is, it has no nontrivial compact subgroup at all). (e) Either H is connected, or H is totally disconnected. (f) Either H is arcwise connected, or the arc components are trivial. (g) If H belongs to LCA and has exponent p then H is totally disconnected.
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Proof. The closure of {1} in H is invariant under topological automorphisms. Thus either it is trivial (and H is a Hausdorff group) or it equals H (and H is indiscrete). Thus assertion (a) is established. Since H is homogeneous, every element in H {1} has the same order. If g ∈ H {1} has finite order n, we write n = pd for a prime divisor p of n, and find that g d has order p. Thus H has exponent p. If H is torsion-free, consider x ∈ H {1} and a positive integer n. Then there is an automorphism α mapping x n to x, and x α is the n-th root of x that we search for. This proves assertion (b). The set {g −1 h−1 gh | g, h ∈ H }, the center and the connected component of H as well as the arc component are of course characteristic subsets of H . Thus assertions (c), (e) and (f) follow. In order to prove assertion (d), assume that H contains a compact nontrivial subgroup C. Then every element of H can be mapped into C by a topological automorphism of H . Thus H is covered by the compact images of C under topological automorphisms. Now assume that H is an Abelian locally compact Hausdorff group. If H is connected then H is divisible by 23.27. But a divisible group with pH = {0} is trivial, since every element x ∈ H can be written as x = py = 0. This proves the last assertion. 2 From now on, we only consider Hausdorff groups. We will need the following result from the theory of locally compact Abelian groups. 27.3 Lemma. Let C be a compact Abelian group. Then the following hold: (a) The group C is torsion-free if, and only if, there numbers c and exist cardinal ∗ )c × dp . (Q (Z ) dp for each prime p such that C ∼ = p∈P p (b) If C is torsion-free and connected, then we have dp = 0 for each prime p. (c) If C is torsion-free and totally disconnected, then c = 0. Proof. Assertion (a) follows from 23.24 and observation that (Q∗ )c × {0} 4.24. The ∗ c d p 2 is the connected component of (Q ) × p∈P (Zp ) yields the rest.
Homogeneous Topological Abelian Groups In the sequel, we concentrate onAbelian groups. Non-Abelian homogeneous groups have been studied in [61]. 27.4 Lemma. Let H be a homogeneous topological Abelian group.
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(a) If H is torsion-free, then H is uniquely divisible; that is: for each natural number n, the endomorphism nH : g → ng is bijective. (b) If pH = {0} for a prime p, then H is uniquely divisible by each natural number n which is not divisible by p; that is: for each of these n, the endomorphism nH is bijective (in fact, a topological automorphism). Proof. Let n be a natural number, and assume that n is not divisible by p in case (b). Then nH is surjective by Lemma 27.2 (b). The kernel of nH is {h ∈ H | nh = 0} = {0}, whence nH is injective. If pH = {0}, then nH = n + pzH for each z ∈ Z. Thus we may interpret n as an element of Z/pZ, and obtain (nH )p−1 = id; recall that we assume that p does not divide n. Thus (nH )−1 = (nH )p−2 is continuous, and nH is a topological automorphism. 2 27.5 Remark. Even if nH is bijective, it need not be a topological automorphism of H , see Remark 27.14 below. Every Abelian group is a Z-module in a natural way. If pH = {0}, we obtain a multiplication by scalars from Fp := Z/pZ. If H is uniquely divisible, we put m μ−1 n for m, n ∈ N; this defines a multiplication by scalars from Q. Thus n g := mg we obtain: 27.6 Corollary. (a) If H is torsion-free, then H is a vector space over Q. (b) If pH = 0, then H is a vector space over Fp . We have thus determined all discrete homogeneous Abelian groups: these are just the additive groups of vector spaces. In the case where H is not discrete and torsion-free, it remains open at this point whether multiplication by n1 is continuous for each positive integer n; that is, whether H is a topological vector space over the discrete field Q. In the next section, we will construct examples of homogeneous locally compact Abelian groups that fail to form topological vector spaces. Theorem 27.21 below characterizes those locally compact groups that are vector spaces over non-discrete locally compact fields; see also Proposition 27.19 (d) and Theorem 27.23.
A Class of Examples As minimal divisible extensions of (Zp )d turn out to play a crucial role in the classification of homogeneous locally compact groups, we study these in detail.
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A divisible group D is called a minimal divisible extension of G if G is a subgroup of D, and no divisible proper subgroup of D contains G. If G is Abelian and torsionfree, then there exists a minimal divisible extension of G, and between any two minimal divisible extensions of G there is an isomorphism that fixes G pointwise; see, for example, [15], A.15, A.16; or form the tensor product of the Z-module G with Q. The minimal divisible extension D of (Zp )d has been determined in [15], 25.32b,c: we have D ∼ = D(p, d) := {(qi )i∈d ∈ (Qp )d | supi∈d |qi |p < ∞}. If d is finite then D is isomorphic to (Qp )d . Using 3.23 (a), we construct a group topology on D(p, d), as follows. 27.7 Lemma. Let d be an arbitrary cardinal. Then there is a unique group topology on D(p, d) such that C := (Zp )d is open and retains its original topology. With this topology, the group D(p, d) is locally compact. 2 Whenever the group D(p, d) is considered in the present treatment, we will topologize it as in 27.7. If d is finite, then this topology coincides with the product topology on D(p, d) = (Qp )d . It will turn out that homogeneity alone does not suffice to single out the locally compact vector spaces over Qp among locally compact Abelian groups. Additional topological properties will help to achieve this aim. Recall that the weight of a topological space is the minimum cardinality of a basis for the topology. 27.8 Lemma. Let p be a prime, and let d be a cardinal number. Then the following are equivalent: (a) The cardinal number d is finite. (b) The weight of D(p, d) is countable. (c) The group D(p, d) is σ -compact. Proof. If D(p, d) is σ -compact then the quotient Q of D(p, d) modulo (Zp )d inherits this property. Now Q is discrete because (Zp )d is open. Therefore, the quotient Q is σ -compact exactly if it is countable. As the set Q contains a copy of {0, 1}d , it is countable exactly if d is finite. If D(p, d) has a countable basis, then Q has a countable basis as well, because it is the image of D under a continuous open surjection. 2 Note, by way of contrast, that (Zp )ℵ0 has countable weight since the dual group is countable; compare [15], 24.15. 27.9 Definitions. The following maps will turn out to form automorphisms of the topological group D(p, d) that help to see that D(p, d) is a homogeneous group, for each choice of p, d.
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(a) For any family α = (αi )i∈d of invertible elements αi of the ring Zp , we define α α : D(p, d) → D(p, d) by stipulating ((xi )i∈d ) = (xi αi )i∈d . (b) For any element j ∈ d, let τj : D(p, d) → D(p, d) be the “transvection” defined by ((xi )i∈d )τj = (ti )i∈d , where ti = xi + xj if i = j , and tj = xj . (c) For any element j ∈ d, let γj : D(p, d) → D(p, d) be defined by ((xi )i∈d )γj = (bi )i∈d , where bi = xi if i = j , and bj = pxj . d 27.10 Lemma. For every α ∈ (Z× α , τj , and γj are p ) and every j ∈ d, the maps topological automorphisms of D(p, d).
Proof. It is easily seen that the maps α , τj , and γj are additive. In any of the cases, the inverse map is rather obvious. Thus it remains to show that these maps and their inverses are continuous at 0. As the open subgroup C is invariant under α and τj , it suffices to show that the induced bijections of the compact Hausdorff space C are continuous. But C carries the product topology, and continuity of α and τj follows from the easy observation that composition with the natural projections yields continuous maps. We have to be a little more careful when dealing with γj since C = C γj . However, the image C γj is open again, and the induced bijection from C onto C γj is clearly continuous. 2 27.11 Theorem. For every prime p and every cardinal number d, the group D(p, d) is homogeneous. Proof. Let x = (xi )i∈d and y = (yi )i∈d be nonzero elements of D(p, d). Then there are elements j, k ∈ d such that xj = 0 = yk . Now there are natural numbers m, n such that supi∈d |xi |p < |p−m xj |p = |p−n yk |p > supi∈d |yi |p . The ultrametric inequality for the p-adic norm yields |p−m xj |p = |xi +p−m xj |p for each i ∈ d {j }. For each a ∈ Qp with |a|p = |p−m xj |p , we thus find a family α = (αi )i∈d of invertible elements αi ∈ Zp such that (xi + p −m xj )αi = a for each α maps i ∈ d {j }, and p−m xj αj = a. This means that the automorphism γj−m τj x to the constant family (a)i∈d . Similarly, we find a family β = (βi )i∈d such that maps y to the very same constant. 2 γk−n τk β We conclude this section with some additional information about endomorphisms of D(p, d). 27.12 Lemma. The topological group D(p, d) is a topological Zp -module; that is, the map μ : Zp × D(p, d) → D(p, d) : (a, (xi )i∈d ) → (axi )i∈d is continuous. Proof. As D(p, d) is a topological group, it suffices to show that the restriction of μ to Zp × C is continuous. This is obvious because C = (Zp )d carries the product topology. 2
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27.13 Corollary. Every continuous endomorphism of D(p, d) is Qp -linear. Proof. Every endomorphism of the additive group of a vector space over Q is Q-linear. By continuity of μ, such an endomorphism is also Zp -linear, and the 2 assertion follows from Qp = Zp Q. 27.14 Remark. If d is infinite, then multiplication by p1 is not continuous in D(p, d), since pC is not open in C. Thus D(p, d) is not a topological vector space (even if we try Q with the discrete topology), and there exist Qp -linear bijections that are continuous but not open.
Homogeneous Locally Compact Abelian Torsion Groups 27.15 Lemma. Let H be a locally compact Abelian group, and assume that there exists a prime p such that pH = {0}. Then there exists a cardinal c and a continuous injective homomorphism μ : H → (Z/pZ)c . If H is compact, then this homomorphism is an embedding, and μ may be chosen as a topological isomorphism. Proof. We claim that the set C of all continuous group homomorphisms from H to P := Z/pZ separates the points of H . In fact, the group H is totally disconnected by Lemma 27.2 (g). This implies that every neighborhood U of 0 in H contains a compact open subgroup SU of H ; see 4.13. The quotient H /SU is a discrete Abelian group of exponent p. Thus the (continuous) group homomorphisms from H /SU to P separate the points of H /SU . For every h ∈ H {0} we find a neighborhood U of 0 such that h ∈ / U , and our claim follows. We obtain an injective group homomorphism η from H to P C by putting η(g) := (γ (g))γ ∈C . The product topology on P C renders η continuous. We identify P C with P c , where c is the cardinality of C. Continuous injections of compact spaces into Hausdorff spaces are always embeddings; that is, homeomorphisms onto the image. Therefore, it remains to show that every closed subgroup H of P c is topologically isomorphic to P d for some cardinal d. The dual V of P c is just a discrete vector space over Fp . The annihilator H ⊥ is a subgroup and therefore a subspace of V . Thus the dual of H is isomorphic to the vector space V /H ⊥ . If d is the dimension of this vector space, we obtain that H is isomorphic to P d . 2 27.16 Corollary. If C is a compact totally disconnected Abelian subgroup of a homogeneous topological group H , then there exist a prime p and a cardinal d such that C is isomorphic either to (Z/pZ)d or to (Zp )d .
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Proof. If H is torsion-free, we know from Lemma 27.3 that C ∼ = p∈P (Zp )dp holds for some sequence (dp )p∈P of cardinal numbers. If dp = 0 for some p, we find an element x ∈ C such that the sequence (p n x)n∈N converges to 0, but (q n x)n∈N does not for each prime q different from p. Since H is homogeneous, this implies C ∼ = (Zp )d . 2 If H is not torsion-free, then Lemma 27.15 applies. 27.17 Lemma. For every cardinal c, the group (Z/pZ)c is homogeneous. Proof. We abbreviate P := Z/pZ. The group C := P c is compact, its dual V := (P c )∗ is (isomorphic to) the discrete group of all maps from c to P with finite support. Now V is the additive group of a vector space over Fp . Consequently, the group Aut(V ) acts transitively on the set of hyperplanes in V . For x ∈ P c {0}, the annihilator x ⊥ := {γ ∈ V | γ (x) = 0} is one of these hyperplanes; note that x may be interpreted as a linear form on V . Every automorphism of V is the adjoint αˆ of some α ∈ Aut(C). For every y ∈ C {0}, there exists, therefore, a topological automorphism α of C such that α(x ˆ ⊥ ) = y ⊥ . We infer α((x ⊥⊥ )εC ) = (y ⊥⊥ )εC . ε ⊥ ⊥ Since (x ) C is just the closed subgroup generated by x, see 20.20 and 23.6, we have α(Fp x) = Fp y. Now multiplication by a ∈ Fp {0} is of course a topological automorphism of C, and we conclude that C is homogeneous. 2 We will see in the last section that in fact the groups (Z/pZ)c are the only homogeneous compact groups (irrespective of commutativity); see 27.25. We are now in a position to determine all homogeneous locally compact Abelian torsion groups, because every such group satisfies the assumptions of the following theorem. 27.18 Theorem. Let H be a locally compact Abelian group. If there is a prime p such that pH = {0} then there exists cardinals c and d such that the group H is topologically isomorphic to P c × V , where P = Z/pZ, the group V is a discrete vector space of dimension d over Fp , and P c carries the product topology. For every choice of cardinals c and d, the group P c × V is homogeneous. Proof. According to Lemma 27.2 (g), the group H is totally disconnected. Therefore, we find a compact open subgroup C in H . The group C is isomorphic to P c for some cardinal c by Lemma 27.15. Since H is a vector space over Fp , we find a subgroup V of H such that H = C + V and C ∩ V = {0}. Then V ∼ = H /C is discrete, and therefore closed in H . Thus H is topologically isomorphic to the direct product of C and V . Every automorphism α of C extends to an automorphism α of H by putting α |V = id. Continuity of α is decided in the neighborhood C. Thus α is a topological automorphism of H if α is a topological automorphism of C. If C is finite, we may assume C = {0}. Then H = V is a discrete vector space, and therefore homogeneous. If H /C is finite then H is compact; we may then take C = H , and infer from 27.17 that H is homogeneous.
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It remains to consider the case where both C and H /C are infinite. Let x and y be elements of H {0}. If both x and y belong to C then Lemma 27.17 asserts that we find a topological automorphism α of C such that α(x) = y. The reasoning above shows that α extends to a topological automorphism of H . Therefore, it suffices to show that any element z ∈ H C can be mapped to an element of C by a topological automorphism of H . As both C and H /C are infinite, we have isomorphisms of topological groups from C onto C + := C + Fp z, and from V onto H /C + . Therefore, we find a topological automorphism of H that maps C 2 onto C + . The following observations show that the additive groups of vector spaces over non-discrete locally compact fields form a quite small subclass of the class of homogeneous non-discrete non-compact locally compact Abelian torsion groups. 27.19 Proposition. Let p be a prime. For arbitrary cardinals c and d, put C = (Z/pZ)c and let V be the additive group of a discrete vector space of dimension d over Fp . Then H = C × V is a homogeneous locally compact Abelian group, and the following hold. (a) The group H is non-discrete if and only if c is infinite. (b) The group H is non-compact if and only if d is infinite. (c) The group H is σ -compact if and only if d is countable. (d) The weight of H is countable if and only if both c and d are countable. (e) If c = d = ℵ0 then H is the additive group of a non-discrete locally compact field of characteristic p. (f) If max{c, d} > ℵ0 then H is not the additive group of a topological vector space over a non-discrete locally compact field. Proof. Assertion (c) follows from the observation that H is σ -compact exactly if V is σ -compact: as V is discrete, this means that d is countable. A product (Z/pZ)c has countable weight exactly if c is countable. The discrete vector space V has countable weight exactly if it has countable dimension. This proves assertion (d). If K is a non-discrete locally compact field, then K is a finite extension either of the field of real numbers, or of a field of p-adic numbers, or of a field of formal Laurent series over a finite field, see 26.43 (or cf. [67], Chap. I). In any case, the weight of K is countable, and every nontrivial vector space of finite dimension over K shares this property. Now assume that K has characteristic p. Since the additive group of K is homogeneous and neither discrete nor compact, we obtain that it is isomorphic to a product of (Z/pZ)c by a discrete vector space of dimension d over Fp , where c
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and d are both infinite. If one of these cardinals were uncountable, then this product would have uncountable weight. Thus c = d = ℵ0 , and assertion (e) follows. Locally compact vector spaces over K have finite dimension, see 26.40 (c). Thus the assumption max{c, d} > ℵ0 implies that H has uncountable weight. This yields assertion (f). The rest is clear. 2
Homogeneous Locally Compact Abelian Groups: the Torsion-Free Case 27.20 Theorem. Let H be a locally compact Abelian group. If H is homogeneous, non-discrete and torsion-free, then either H is isomorphic to Rn for some natural number n, or there exist a prime p and a cardinal d such that H is isomorphic to the minimal divisible extension D(p, d) of (Zp )d , topologized as in 27.7. Conversely, all these groups are homogeneous. Proof. Assume first that H is not totally disconnected. Then H is connected by Lemma 27.2. As a connected locally compact Abelian group, the group H is isomorphic to the product of a compact group C and Rn for some natural number n; see 23.11. Since C is a characteristic subgroup of H , we infer that either H ∼ = Rn or H = C holds true. Now Lemma 27.3 (b) asserts that H = C implies that H ∼ = (Q∗ )c holds for some cardinal c. Since the group Q∗ is not arcwise connected but has nontrivial arc component (compare [15], 10.13, 10.15, and 25.4), Lemma 27.2 yields c = 0. Now assume that H is totally disconnected. Then there exists a compact open subgroup C of H , see 4.13. By Corollary 27.16, we have C ∼ = (Zp )d for some cardinal d and some prime p. We claim that H is a minimal divisible extension of C. In order to see this, let D be a divisible subgroup of H containing C, and let g be an arbitrary element of H {0}. Then there is a topological automorphism α of H such that α(g) ∈ C. This implies that the closed subgroup S generated by g is compact. Consequently, the quotient SC/C is compact as well. Since C is open, we have that H /C is discrete. Therefore, the group SC/C is finite, and there exists a natural number n such that ng ∈ C. Since H is uniquely divisible, we conclude g ∈ D, and the claim is established. Being a minimal divisible extension of C ∼ = (Zp )d , the group H is algebraically isomorphic to D(p, d). Since C is open in H , the group H has the topology introduced in 27.7. We have seen in 27.11 that D(p, d) is homogeneous. 2 In 27.14 we have shown that D(p, d) is not a topological vector space if d is infinite. However, the additional assumption that the group in question has
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countable weight rules out the minimal divisible extension of (Zp )d for infinite d; see Lemma 27.8. Combining this observation with Proposition 27.19, we obtain the following. 27.21 Theorem. Let H be a homogeneous locally compact Abelian group. If H has countable weight, and if H is neither compact nor discrete, then H is the additive group of a vector space of finite dimension over some non-discrete locally compact field. 27.22 Proposition. Every homogeneous compact Abelian Hausdorff group is totally disconnected. Proof. According to 27.2 (e), it suffices to consider a connected compact homogeneous group. Now 27.2 (g) yields that H is torsion-free, and H is trivial by Theorem 27.20. 2 27.23 Theorem. If H is a homogeneous connected locally compact Abelian group, then there exists a natural number n such that H is isomorphic to the additive group Rn . Proof. If the group H is compact, it is isomorphic to R0 by Proposition 27.22. If H is not compact, we find a maximal compact subgroup C and a vector subgroup V ∼ = Rn such that H = C ⊕ V , compare 23.11. Now C = Comp(H ) = H implies C = {0}; see 27.2 (d). 2
Subgroups of Homogeneous Groups Now that we know all homogeneous locally compact Abelian groups, we can infer the following. 27.24 Theorem. Let H be a locally compact homogeneous Abelian group. (a) If H is a torsion group (in particular, if H is compact) then every closed subgroup of H is homogeneous. (b) If H is torsion-free then a closed subgroup S of H is homogeneous exactly if it is divisible. Proof. If H is a torsion group then it is elementary Abelian. Every closed subgroup is an elementary Abelian locally compact group again, and thus homogeneous by 27.18. Now assume that H is torsion-free. Every homogeneous subgroup is divisible by 27.2 (b). Conversely, let S be a closed divisible subgroup of H . Then S is a closed
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Q-subspace; and therefore an R-subspace if H ∼ = Rn . In the remaining cases, we ∼ have H = D(p, c) by 27.20. As S is locally compact and totally disconnected, we find a compact open subgroup U of S. According to 27.16, there is a cardinal d such that U ∼ = (Zp )d . The divisible hull V = QU is an open subgroup of S. Picking a vector space complement W to V in S, we observe that W is discrete, and S = V ⊕ W . As S is closed, we have that S is a Zp -submodule of H , see 27.12. This means W = {0}, and S = V is the divisible hull of U ∼ = (Zp )d . According to 27.11, the group S is homogeneous. 2 Compactness is another strong topological condition to be imposed on homogeneous groups, it has consequences that are analogous to those of connectedness. 27.25 Theorem. Let H be a compact group. Then the following are equivalent: (a) H is homogeneous. (b) H is elementary Abelian. (c) H is isomorphic to (Z/pZ)c for some prime p and some cardinal c. Proof. Let H be a compact Abelian group. Without loss, we assume that H is nontrivial. Assume first that H is homogeneous. Then H is totally disconnected by Proposition 27.22. Therefore, we find a proper open normal subgroup N in H . Being compact and discrete, the quotient H /N is finite. For every h ∈ H , there exists a prime p such that the closed subgroup generated by h is either isomorphic to Zp or to Z/pZ; compare 27.16. Since Zp has no closed subgroups apart from p n Zp for n ∈ N, we conclude that H /N is a p-group. Therefore, the group H /N is nilpotent, and we conclude from 27.2 (c) that H is Abelian. If the group H were torsion-free, then it would be trivial by Theorem 27.20. According to Lemma 27.2, there exists a prime p such that H has exponent p. Thus we have shown that assertion (a) implies assertion (b). We have seen in 27.15 that assertion (b) implies assertion (c). According to Lemma 27.17, the groups (Z/pZ)c are homogeneous. Therefore, assertion (c) implies assertion (a). 2
Résumé 27.26 Theorem: Homogeneous locally compact groups. The class of homogeneous locally compact Abelian groups is the union of the following subclasses: Conn: additive groups of vector spaces of finite dimension over the reals. Vect: discrete vector spaces over the rationals.
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TopVect: additive groups of vector spaces of finite dimension over a non-discrete
locally compact field. DivExt: minimal divisible extensions of arbitrary powers of the compact group
of p-adic integers, for a prime p. Comp: compact elementary Abelian groups, see 27.17. TorAb: locally compact elementary Abelian groups, see 27.18.
Notice that Conn ∪ DivExt ∪ TorAb properly contains TopVect, and that TorAb properly contains Comp. Our results say that, among the homogeneous locally compact Abelian groups, • the connected ones are exactly those in Conn, • the compact ones are exactly those in Comp, • the torsion groups are exactly those in TorAb, • the torsion-free groups are exactly those in Conn ∪ DivExt ∪ Vect, • the non-discrete, non-compact ones of countable weight are just those in TopVect. 27.27 Remark. Using deep results about approximation of locally compact groups by Lie groups, the results of the present section have been extended to non-Abelian locally compact homogeneous groups, see [61]. In particular, it turns out that such a group has to be totally disconnected and non-compact (another instance where connectedness and compactness turn out to be rather restrictive). Discrete nonAbelian homogeneous groups have been constructed by G. Higman, B. Neumann and H. Neumann, see [17]. We do not know of any non-discrete example of a locally compact homogeneous non-Abelian group.
Chapter G
Locally Compact Semigroups 28 Topological Semigroups Recall from 3.2 that a semigroup is a nonempty set S with an associative binary operation μ : S × S → S. If S is a topological space and μ is continuous, we speak of a topological semigroup. We say that S is a compact, locally compact, or connected semigroup, respectively, if S is a topological semigroup such that the underlying topological space has the property in question. Mainly, we will use multiplicative notation, and write simply S instead of (S, μ). Note that we do not require that (S, μ) has a neutral element. In 28.18 below we introduce a method to attach a neutral element to a topological semigroup.
Basic Examples and Elementary Properties Topological semigroups occur quite naturally as multiplicative structures of topological rings, but also as subsemigroups of topological transformation groups. 28.1 Examples. (a) With the usual addition and multiplication, the set N of nonnegative integers provides us with two (quite different) semigroups (N, +) and (N, ·). (b) Let S be any semigroup, written multiplicatively. With the restricted multiplication, any nonempty subset T ⊆ S with T T ⊆ T forms a subsemigroup of S. Note that, in contrast with subgroups, one cannot expect that equality holds: in general, one has T T = T . (c) Let ω : X × G → X be an action of the group G on the set X; compare 10.1. For any nonempty subset Y ⊆ X, we have the compression semigroup compr G (Y ) := {g ∈ G | ∀y ∈ Y : (y, g)ω ∈ Y }. Although, at a first glance, this looks like the stabilizer of Y in the induced action on the power set (see Exercise 10.1), the subset compr G (Y ) may be larger than this stabilizer. In this case, it forms a proper semigroup. For a concrete example, take the group Z with its regular action (by addition) on X := Z, and consider Y := N. Then compr Z (N) = N.
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The following example shows that the existence of a continuous multiplication imposes less constraint on the topology than the existence of a compatible group structure (compare 6.6, for instance). 28.2 Example. Let X be a nonempty, but otherwise arbitrary topological space. Then the projection π : X × X → X : (x, y) → x makes (X, π ) a topological semigroup. This is a first indication that the existence of an embedding into a topological group imposes strong restrictions on topological semigroups. We will meet a simple algebraic restriction in 28.4 below. 28.3 Definitions. Let S be a semigroup, written multiplicatively. (a) An element p ∈ S is called an idempotent in S if pp = p. (b) An element z ∈ S is called a zero in S if Sz = {z} = zS. (c) An element e ∈ S is called an identity in S if xe = x = ex holds for each x ∈ S. We remark that a semigroup can possess at most one zero, and at most one unit. Clearly, zeros and identities are idempotents. Usually, one denotes a zero by 0, and an identity by 1. (d) An element u ∈ S is called a unit in S if S possesses an identity e and e ∈ uS ∩ Su. (e) The semigroup S is called cancellative from the right if for each r ∈ S the equality xr = yr implies x = y. (f) The semigroup S is called cancellative from the left if for each l ∈ S the equality lx = ly implies x = y. (g) The semigroup S is called cancellative if it is cancellative both from the left and from the right. The example 28.2 shows that a semigroup may have idempotents in abundance, and that a semigroup may be cancellative from the right while being far from cancellative from the left. That example also shows that a semigroup may have many right ideals that are not left ideals. Using inverses of subsemigroup elements inside a surrounding group, one obtains the following. 28.4 Lemma. Subsemigroups of groups are cancellative.
2
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Note, however, that not every cancellative semigroup can be embedded in a group; see [6], 12.30. 28.5 Definitions. Let S be a semigroup. A nonempty subset T ⊆ S is called – a subsemigroup if T T ⊆ T , – a left (semigroup) ideal if ST ⊆ T , – a right (semigroup) ideal if T S ⊆ T , – a (semigroup) ideal if ST ∪ T S ⊆ T . A subsemigroup G of S is called a subgroup of S if it possesses an identity and, as a semigroup in its own right, consists of units. (We leave it as an exercise to show that every subgroup of a semigroup “is” in fact a group.) 28.6 Examples. Let S be a semigroup. (a) If z is a zero in S then {z} is an ideal in S. (b) For any x ∈ S, the set xS is a right ideal, and the set {x} ∪ xS is the smallest right ideal in S containing x. (c) For any x ∈ S, the set Sx is a left ideal, and the set {x} ∪ Sx is the smallest left ideal in S containing x. (d) For any x ∈ S, the set SxS is an ideal, and the set {x} ∪ Sx ∪ xS ∪ SxS is the smallest ideal in S containing x. The precautions taken in the construction of the smallest (left, right) ideal containing x are only needed if S does not have an identity. However, examples like those constructed in 28.2 shows that these precautions are necessary in general. Proceeding as in the proofs of 4.2 and 4.3, one obtains the following. 28.7 Lemma. Let S be a topological semigroup, and let T ⊆ S be a subset. (a) If T is a subsemigroup then the induced topology turns T into a topological semigroup. (b) If T is a subsemigroup then the closure T is a subsemigroup, as well. (c) If T is a (left, right) ideal in S then T is a (left, right) ideal in S, as well. (d) If the topology of S belongs to the class T1 and T is a commutative subsemigroup then T is commutative, as well. 2 The passage to quotients is much more delicate. We will have to revise our point of view of the kernel of a homomorphism.
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Semigroup Homomorphisms and Congruences Of course, a semigroup homomorphism ϕ : (S, μ) → (T , ν) is a mapping ϕ : S → T satisfying the following condition: ∀x, y ∈ S : (x, y)μϕ = (x ϕ , y ϕ )ν or, using multiplicative notation: ∀x, y ∈ S : (xy)ϕ = x ϕ y ϕ . An inexperienced reader might conjecture that semigroup ideals essentially describe all semigroup homomorphisms, just as ring ideals (or normal subgroups) essentially describe ring (or group) homomorphisms as passage to the quotient by the kernel. However, this is not the case: for instance, full pre-images of the images of different elements of S under a semigroup homomorphism ϕ : S → T may have different sizes. Examples may be constructed using the semigroup (X, π ) defined in 28.2: in fact, every map ϕ : X → X gives a semigroup homomorphism ϕ : (X, π ) → (X, π). The kernel of a group homomorphism provides a simple and economical way to describe the so-called kernel relation, which can be defined for any map ϕ : X → Y between sets X and Y : the kernel relation ∼ϕ is the binary relation on X defined by a ∼ϕ b ⇐⇒ a ϕ = bϕ . 28.8 Definition. Let S be a semigroup. A binary relation ∼ on S is called a congruence on S if it is an equivalence relation and ac ∼ bd holds whenever a, b, c, d ∈ S satisfy a ∼ b and c ∼ d. Clearly, the kernel relation ∼ϕ for a semigroup homomorphism ϕ : S → T is a congruence on the semigroup S. The interesting fact is that the converse is also true. 28.9 Lemma. Let (S, μ) be a semigroup, and let ∼ be a congruence on S. Then the set S/∼ = {[x]∼ | x ∈ S} of equivalence classes is a semigroup with the multiplication μ∼ mapping ([x]∼ , [y]∼ ) to [(x, y)μ ]∼ . If S is written multiplicatively, we will use multiplicative notation for S/∼, as well, writing [x]∼ [y]∼ := [xy]∼ . Proof of 28.9. It is an immediate consequence of the definition of a congruence that μ∼ is well defined. The associative law for μ∼ follows directly from the fact that μ is associative. 2
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28.10 Examples. (a) Every ideal J in a semigroup S defines a congruence ∼J on S: we put ∼J := (x, y) ∈ S 2 x, y ∈ J ∪ {(x, x) | x ∈ S}. In other words, two elements of S fulfill the relation if either they are equal or both belong to the ideal. The semigroup S/∼J is obtained by “shrinking the ideal to a single point”, and this point is a zero in S/∼J . (b) Whenever more than one equivalence class of some congruence ∼ on a semigroup S consists of more than one element, the congruence cannot be obtained from an ideal as in a. The same is true if S/∼ does not have a zero. While the passage to quotients of topological semigroups by congruence relations may be topologically delicate, everything works out fine in the case of locally compact semigroups, that is, topological semigroups where the underlying space is locally compact. 28.11 Theorem. Let (S, μ) be a locally compact semigroup, and let ∼ be a congruence on (S, μ). If S/∼ is a locally compact space then (S/∼, μ∼ ) is a topological semigroup (and thus a locally compact semigroup). Proof. Let π : S → S/∼ denote the natural map. We decompose the natural map from C × C onto C/∼ × C/∼ as a composition of the two maps α : C × C → C × C/∼ and β : C × C/∼ → C/∼ × C/∼ defined by (x, y)α := (x, [y]∼ ) and (x, [y]∼ )β := ([x]∼ , [y]∼ ). According to Whitehead’s Theorem 9.12, the maps α and β are quotient maps. Now continuity of μπ = αβμ∼ implies that μ∼ is 2 continuous. 28.12 Remark. Note that we did not need Whitehead’s Theorem for the discussion of quotients of topological groups in 6.2, where we knew that the quotient maps modulo normal subgroups (describing the congruences) are open maps. For topological semigroups in general, congruences are not as well-behaved.
Subgroups in Semigroups Every subgroup in a semigroup contains an idempotent: namely, the neutral element of the subgroup. Conversely, every idempotent defines a subgroup, as follows. 28.13 Definition. Let S be a semigroup, and let e be an idempotent in S. We put ⎫ ⎧ ae a = e = aae ⎬ ⎨ G(e) := GS (e) := a ∈ S ∃ae ∈ S : ae = a = ea . ⎭ ⎩ ae e = ae = eae
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The subsets G(e) form maximal subgroups in the following sense: 28.14 Proposition. Let e be an idempotent in a semigroup S. (a) The set G(e) forms a group, with neutral element e, and ae being the inverse for a ∈ G(e). (b) If G is any subgroup of S having nonempty intersection with G(e) then G is contained in G(e). (c) If f is also an idempotent in S, we have G(e) ∩ G(f ) = ∅ ⇐⇒ e = f . Proof. Assertion (a) is checked by trivial verification. In order to prove assertion (b), we consider the neutral element n of G. Clearly this is an idempotent in S, and G ⊆ GS (n) follows (we use the inverse of g ∈ G inside the group G for gn ). Now consider a ∈ G ∩ GS (e) and observe n = an a = an (ae) = (an a)e = ne = n(aae ) = (na)ae = aae = e. The last assertion follows by combination of the previous ones. 2 In general, the closure of a subgroup of a topological semigroup need not be a subgroup, but only a subsemigroup (this is different in compact semigroups, see 30.7 below). Thus one should not expect that the subgroups GS (e) are closed. In fact, the semigroup obtained on the interval [0, ∞[ with the usual topology and multiplication provides an example: the subgroup G(1) is not closed. We are going to introduce a binary relation that will be used for an embedding of suitable semigroups into groups in 29.2 below. 28.15 Definition. Let S be a semigroup, and define the binary relation on S by x y ⇐⇒ {x} ∪ xS ⊆ {y} ∪ yS. This relation is called the right invariant pre-order on S. One checks easily that is in fact a pre-order, that is, a reflexive and transitive relation. The semigroup S is called directed (with respect to the right invariant pre-order) if (S, ) is a directed set. Note that it is a quite restrictive assumption on the semigroup S that (S, ) be directed. The following statements are easily verified, we leave the details for the exercises. 28.16 Examples. (a) A semigroup S is directed if, and only if, any two right ideals in S have nonempty intersection. (b) Every commutative semigroup is directed.
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(c) Every group, considered as a semigroup, is directed. (d) Subsemigroups of directed semigroups need not be directed. For instance, the subset S := a0 b1 a, b ∈ N, a = 0 of GL(2, Q) carries a semigroup which is not directed with respect to the right invariant pre-order. (e) Of course, one may also introduce a left invariant pre-order on any semigroup by considering left ideals instead of right ideals. It does happen that a semigroup is directed with respect to the right invariant pre-order, but not with respect to the left invariant pre-order. For instance, consider the semigroup introduced in assertion (d).
Attaching an Identity or a Zero 28.17 Definitions. Let S be a topological semigroup. (a) Assume that there is no identity in S. Without loss (that is, up to a change of notation), we may assume 1 ∈ / S. We endow S 1 := S ∪ {1} with the sum topology and extend the multiplication of S to a binary operation μ : S 1 ×S 1 → S 1 by putting (x, 1)μ = x = (1, x)μ . (b) Now assume that there is no zero in S. Without loss, we may now assume 0 ∈ / S. We endow S 0 := S ∪ {0} with the sum topology and extend the multiplication of S to a binary operation μ : S 0 × S 0 → S 0 by putting (x, 0)μ = 0 = (0, x)μ . If S contains an identity, we simply put S 1 := S. Similarly, if S contains a zero, we put S 0 := S. We will assume in these cases that the identity is called 1 and the zero is called 0. (Note, however, that one should be careful with these conventions if the semigroup is not written multiplicatively!) The symbol S 0 should not be confused with S ◦ , the latter denoting the interior of S with respect to some fixed topological space surrounding S. Easy verification yields: 28.18 Proposition. Let S be a topological semigroup. Then S 1 and S 0 are topological semigroups with identity 1 and zero 0, respectively. Moreover, we have (S 1 )0 = (S 0 )1 . 2 28.19 Lemma. Let S be a semigroup. (a) If S is a directed semigroup then S 1 is directed, as well. (b) If S is cancellative then S 1 is also cancellative.
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Proof. Assertion (a) follows immediately from the observation that the definition of the right invariant pre-order uses the semigroup S 1 rather than S. In order to prove assertion (b), we assume that S is cancellative, but there are x, y, r ∈ S 1 with x = y and xr = yr. Then r = 1, and cancellativity of S implies 1 ∈ {x, y}. Without loss of generality, we assume x = 1. For any s ∈ S, we have sr = s(1r) = s(yr) = (sy)r, and cancellativity from the right yields s = sy. In particular, we have yy = y. Now we obtain ys = yys and s = ys since S is cancellative from the left. This means that y is an identity in S, and S = S 1 . This is the contradiction 2 we were aiming at. The proof also indicates why S 1 was defined to be S if S possesses an identity. Assertion 28.19 (b) cannot be generalized to cancellativity from the right: for instance, consider a semigroup as in 28.2.
Exercises for Section 28 Exercise 28.1. Let ω : X × G → X be an action of the group G on the set X, and let Y ⊆ X be a nonempty subset. (a) Verify that compr G (Y ) := {g ∈ G | ∀y ∈ Y : (y, g)ω ∈ Y } carries a subsemigroup of G. (b) Prove that compr G (Y ) is a subgroup of G if Y is finite. (c) Find an example where compr G (Y ) is not a subgroup of G. Exercise 28.2. Verify that the multiplication π defined in 28.2 is associative. Determine the idempotents in (X, π ), and discuss cancellativity properties of that semigroup. Exercise 28.3. Let (Sα )α∈A be a family of topological semigroups. Prove that the cartesian product Xα∈A Sα forms a topological semigroup if endowed with the product topology and component-wise multiplication. What about a universal property? Exercise 28.4. Prove that a semigroup can have at most one zero, and at most one identity. Exercise 28.5. Assume that S is a semigroup with identity 1. (a) Show that for each unit u ∈ S there is exactly one element u ∈ S that satisfies u u = 1 = uu . (b) Verify that the set of all units in S forms a subgroup of S. Exercise 28.6. Let (S, μ) be a semigroup, and assume that G is a subgroup of this semigroup. Let μ : G2 → G denote the restriction of μ. Show that there exist a unary operation ι : G → G and a nullary operation ν : {0} → G such that (G, μ , ι, ν) is a group, compare 3.1. Exercise 28.7. Let S be a locally compact semigroup. For a ∈ S, define maps ρa and λa from S to itself by putting x ρa = xa and x λa = ax. Recall from 9.5 that C(S, S) is a topological semigroup.
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(a) Show that ρ is a continuous semigroup homomorphism from S to C(S, S), and that λ is a continuous “anti-homomorphism” from S to C(S, S). (b) Show that the semigroup S is cancellative from the right if, and only if, its image S ρ consists of injective maps. (c) Give an example to show that ρ need not be injective, even if S is right cancellative. (d) Show that ρ and λ are embeddings of topological spaces (in the sense of 1.1) if S contains an identity.
29 Embedding Cancellative Directed Semigroups into Groups In this section, we are going to discuss a sufficient (but not necessary) condition for a semigroup to be a subsemigroup of a group. This condition goes back to O. Ore [44]. Of course, we are interested in embeddings of topological semigroups into topological groups. For the present section, we will have to be satisfied with an embedding as an open subsemigroup in a group with continuous multiplication. We will see later in 31.2 that every group whose multiplication is continuous with respect to a locally compact Hausdorff topology has continuous inversion, as well. 29.1 Proposition. Let G be a group, and let S be a subsemigroup of G. (a) If SS −1 := {xy −1 | x, y ∈ S} is a subgroup of G then S is a directed semigroup. (b) Conversely, if the semigroup S is directed then SS −1 is a subgroup. Proof. Assume that SS −1 is a subgroup, and consider x, y ∈ S. Since SS −1 contains x = x 2 x −1 and y = y 2 y −1 , we have x −1 y ∈ SS −1 . This means that there are s, t ∈ S such that x −1 y = st −1 , and we obtain xs = yt ∈ xS ∩ yS. This proves assertion (a). Now assume that S is directed. Clearly the set SS −1 contains the neutral element, and is closed under taking inverses. It remains to show that products of elements of SS −1 belong to SS −1 , as well. For a, b, c, d ∈ S we find x, y ∈ S with bx = cy because S is directed, and compute (ab−1 )(cd −1 ) = ab−1 bx(cy)−1 cd −1 = (ax)(dy)−1 ∈ SS −1 , as required. 2 If S is a directed subgroup of a topological group G and S has nonempty interior S ◦ in G then SS −1 is an open subgroup. However, it may happen that SS −1 is an open subgroup although S ◦ = ∅: for instance, consider the subsemigroup N {0} in Q× , with the usual topology. 29.2 Theorem. Let S be a cancellative and directed topological semigroup, and assume that S satisfies the following condition:
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(OM)
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For each a ∈ S, the maps ρa and λa from S to itself defined by putting x ρa = xa and x λa = ax are open maps.
Then there is a group G with a topology on G in such a way that the multiplication is continuous and that S is embedded as an open subsemigroup of G. Note that it is not clear whether inversion in the group G is continuous. Indeed, there are examples of topological semigroups that are groups, but where inversion is not continuous, see Exercise 29.3. In 29.4 below we will prove that if the semigroup S in 29.2 is locally compact Hausdorff then G is locally compact Hausdorff, as well, and inversion is continuous. Proof of 29.2. For each x ∈ S, we put Dx := S × {x}. Thus we obtain a family := (Dx )x∈S of disjoint sets. The union G x∈S Dx is nothing but the cartesian product S × S. However, we give it the sum topology, such that each of the “horizontals” Dx is embedded as an open subset. In fact, the map ιx : Dx → G defined by s ιx := (s, x) is an open embedding. we define a binary relation ≡ by On the set G (s, x) ≡ (t, y) ⇐⇒ ∃a, b ∈ S : (sa, xa) = (tb, yb) and claim that ≡ is an equivalence relation. Indeed, the relation is clearly symmetric and reflexive. In order to see that ≡ is transitive, consider (s, x), (t, y), (u, z) ∈ with (s, x) ≡ (t, y) and (t, y) ≡ (u, z). Then there are a, b, c, d ∈ S with G (sa, xa) = (tb, yb) and (tc, yc) = (ud, zd). As S is directed, we find e, f ∈ S such that be = cf , and obtain (sae, xae) = (tbe, ybe) = (tcf, ycf ) = (udf, zdf ). Thus (s, x) ≡ (u, z) holds as required. and [s, x] := [(s, x)]≡ . On the set G, we define a binary We write G := G/≡ operation μ as follows. For [s, x]≡ , [t, y]≡ ∈ G we pick u, v ∈ S with xu = tv and put ([s, x]≡ , [t, y]≡ )μ := [su, yv]≡ . A unary operation ι on G is simply obtained by putting [s, x]ι≡ := [x, s]≡ . We have to check that these operations are well defined. Assume that [s, x]≡ = [s , x ]≡ and [t, y]≡ = [t , y ]≡ , and that x u = t v . Then there are a, b, c, d ∈ S such that (sa, xa) = (s b, x b) and (tc, yc) = (t d, y d). We pick e, f, g, h ∈ S such that (xa)e = (tc)f and ug = (ae)h. Then we have tvg = xug = xaeh = tcf h, and cancellativity yields vg = cf h. Now we pick k, l ∈ S with u k = (beh)l. This leads to t df hl = tcf hl = xaehl = x behl = x u k = t v k, and cancellativity yields df hl = v k. We obtain (su, yv) ≡ (sug, yvg) = (saeh, ycf h) ≡ (sae, ycf ) = (s be, y df ) ≡ (s behl, y df hl) = (s u k, y v k) ≡ (s u , y v ), and see that μ is well defined. In order to check that ι is well defined, we just note that (s, x) ≡ (s , x ) is equivalent to (x, s) ≡ (x , s ). We will write μ multiplicatively from now on. The next aim is to show that μ is associative. To this end, consider elements [s, x]≡ , [t, y]≡ , [w, z]≡ ∈ G, and pick
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u, v, p, q ∈ S such that xu = tv and yvp = wq. Now we compute ([s, x]≡ [t, y]≡ ) [w, z]≡ = [su, yv]≡ [w, z]≡ = [sup, zq]≡ and [s, x]≡ ([t, y]≡ [w, z]≡ ) = [s, x]≡ [tvp, zq]≡ = [sup, zq]≡ . For any a ∈ S, the element [a, a]≡ clearly is an identity in the semigroup (G, μ), and [s, x]≡ [x, s]≡ = [sx, sx]≡ = [a, a]≡ shows that [s, x]ι≡ is an inverse of [s, x]≡ in this semigroup. Thus G is a group. We have to introduce a topology on G such that μ is continuous. We use the → G given by (s, x)q = quotient topology with respect to the natural map q : G with the quotient map q, [s, x]≡ . Composing the open embedding ιx : Dx → G we obtain a continuous map ιx := ιx q : Dx → G. We claim that ιx is an open embedding. We have to check that ιx is injective, and an open map. Equality (s, x)ιx = (t, x)ιx of images means (s, x) ≡ (t, x) and is thus equivalent to the existence of a, b ∈ S with sa = tb and xa = xb. Using cancellativity twice, we obtain first a = b and then s = t, and see that ιx is injective. Now let U be open in Dx . This means that there is an open set V in S such that U = V × {x}. In order to see that U ιx = U q is open in G, we have to show that ← For s ∈ S, we use the continuous and open the full pre-image U qq is open in G. maps λs and ρs from S to itself, and write ← | ∃s ∈ V : (s, x) ≡ (t, y) U qq = (t, y) ∈ G | ∃s ∈ V ∃a ∈ (xS)λy : (ta, ya) = (sdx,ya , xdx,ya = (t, y) ∈ G ) ) ← = (t, y) ∈ Dy | t ∈ (V dx,ya )ρa . y∈S a∈(xS)λy ←
Because the sets V dx,ya are open by our assumption (OM), we see that U qq is carries the sum topology, this argument also shows that q is an open open. Since G map. In order to show that μ is continuous at ([s, x]≡ [t, y]≡ ), it suffices to consider a suitable neighborhood. We fix an element r ∈ (xS)λt ← . According to our assumption (OM), the map λx induces a homeomorphism from S onto xS. We denote the inverse of this homeomorphism by ϕ. For each u ∈ S and each v ∈ ← {a | ar ∈ xS} = (xS)ρr , we have [u, x]≡ [v, y]≡ = [u(vr)ϕ , yr]. This shows that ← μ is continuous on the neighborhood S ιx × (xS)ρr ιy of ([s, x]≡ [t, y]≡ ) in G. In order to obtain an embedding of the semigroup S into the group G, we pick an arbitrary element a ∈ S and define ηa : S → G by s ηa := [sa, a]≡ . (If S has an identity, one would of course prefer to choose that element for a). In order to
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see that ηa is a semigroup homomorphism, we consider elements s, t ∈ S, pick x, y ∈ S such that ax = ty, and compute s ηa t ηa = [s, a]≡ [t, a]≡ = [sax, ay]≡ = [sty, ay]≡ = [st, a]≡ = (st)ηa . 2 η
29.3 Proposition. In the group G constructed in 29.2, the subsemigroup T := Sa satisfies T T −1 = G; compare 29.1. For each g ∈ G, we have that T g is an open neighborhood of g in G. Proof. Multiplication by g −1 is continuous, and T g is the pre-image of the open subset T . This proves the last assertion, the rest is clear. 2 29.4 Corollary. If S is locally compact Hausdorff then G is locally compact Hausdorff, too. 2
Exercises for Section 29 Exercise 29.1. Prove the assertions made in 28.16: (a) Prove that a semigroup S is directed if, and only if, any two right ideals in S have nonempty intersection. (b) Verify that every commutative semigroup is directed. (c) Verify that every group, considered as a semigroup, is directed. (d) Show that the subset S := a b a, b ∈ N, a = 0 of GL(2, Q) carries a semigroup 0 1
which is not directed with respect to the right
invariant pre-order. Hint. Consider the elements 02 01 and 02 11 of S. (e) Introduce a left invariant pre-order on any semigroup by considering left ideals instead of right ideals. Produce an example of a semigroup that is directed with respect to the right invariant pre-order, but not with respect to the left invariant pre-order. Exercise 29.2. Let S be a topological semigroup satisfying condition (OM) of 29.2 (for instance, a topological group), and let T be a subsemigroup with nonempty interior T ◦ . (a) Show that T ◦ is a subsemigroup of S, and that T ◦ satisfies (OM). (b) Show that T ◦ is an ideal in T . (c) Prove that T ◦ is directed if T is directed. Exercise 29.3. Let T denote the topology generated on R by the set {[a, b[ | a, b ∈ R, a < b} of half-closed intervals. Show that (R, T , +) is a topological semigroup and a group, but inversion is not continuous. Verify that this topological semigroup satisfies condition (OM) of 29.2. Exercise 29.4. Exhibit an example of a topological group G with a subsemigroup S satisfying condition (OM) of 29.2 such that S ∪ {1} does not satisfy that condition.
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30 Compact Semigroups Recall that a compact semigroup is a topological semigroup where the underlying space is compact. 30.1 Lemma. Let C be a compact semigroup. Then the set J of all closed ideals of C contains a unique minimal element, namely M := M(C) := J ∈J J . If C is Hausdorff then this ideal is in contained in each ideal of C, irrespective of closedness. Proof. Clearly M is closed in C. In order to show that M forms an ideal, the main task is to verify that M is not empty: This follows from the fact that J forms a filterbasis of closed subsets of a compact space; see 1.23. We leave the details for an exercise. Now assume that C is Hausdorff. For each a ∈ C the set CaC then is closed (being compact), and belongs to J. If J is any ideal of C, we pick a ∈ J and find M ⊆ CaC ⊆ J , as required. 2 30.2 Lemma. Let S be a semigroup, and assume that M is a minimal ideal in S. If M is commutative then M is a subgroup of S. Proof. Let a be any element of M. Then aM ⊇ aMM = MaM ⊇ SMaMS ⊇ S(aM)S shows that aM is an ideal in S. As M ⊇ aM is a minimal ideal, we have equality M = aM. Similarly, we obtain M = Ma. This means that we find e, f ∈ M such that ae = a = f a. For any b ∈ M, we find c, d ∈ M such that ca = b = ad. We compute be = cae = ca = b. Analogously, we compute f b = f ad = b. Choosing b ∈ {e, f } ⊆ M, we obtain e = f e = f , and e is an identity in M. It remains to find inverses: for any b ∈ M, we find elements bl , br ∈ M with bl b = e = bbr , and computing bl = bl (bbr ) = (bl b)br = br we obtain that they 2 are equal. 30.3 Corollary. Each compact Hausdorff semigroup contains at least one idempotent. Proof. Choose an arbitrary element a, and let A be the closure of {a n | n ∈ N {0}}. Then A is a commutative compact Hausdorff semigroup, and 30.2 yields that M(A) is a group. The identity in M(A) is an idempotent in any semigroup containing A as a subsemigroup. 2 Note that the identity in a sub(semi)group T of a semigroup S need not be an identity in S: consider the examples constructed in 28.2. Using 30.1, 30.2, or 30.3 one can show easily that (infinite subsemigroups of) the semigroup N cannot be endowed with compact Hausdorff topologies such that the usual multiplication becomes continuous, compare Exercise 30.10.
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30.4 Lemma. Let C be a compact Hausdorff semigroup, and assume that C contains an identity. If this identity is the only idempotent in C then C is a (sub)group (of C). Proof. Consider an arbitrary element a ∈ C. As in the proof of 30.3, let A be the closure of {a n | n ∈ N {0}}. By 30.2, the minimal ideal M(A) is a subgroup of C, and contains the unique idempotent 1. According to 28.14 (b), we have M(A) ⊆ GC (1). Now 1 ∈ M(A) ⊆ A yields a = 1a1 ∈ AaA ⊆ M(A) ⊆ GC (1). As a was an arbitrary element of C, we obtain C ⊆ GC (1). 2 30.5 Theorem. Let (G, μ, ι) be a group, and assume G carries a compact Hausdorff topology such that μ is continuous. Then ι is continuous, as well. ←
Proof. The graph ι = {(x, x ι ) | x ∈ G} coincides with the pre-image {1}μ = {(x, y) ∈ G × G | xy = 1}. As G is Hausdorff, this yields that ι is closed in G×G. 2 Now the Closed Graph Theorem 1.21 yields that ι is continuous. We will generalize this result in 31.2 below to the locally compact case. 30.6 Theorem. Let C be a compact Hausdorff semigroup, and assume that C is cancellative. Then C is a compact group. Proof. According to 30.3, there exists an idempotent e in C. For each c ∈ C, we have cee = ce and eec = ec, and cancellativity yields that e is an identity in C. Now 30.4 yields that C is a group, and 30.5 shows that inversion in that group is continuous. 2 30.7 Theorem. Let C be a compact Hausdorff semigroup, and let G be a subgroup of C. Then the closure G is a subgroup, as well, and inversion in G is continuous. Proof. The identity in G will be denoted by e, and inverses in G by g ι . We will first extend the map ι : G → G to a map from G to itself. Let x ∈ G. According to 1.42 there is a net a = (aj )j ∈J in G that converges to x in C. Inverting the elements aj in G, we obtain another net b := (bj )j ∈J by putting bj := ajι . As C is compact, the net b accumulates at some point y in C; see 1.43. According to 1.44, the net ((aj , bj ))j ∈J accumulates at (x, y) in C × C. Continuity of multiplication in C yields that the net c := (aj bj )j ∈J accumulates at xy. On the other hand, the net c is a constant: we have aj bj = aj ajι = e for each j ∈ J . If xy were different from e, we could choose a neighborhood U of xy such that e ∈ / U because C is Hausdorff. This would contradict the fact that the constant net c accumulates at xy. Thus we have xy = e. Similarly, we obtain yx = e. It remains to show that e is a neutral element in G: this follows from the fact that the constant net c converges to e, whence the net a = (aj )j ∈J = (aj e)j ∈J = (eaj )j ∈J converges to x, to xe and to ex. As C is Hausdorff, this gives x = xe = ex. 2 Continuity of ι on G follows from 30.5.
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30.8 Corollary. In every compact ring the group of invertible elements is a closed subgroup of the multiplicative semigroup, and inversion is continuous in this group. 2 The ring of n × n matrices over a compact ring is a compact ring, as well. The group of invertible elements in the multiplicative semigroup of that ring is GL(n, R), and we obtain the following. 30.9 Corollary. If R is a compact ring then GL(n, R) is a compact group, for each natural number n ∈ N. 2 In the multiplicative semigroup of a ring R, one has the idempotents 0 and 1. In view of 30.4, the following observation is interesting. 30.10 Lemma. Let R be a ring. If R {0} is a subsemigroup of the multiplicative semigroup of R (that is, if R has no “divisors of zero”) then 0 and 1 are the only idempotents in the multiplicative semigroup of R. Proof. Let e ∈ R be an idempotent. Then ee = e implies 0 = ee − e = e(e − 1). Our assumption yields 0 ∈ {e, e − 1}, and e ∈ {0, 1}. 2 30.11 Theorem. Let R be a locally compact Hausdorff ring, and assume that J is a compact open right ideal in R with J = R. Then the set 1 + J = {1 + x | x ∈ J } carries a compact open subsemigroup of R. If 0 and 1 are the only idempotents in the multiplicative semigroup of R then 1 + J is a subgroup of the multiplicative semigroup. In this case, the group R × of units is open in R, and forms a topological group (that is, inversion is continuous in R × ). Proof. For x, y ∈ J we compute (1 + x)(1 + y) = 1 + x + y + xy ∈ 1 + J . This shows that 1 + J is a subsemigroup of the multiplicative semigroup of R. Clearly, the set 1 + J is both open and compact since J has these properties. We have 0 ∈ / 1 + J since −1 ∈ J would imply R = (−1)R ⊆ J R ⊆ J , contradicting our assumption J = R. Thus 1 is the only idempotent in the compact semigroup 1 + J , and 30.4 yields that 1 + J is a group. According to 30.5, inversion in 1 + J is continuous. In order to see that R × is open in R and that inversion is continuous on R × , we consider a ∈ R and the maps λa : R → R and ρa : R → R defined by x λa = ax and x ρa = xa, respectively. These maps are continuous. Now R × contains the set λ← (1 + J ) a−1 = a(1 + J ) since R × is a group. As λa −1 is continuous, we have that a(1 + J ) is open. This shows that R × is open in R. Inversion ι in R × is continuous in the neighborhood 1 + J of 1. Using ι = ρa ιλa we conclude that ι is continuous at a. 2 If R is any compact Hausdorff ring, we find lots of compact open ideals in R, see 26.10. For each positive integer n, the matrix ring R n×n is a compact ring, as
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well. Each compact open ideal J in R yields a compact open ideal J n×n in R n×n . After 30.11, we are interested in idempotents in 1 + J n×n . The following two lemmas will come handy. 30.12 Lemma. Let A be a commutative group, written additively, and let e be an endomorphism of A. If e is an idempotent then either e = idA or the kernel of e is not trivial. Proof. For any endomorphism e of A, the sets Ae and ker e are subgroups of A. If e is an idempotent then (a e )e = a ee = a e shows that e induces the identity on Ae . Decomposing a = a e + (a − a e ) and noting a − a e ∈ ker e we obtain that A is the 2 direct sum of Ae and ker e. If the kernel is trivial then e = idAe = idA . 30.13 Lemma. Let R be a ring, let J = R be an ideal in R, and let n be a positive integer. Inductively, we put J0 := R and Jk+1 := {xy | x ∈ Jk , y ∈ J }. If k∈N Jk = {0} then the identity matrix 1 is the only idempotent in the subsemigroup 1 + J n×n of the matrix ring R n×n . Consequently, the group of invertible elements of the ring R n×n is open in R n×n , and forms a topological group. Proof. Consider the additive group of R n×n . Aiming at an application of 30.12, we study the kernel ker(1 + S), where S = (sij )i,j ∈n is an element of J n×n , and 1 + S maps a = (aj )j ∈n ∈ R to a + aS = (aj + i∈n ai sij )j ∈n . Assume a = 0. Our assumption k∈N Jk = {0} implies that for each j ∈ n with aj = 0 the set {k ∈ N | aj ∈ Jk } has a maximal element mj . We choose j so that mj becomes minimal. For each i ∈ n {j }, we then have ai ∈ Jmj and ai sij ∈ Jmj +1 . Now aj ∈ / Jmj +1 yields aj + i∈n ai sij = 0, and a ∈ / ker(1 + S). 2 According to 30.12, we obtain that 1 + S is an idempotent only if S = 0. Examples of locally compact rings with an ideal J such that k∈N Jk = {0} occur within locally compact, totally disconnected fields, see 26.23. 30.14 Example. Let F be a finite field, with the discrete topology, and let c be any nonempty set. Then the pointwise operations and the product topology turn the set F c into a compact ring. For each subset d ⊆ c, the characteristic function χd is an idempotent in F c . The group (F c )× of invertible elements equals (F × )c . If c is finite then F c is discrete and (F c )× is open in F c . However, if c is infinite then (F c )× is not open in F c .
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Exercises for Section 30 Exercise 30.1. Let b be a positive real number with b ≤ 1. Show that the intervals [0, b] and [−b, b] form compact subsemigroups of the multiplicative semigroup R. What are the idempotents, and what can be said about subgroups? Are you able to determine all ideals in these semigroups? Exercise 30.2. Let C be a compact subset of R. For x, y ∈ C, put x ∧ y := min{x, y}. Verify that (C, ∧) is a compact semigroup, and determine the idempotents. Exercise 30.3. Find an example of a compact Hausdorff semigroup S such that S is not a group, but contains only one idempotent. Exercise 30.4. Let q be a positive integer, put Cq := Xn∈N Z/q n Z, and let a ∈ Cq {0}. Show that Na is not a subgroup of (Cq , +). Determine the closure of Na and its minimal ideal. Exercise 30.5. For a ∈ R/Z, determine the closure of Na. Does it differ from the closure of the set {na | n ∈ N, n > k} if k is a positive integer? Exercise 30.6. Let S be a semigroup, and let L be a collection of ideals. (a) Show that L∈L L is an ideal of S if L is finite. (b) Produce an example where L∈L L is not an ideal. Exercise 30.7. Let C be a compact semigroup, and consider the set J of all closed ideals. (a) For I, J ∈ J, verify that I ∩ J contains I J . (b) Show that J forms a filterbasis. Exercise 30.8. Produce an example of a compact semigroup C with an element a ∈ C such that CaC is not closed in C. Hint. You should, of course, consider non-Hausdorff spaces. Exercise 30.9. Let C be a compact Hausdorff semigroup. Show that C/M(C) is isomorphic to (K, μ), where K is the one-point-compactification of C M(C), and μ extends the multiplication induced from C in such a way that the new point attached (at infinity) to C M(C) acts as a zero. Exercise 30.10. Let m ∈ N, and put Sm := {n ∈ N | n ≥ m}. (a) Show that Sm is a subsemigroup of the semigroup N, endowed with the usual multiplication. (b) Prove that there is no compact Hausdorff topology on Sm that turns Sm into a topological semigroup.
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31 Groups with Continuous Multiplication 31.1 Lemma. Let G be a group, and consider the maps λa : G → G and ρa : G → G defined by x λa = ax and x ρa = xa. If G carries a topology such that λa and ρa are continuous for each a ∈ G then these maps are homeomorphisms from G onto G. As a consequence, each neighborhood basis U at an element x ∈ G yields neighborhood bases aU := {aU | U ∈ U} and Ua := {U a | U ∈ U} at the elements ax and xa, respectively. Proof. It suffices to note that the inverses of these maps are given as λ−1 a = λa −1 and ρa−1 = ρa −1 , respectively. 2 We have seen in 30.5 that groups whose multiplication is continuous with respect to some compact Hausdorff topology have continuous inversion, as well. The next theorem generalizes this to the locally compact case. 31.2 Theorem. Let (G, μ, ι) be a group, endowed with a locally compact Hausdorff topology such that the multiplication μ is continuous. Then ι is continuous, as well, and G is a topological group. Proof. It suffices to show that ι is continuous at 1 because ι = ρa ιλa is then continuous at a, as well. Aiming at a contradiction, we assume that ι is not continuous at 1. Then there exists a neighborhood U of 1 such that for each neighborhood V of 1, we have V ι U .
(∗)
Without loss of generality, we assume that U is compact. As μ is continuous at (1, 1) and the locally compact Hausdorff space G belongs to the class T3 by 1.26, we find inductively a sequence (Un )n∈N of neighborhoods of 1 such that U0 = U and Un+1 Un+1 ⊆ Un holds for each n ∈ N. According to (∗), we find for each n ∈ N some xn ∈ Un such that xn−1 ∈ / U. Inductively, we define y0 := x1 and yn+1 := yn xn = x1 x2 . . . xn . Our choice of Un secures yn+1 ∈ U1 U2 . . . Un−1 Un ⊆ U1 U2 . . . Un−1 Un−1 ⊆ U1 U2 . . . Un−2 ⊆ · · · ⊆ U1 U2 U2 ⊆ U1 U1 ⊆ U for each natural number n ∈ N. As U is compact, the sequence (yn )n∈N accumulates at some point y ∈ U . For each n ∈ N, the set yUn is a neighborhood of y, according to 31.1. Fixing n, −1 −1 we thus find k > n such that yk ∈ yUn , and obtain xk−1 = yk+1 yk ∈ yk+1 yUn . −1 For each j > k, we have yk+1 yj = xk+1 xk+2 . . . xj ∈ Uk+1 . . . Uj −2 Uj −1 Uj ⊆ −1 Uk+1 . . . Uj −2 Uj −2 ⊆ · · · ⊆ Uk . Therefore, the accumulation point yk+1 y of the −1 sequence (yk+1 yj )j ∈N belongs to Uk ⊆ Uk−1 , and we arrive at the contradiction −1 xk−1 ∈ yk+1 yUn ⊆ Uk−1 Un ⊆ Un Un ⊆ U . 2 31.3 Theorem. Let F be a field, and let T be a locally compact Hausdorff topology such that F is a topological ring. Then F is a topological field.
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Proof. It suffices to note that the multiplicative group is the complement of a single point, and thus open in (F, T ): this yields that the multiplicative group is a locally compact semigroup, and 31.2 applies. 2 A variant of the following result has already been obtained in Section 5, by rather elementary means. Here we take the opportunity to show how the results of the preceding section may work: 31.4 Example. Let F be a locally compact field, and let n be a natural number. Then GL(n, F ) is a locally compact group. Proof. It is clear that multiplication of matrices with entries in a topological ring is continuous. Thus GL(n, F ) is a topological semigroup. In order to see that GL(n, F ) is locally compact, we show that GL(n, F ) is open in the space of all n × n matrices over F . To this end, we distinguish two cases: If F is connected then F is a real algebra of finite dimension by 26.8, and F n×n is embedded into some matrix algebra Rm×m . Clearly, every element of GL(n, F ) has nonzero determinant in Rm×m . Conversely, if A ∈ F n×n has nonzero determinant in Rm×m then the Cayley–Hamilton Theorem says that the inverse of A in Rm×m is a polynomial in A. Thus A is invertible in the algebra F n×n . Now it remains to note that the determinant function is continuous. Now assume that F is totally disconnected. By 26.22 and 26.10, we find an open compact ideal J in an open compact subring R of F . According to 26.23, we have k∈N Jk = {0}, and 30.13 shows that GL(n, F ) contains the open subgroup 1 + J n×n . This yields that GL(n, F ) is open. In both cases, inversion in GL(n, F ) is continuous by 31.2. 2
Chapter H
Hilbert’s Fifth Problem 32 The Approximation Theorem The class of locally compact (not necessarily Abelian) Hausdorff groups admits a strong structure theory. This is due to the fact that – via approximation (projective limits, see Section 17) – important parts of the theory of Lie groups and Lie algebras carry over. This phenomenon becomes particularly striking if one assumes, in addition, that the groups under consideration are connected and of finite dimension. The aim of the present chapter is to collect results and to show that Lie theory yields complete information about the rough structure (i.e., the lattice of closed connected subgroups) of locally compact finite-dimensional groups. Moreover, we shall describe the possibilities for locally compact connected non-Lie groups of finite dimension. We will not give complete proofs in this chapter, but will often provide references to the literature. We shall only consider Hausdorff groups (and shall, therefore, form quotients only with respect to closed subgroups – except for the warning Example 33.4). The present chapter (Sections 32–40) grew out of an appendix to the author’s Habilitationsschrift [59], and a survey paper [60]. Hilbert’s Fifth Problem asks for a topological characterization of Lie groups. In this section, we present a solution, and discuss consequences. If G is a locally compact group such that G/G1 is compact, then there exist arbitrarily small compact normal subgroups such that the factor group is a Lie group. To be precise: 32.1 Approximation Theorem. Let G be a locally compact group such that G/G1 is compact. (a) For every neighborhood U of 1 in G there exists a compact normal subgroup N of G such that N ⊆ U and G/N admits local analytic coordinates that render the group operations analytic. (b) If, moreover, ind G < ∞, then there exists a neighborhood V of 1 such that every subgroup H ⊆ V satisfies ind H = 0. That is, there is a totally disconnected compact normal subgroup N such that G/N is a Lie group with ind G = ind G/N. Proof. [39], Chap. IV, [12] Th. 9, see also [35], II.10, Th.18.
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The dimension function ind that we use in 32.1 (b) is small inductive dimension.
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See Section 2 for its basic properties and Section 33 for properties related to grouptheoretic concepts. 32.2 Remarks. For locally compact groups in general, one knows that there always exists an open subgroup G such that G/G1 is compact, cf. 4.13. For the special case of locally compact Abelian groups, we have proved a version of the Approximation Theorem in 21.16. The fact that unitary representations on finite-dimensional complex Hilbert spaces (see 14.33) is the key to the proof of the Peter–Weyl-Theorem 19.1: this is the version of the Approximation Theorem for the special case of compact groups. The general case, however, is much harder. We obtain a useful criterion. 32.3 Theorem. A locally compact group G is a Lie group if, and only if, every compact subgroup of G is a Lie group. If G is a locally compact group such that G/G1 is compact, then we can say even more: in this case, the group G is a Lie group if, and only if, every compact normal subgroup is a Lie group. Proof. Closed subgroups of Lie groups are Lie groups; see, e.g., [18], VIII.1. Conversely, assume that every compact subgroup of G is a Lie group. According to 32.2, there exists an open subgroup H of G such that H /H1 is compact. Let N be a compact normal subgroup of H such that H /N is a Lie group. Then N is a Lie group by our assumption, and has, therefore, no small subgroups. Consequently, there exists a neighborhood U in H such that every subgroup M ⊆ N ∩ U is trivial. Let M be a compact normal subgroup of H such that M ⊆ U and H /M is a Lie group. Then H /(M ∩ N) is a Lie group as well [12], 1.5, but M ∩ N = {1}. Thus H is a Lie group, and G is a Lie group as well, since H is open in G. If G/G1 is compact, our proof works for H = G, yielding the second part of our assertion. 2 For the case where G/G1 is compact, the criterion 32.3 can also be deduced from the fact that the class of Lie groups is closed with respect to extensions [31], Th. 7. 32.4 Corollary. Let G be a locally compact group, and assume that G is connected and of finite dimension. Then G is a Lie group if, and only if, the center of G is a Lie group. Proof. According to 32.1 (b), the question whether or not G is a Lie group is decided in some totally disconnected normal subgroup N . Since G is connected, this subgroup is contained in the center of G. 2 Compact subgroups play an important role in the theory of locally compact groups. They are understood quite well (see also Sections 19 and 36 on compact groups), especially in the connected case. 32.5 Malcev–Iwasawa Theorem. Let G be a locally compact group such that G/G1 is compact. Then the following hold.
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(a) Every compact subgroup of G is contained in some maximal compact subgroup of G. (b) The maximal compact subgroups of G form a single conjugacy class. (c) There exists some natural number n such that the underlying topological space of G is homeomorphic to Rn × C, where C is one of the maximal compact subgroups of G. (d) In particular, every maximal compact subgroup of a locally compact connected group is connected. Proof. [31], §4, Th. 13, cf. also [18], Th. 3.1, and [27].
2
The Splitting Theorem 23.11 is a special case of 32.5. Considering, for example, a discrete infinite torsion group, one easily sees that some connectedness assumption is essential for the mere existence of maximal compact subgroups. Note that 32.5, in combination with the solution of D. Hilbert’s Fifth Problem, provides another proof for 32.3. There is, in general, no natural choice of a normal subgroup N such that G/N is a Lie group. However, we have: 32.6 Theorem. Let G be a locally compact connected group of finite dimension. If both N1 and N2 are closed normal subgroups such that ind Ni = 0 and G/Ni is a Lie group, then G/N1 is locally isomorphic to G/N2 . Proof. The factor group G/(N1 ∩ N2 ) is also a Lie group, cf. [12], 1.5. Now Ni /(N1 ∩ N2 ) is a Lie group of dimension 0, and therefore discrete. This implies 2 that G/(N1 ∩ N2 ) is a covering group for both G/N1 and G/N2 . It is often more convenient to work with compact normal subgroups than with arbitrary closed normal subgroups (for instance, the quotient map modulo a compact subgroup is a closed map, see 6.5). The general case may be reduced to the study of quotients with respect to compact kernels. 32.7 Theorem. Let G be a locally compact connected group of finite dimension. If N is a closed normal subgroup such that ind N = 0 and G/N is a Lie group, then there exists a compact normal subgroup M of G such that M ≤ N and G/M is a Lie group. The natural map π : G/M → G/N is a covering. Proof. Choose a compact neighborhood U of 1 in G. According to 32.1 (b), there exists a compact normal subgroup N such that N ⊆ U and G/N is a Lie group. Now M := N ∩ N has the required properties; in fact, G/M is a Lie group by [12], 1.5, and the kernel of the natural map π : G/M → G/N is a totally disconnected Lie group, hence discrete. 2
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A main reason why, in general, quotients with respect to compact subgroups behave better than quotients with respect to arbitrary closed subgroups is the following. 32.8 Lemma. Let G be a topological group, and let H be a compact subgroup of G. Then the natural map π : G → G/H is a perfect map, i.e., for every compact −1 subset C ⊆ G/H the preimage C π is also compact. Proof. Since H is compact, the natural map π is closed, see 6.5. Now π is a closed map with compact fibers, and therefore perfect, cf. [11], XI.5. 2
33 Dimension of Locally Compact Groups We need a notion of (topological) dimension. Several topological dimension functions have been introduced in order to solve the problem of distinguishing euclidean spaces Rn and Rm topologically. While these dimension functions show all kinds of pathological behavior on arbitrary topological spaces, they behave well for separable metric spaces, and for locally compact groups. In particular, the most commonly used dimension functions coincide on these classes. We formulate our results for small inductive dimension, denoted by ind, see Section 2. For the reader’s convenience, we repeat the definition here: 33.1 Definition. Let X be a topological space. We say that ind X = −1 if, and only if, X is empty. If X is non-empty, and n is a natural number, then we say that ind X ≤ n if, and only if, for every point x ∈ X and every neighborhood U of x in X there exists a neighborhood V of x such that V ⊆ U and the boundary ∂V satisfies ind ∂V ≤ n − 1. Finally, let ind X denote the minimum of all n such that ind X ≤ n; if no such n exists, we say that X has infinite dimension. Obviously, ind X is a topological invariant: for each space Y homeomorphic to X one has ind Y = ind X. A non-empty space X satisfies ind X = 0 if, and only if, there exists a neighborhood basis consisting of closed open sets. Consequently, a T1 space of dimension 0 is totally disconnected. See [30] for a study of the properties of ind for separable metric spaces. Although it is quite intuitive, our dimension function does not work well for arbitrary spaces. Other dimension functions, notably covering dimension [47], 3.1.1, have turned out to be better suited for general spaces, while they coincide with ind for separable metric spaces. See [47] for a comprehensive treatment. Note, however, that small inductive dimension coincides with large inductive dimension and covering dimension, if applied to locally compact groups [1], [45], compare [54], 93.5. The duality theory for compact Abelian groups uses covering dimension rather than
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inductive dimension, cf. [40], pp. 106–111, [15], 3.11. For this special case, we shall prove the equality in 36.7 below. While it is rather easy to show ind Rn ≤ n, for any positive integer n, it is much harder to prove the reverse inequality ind Rn ≥ n (and thus ind Rn = n). See 2.19. Let G be a locally compact group. Finiteness of ind G allows to obtain analogues of counting arguments, as used in the theory of finite groups. In particular, we have the following: 33.2 Theorem. Let G be a locally compact group, and assume that H is a closed subgroup of G. Then ind G = ind G/H + ind H . If G/H has finite dimension then the canonical projection πH : G → G/H has local sections at every point of G/H : that is, for every g ∈ G there are a neighborhood U of H g in G/H and a continuous map σU : U → G such that σU πH = idU . Proof. The first assertion follows from [41], Sect. 5, Cor 2, since by 2.19 and 2.20 inductive dimension has the following properties that [41], p. 64f, requires for a dimension function Dim: (a) Dim E n = n, and Dim C n = n, where E n is euclidean n-space, and C n is the closed n-cell. (b) If A ⊆ B, then Dim A ≤ Dim B. An arbitrary cartesian product of finite discrete spaces is 0-dimensional. (c) If E is an Euclidean space, and Z is a compact 0-dimensional space, then Dim(E × Z) = Dim E. (d) Dim is invariant under homeomorphisms. (e) For every locally compact group G, we have Dim G = Dim U where U is any open neighborhood in G. The existence of local sections has also been proved in [41], Thm. 8, using the 2 Approximation Theorem 32.1. 33.3 Remark. The same conclusion holds if we replace small inductive dimension by large inductive dimension, or by covering dimension, see [46]. 33.4 Example. The closedness assumption on H is indispensable in 33.2. For instance, consider the additive group R. Since Q is dense in R, the factor group R/Q has the indiscrete topology. Hence ind R/Q = 0 = ind Q, but ind R = 1. We claim that the dimension of a locally compact group G is determined by the connected component G1 of G. In fact, using 2.20 (e), we obtain: 33.5 Theorem.
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(a) If G is a locally compact group, then ind G = ind G1 . (b) If H is a closed subgroup of a locally compact group G, and if ind H = ind G < ∞, then G1 ≤ H . Proof. Assertion (a) follows from 33.2 and the fact that G/G1 is totally disconnected, see 2.9 (d). If ind H = ind G < ∞ then ind G/H = 0 by 33.2, and assertion (b) follows from the fact that the connected component can only act trivially on the totally disconnected space G/H . 2 33.6 Theorem. Let G be a locally compact group. Assume that N is a closed normal subgroup, and C is a closed σ -compact subgroup such that ind(C ∩ N ) = 0 and CN = G. Then ind G = ind C + ind N . Proof. From 6.19 we know G/N = CN/N ∼ = C/(C ∩ N ), and 33.2 yields the assertion. 2 33.7 Definition. (a) If, in the situation of 33.6, we have in addition that C and N are connected, we say that G is an almost semi-direct product of C and N. (b) If, moreover, the subgroup C is normal as well, we say that G is an almost direct product of C and N. Note that every locally compact connected group is generated by any compact neighborhood (see 4.10), and is therefore σ -compact. Recall also that a locally compact group G is σ -compact if it is compactly generated; in particular if G/G1 is compact, or if G/G1 is countable. The terminology suggests that every almost (semi-)direct product is a proper (semi-)direct product, ‘up to a totally disconnected normal subgroup’. This may be made precise in two different ways. Either the almost (semi-)direct product is obtained from a proper (semi-)direct product by forming the quotient modulo a totally disconnected subgroup, or one obtains a proper (semi-)direct product after passing to such a quotient. From the first of these viewpoints, our terminology is fully justified. In fact, every almost (semi-)direct product G = CN is isomorphic to the quotient of the proper (semi-)direct product C N modulo K, where the action of C on N is given by conjugation in G, and K = {(g, g −1 ) | g ∈ C ∩ N } is isomorphic to the totally disconnected group C ∩ N. From the second point of view, our terminology is adequate for almost direct products, but almost semi-direct products are more delicate. 33.8 Theorem. Assume that the locally compact group G is an almost direct product of (closed connected) subgroups N1 and N2 . Then the following hold: (a) The intersection N1 ∩ N2 is contained in the center of G.
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(b) The quotient G/(N1 ∩N2 ) is a direct product of N1 /(N1 ∩N2 ) and N2 /(N1 ∩N2 ). (c) We have ind G = ind (G/(N1 ∩ N2 )), and ind Ni = ind (Ni /(N1 ∩ N2 )). Proof. We note first that the subgroup Ni /(N1 ∩ N2 ) is closed in G/(N1 ∩ N2 ) because its pre-image Ni is closed in G. Now the Open Mapping Theorem 6.19 applies, and yields assertion (b), cf. 6.17 and 6.23. Assertion (a) follows from the fact that the connected group G acts trivially on the totally disconnected normal subgroup N1 ∩ N2 , and assertion (c) follows 2 from 33.2. 33.9 Example. For almost semi-direct products G = CN , the intersection C ∩ N need not be a normal subgroup of G. E.g., let N = SO(3, R), let C = T, the circle group, and let γ : C → N be an embedding. Let a be an element of order 4 in C. Now (c, x)(d, y) := (cd, (d −1 )γ xd γ y) defines a semi-direct product G = C N . It is easy to see that Z = (a, (a −1 )γ ) is contained in the center of G. We infer ¯ := G/Z is an almost semi-direct product of C¯ := ZC/Z and N¯ := ZN/Z, that G ¯ and that Z(a, 1) = Z(1, a γ ) belongs to C¯ ∩ N¯ , but Z(1, a) is not central in G. ¯ ¯ ¯ ¯ ¯ Since G is connected and C ∩ N is totally disconnected, normality of C ∩ N would imply that C¯ ∩ N¯ is central. We obtain the following applications of Halder’s Lemma 2.22. 33.10 Corollary. Let G be a locally compact connected group. If G acts on a separable metric space Y , then ind(G/Gy ) = ind y G ≤ ind Y , where y ∈ Y is any point, Gy is its stabilizer, and y G its orbit under the given action. Important special cases are the following. (a) If H is a locally compact group, and α : G → H is a continuous homomorphism, then ind(G/ ker α) = ind Gα ≤ ind H . (b) If G acts linearly on V ∼ = Rn , then ind(G/Gv ) = ind vG ≤ ind V = n, where v ∈ V is any vector, Gv is its stabilizer, and vG its orbit under the given action. Proof. Every locally compact connected group G and every quotient space G/S, where S is a closed subgroup of G, satisfies the assumptions on X in 2.22: in fact, the group G is algebraically generated by every neighborhood of 1. Therefore, assertion (a) follows from the fact that the stabilizer Gv is closed in G. Assertion (b) follows from the fact that the image Gα is contained in the connected component 2 H1 , which is separable metric by 37.1. In general, a bijective continuous homomorphism of topological groups need not be a topological isomorphism; e.g., consider the identity with respect to the discrete and some non-discrete group topology. Locally compact connected groups, however, behave well, see 10.10:
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33.11 Theorem. Let G be a locally compact group, and assume that G is σ compact. Then the following hold: (a) If X is a locally compact space, and ω : (X, G) → X is a continuous transitive action, then the map ϕ : G/Gx → X : g → (x, g)ω is open, for every x ∈ X. (b) If μ : G → H is a surjective continuous homomorphism onto a locally compact 2 group H , then μ is in fact a quotient map.
34 The Rough Structure In this section, we introduce the lattice of closed connected subgroups of a locally compact group, with additional binary operations. We show that this structure is preserved under the forming of quotients modulo compact totally disconnected normal subgroups. For subsets A, B of a topological group G, let [A, B] be the closure of the subgroup that is generated by the set {a −1 b−1 ab | a ∈ A, b ∈ B}. 34.1 Lemma. Let G be a topological group, and let N be a totally disconnected closed normal subgroup of G. If A, B are connected subgroups of G, then [A, B] = {1} if, and only if, [AN/N, BN/N] = {1}. Proof. It is obvious that [A, B] = {1} implies [AN/N, BN/N] = {1}. The set C := {a −1 b−1 ab | a ∈ A, b ∈ B} is a continuous image of the connected set A × B, and thus connected. Clearly, the set C contains 1, and C −1 = C. Putting C1 := C and Cn+1 := Cn C, we find that the subgroup generated by C equals n≥1 Cn , and is connected. Now the closure [A, B] of this subgroup is connected, as well. The condition [AN/N, BN/N] = {1} implies [A, B] ⊆ N , and connectedness of [A, B] yields [A, B] = {1}. 2 Let G be a locally compact group. We are interested in the lattice of closed connected subgroups; i.e., for closed connected subgroups A, B of G, we consider the smallest closed (necessarily connected) subgroup A∨B that contains both A and B, and the biggest connected (necessarily closed) subgroup A ∧ B that is contained in both A and B. Note that A ∧ B = (A ∩ B)1 . Moreover, we are interested in the connected components of the normalizer and the centralizer of B, taken in A 1 (B) and C1 (B), respectively). This gives rise to a binary operations (denoted by NA A 1 1 N and C on the set of closed connected subgroups of G. Finally, recall that the commutator subgroup [A, B] is necessarily connected, while closedness is enforced by the very definition.
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34.2 Definition. (a) For any locally compact group, let Struc(G) be the set of all closed connected subgroups of G, endowed with the five binary operations ∨, ∧, N1 , C1 , [ , ], as introduced above. We call Struc(G) the rough structure of G. (b) Let Cp(G) be the set of compact connected subgroups of G, and let CpFree(G) be the set of compact-free closed connected subgroups of G (in the sense of 23.15: those that have no compact subgroup except the trivial one). 34.3 Remarks. (a) Note that Cp(G) and CpFree(G) are subsets but, in general, not subalgebras of Struc(G). (b) Of course, Struc(G) = Struc(G1 ). We are going to investigate the effect of continuous group homomorphisms on the rough structure. Our results will justify the vague feeling that the quotient of a locally compact group by a compact totally disconnected normal subgroup has ‘roughly the same structure’. 34.4 Proposition. Let G and H be locally compact groups, and let α : G → H be a continuous homomorphism. For every closed connected subgroup A of G, let Aα := Aα be the closure of Aα in H . (a) The map α maps Struc(G) to Struc(H ), and it maps Cp(G) to Cp(H ). (b) For A ≤ B ≤ G, we have that Aα ≤ B α . (c) For every choice of A, B ∈ Struc(G), we have that Aα ∨ B α ≤ (A ∨ B)α and Aα ∧ B α ≥ (A ∧ B)α . 1 (B))α ≤ N1 (B)α , (d) For every choice of A, B ∈ Struc(G), we have that (NA α
and that (C1A (B))α ≤ C1 α (B)α .
A
A
Proof. Assertions (a) and (b) are obvious from the definition of α and the fact that every continuous map preserves compactness. From A, B ≤ C it follows that Aα , B α ≤ C α . This implies that Aα ∨ B α ≤ (A ∨ B)α . The second part of (c) follows analogously. Assertion (d) follows from the well-known inequalities (NA (B))α ≤ NAα (B)α and (CA (B))α ≤ CAα (B)α , combined with the fact that continuous images of connected spaces are connected. 2 Even if α is a quotient morphism with totally disconnected kernel, the map α may be far from being injective. E.g., consider a quotient map from R2 onto T2 . In fact, the rough structure Struc(R2 ) has uncountably many elements, while Struc(T2 ) is countable. However, we have:
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34.5 Theorem. Let G be a locally compact group, let N be a compact totally disconnected normal subgroup, and let π be the natural quotient map from G onto G/N. Then the following hold: (a) For every A ∈ Struc(G), we have that Aπ = Aπ¯ , and that ind A = ind Aπ . (b) The map π induces an isomorphism Struc(G) ∼ = Struc(G/N ). (c) The map π induces bijections of Cp(G) onto Cp(G/N ), and of CpFree(G) onto CpFree(G/N ). Proof. According to 34.3, we may assume that G is connected. Hence G is σ compact, and so is every closed subgroup A of G. In particular, AB/A ∼ = B/(A∩B) for every closed subgroup B of NG (A), see 6.19. Mutatis mutandis, the same assertion holds for subgroups of G/N. (i) Being a quotient map with compact kernel, the map π is closed, see 6.5. Thus π¯ here means nothing else but applying π . For the sake of clarity, we will, however, continue to write π¯ . The restriction of π to A is a closed surjection onto Aπ , hence a quotient map. Therefore, ind A = ind Aπ by 33.2, and assertion (a) is proved. (ii) For every H ∈ Struc(G/N ), let H ψ be the connected component of the ← π-preimage H π . Since π is continuous, and connected components are closed (see 2.9 (b)), we infer that ψ is a map from Struc(G/N ) to Struc(G). The re← striction of the quotient map π to the saturated closed subset H π is a quotient map (see 1.35 (d)), and the composition of quotient maps x → H ψπ x π is an open ← surjection from H π onto H /H ψπ , see 6.2 (a). This yields an isomorphism from ← the totally disconnected group H π /(H ψ N ) onto the group H /(H ψπ ). Hence H ψπ ≥ H1 = H , and H ψπ = H ψ π¯ = H follows. Every A in Struc(G) is connected, and therefore centralizes the totally discon← nected normal subgroup N. Thus A is a normal closed subgroup of AN = Aπ π . The (compact) totally disconnected quotient AN/A is isomorphic to N/(N ∩ A), see 6.17, and we infer A = (AN )1 = Aπ ψ = Aπ¯ ψ . Thus we have proved that π¯ is invertible (with inverse ψ). (iii) The map ψ is monotone. In fact, let H ≤ K in Struc(G/N ). Then H ψ is a ← ← connected subgroup of H π ≤ K π , hence H ψ ≤ K ψ . In view of 34.4 (b), this shows that π respects the binary operations ∨ and ∧. (iv) From 34.1, we infer that π respects the operations [ , ] and C1 . We have seen in step (ii) that B is the connected component of BN . This yields that NA (B) = {a | a ∈ A, a −1 Ba ⊆ B} = {a | a ∈ A, a −1 Ba ⊆ BN }, and NA (B)π ≤ NAπ (B π ) 1 (B))π¯ ≤ N1 (B π ). Conversely, every element x ∈ follows. This means (NA Aπ ← π ψ −1 (NAπ (B )) satisfies (x Bx)π ⊆ B π , and x −1 Bx ⊆ (B π π )1 = (BN )1 = B 1 π ψ π ψ follows, leading to (NAπ (B )) ≤ NG (B) ∩ A = NA (B). Thus π¯ respects the operation N1 , as well.
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(v) The natural quotient morphism π is a perfect continuous map, see 32.8. Thus π is a bijection of Cp(G) onto Cp(G/N ). For A ∈ CpFree(G), we obtain that A ∩ N = {1}, hence A ∼ = Aπ ∈ CpFree(G/N ). Conversely, assume that A ∈ Struc(G) CpFree(G). According to 32.5 (d), there exists a connected nontrivial compact subgroup C of A. Now C π is a non-trivial compact subgroup of 2 Aπ . This completes the proof of assertion (c). An interesting feature of locally compact connected Abelian groups is the fact that the lattice of closed connected subgroups is complemented: 34.6 Theorem. Let A be a locally compact connected Abelian group, and assume that B is a closed connected subgroup of A. Then there exists a closed connected subgroup K of A such that A = BK and ind(B ∩ K) = 0 (i.e., B ∩ K is totally disconnected). Proof. It suffices to show the existence of a closed subgroup S such that BS = A and ind(B ∩ S) = 0; in fact, connectedness of A implies that BS1 = A (consider the action of A on the totally disconnected homogeneous space A/(BS1 )). (i) Assume first that A is compact. Then the dual group A∗ is discrete (cf. 20.6) and torsion-free (since A is connected, see 23.18). Consequently, A∗ embeds in Q := Q ⊗ A∗ , taken with the discrete topology. Since A∗ spans the Q-vector space Q, there exists a basis E ⊂ A∗ for Q. Moreover, we can choose E in such a way that E ∩ B ⊥ is a basis for the subspace U spanned by B ⊥ . Now E B ⊥ spans a complement V of U in Q. Writing L := V ∩A∗ , we infer that B ⊥ ∩L = {1}. Since E ⊂ B ⊥ ∪ L, the factor group Q/(B ⊥ L) is a torsion group, and so is A∗ /(B ⊥ L). We conclude that BL⊥ = A, and ind(B ∩ L⊥ ) = 0. (ii) In the general case, we write A = R × C and B = S × D with compact groups C, D, where R ∼ = Rb . According to 24.8, there exists a = Ra and S ∼ continuous homomorphism α : R → C such that S is contained in the graph α , and A = α × C by 24.8. Therefore, we may assume that S = Rb ≤ R = Ra . For any subgroup Z ∼ = Za of Ra such that B ∩ Z ∼ = Zb , the group A/Z is compact, and BZ/Z is a compact, hence closed, subgroup. Now (i) applies, and we infer that there exists a closed subgroup S of A such that A = BS and ind(B ∩ S) = 0. 2 34.7 Remarks. (a) The example of a two-dimensional indecomposable group in [48], Bsp. 68 shows that, in general, a complement for a closed connected subgroup need not exist. (b) Complements do exist in Abelian connected Lie groups; this can be derived from the fact that, in this case, the dual group is isomorphic to Ra × Zc .
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(c) If A is a locally compact Abelian group, and B is a closed connected subgroup of A such that B is a Lie group (i.e., B is isomorphic to Ra × Tc for suitable cardinal numbers a < ∞ and c), then there exists a complement for B in A, see [2], 6.16. (d) The assertion of 34.6 can also be derived from (b) and 34.5.
35 Notions of Simplicity We are now going to introduce the concepts ‘almost simple’, ‘semi-simple’, ‘minimal closed connected Abelian normal subgroup’, ‘solvable radical’ in the context of locally compact connected groups of finite dimension. See Section 7 for a comparison of the concepts of solvability and nilpotency in topological groups and in discrete groups. A locally compact connected non-Abelian group G is called semi-simple if it has no non-trivial closed connected Abelian normal subgroup; the group G is called almost simple if it has no proper non-trivial closed connected normal subgroup. Let (Gi )i∈I be a family of normal subgroups of a topological group G. Assume generthat G isgenerated by i∈I Gi and the intersection of Gj with the subgroup ated by i∈I {j } Gi is totally disconnected. Then each Gj centralizes i∈I {j } Gi . Generalizing 33.7 (b), we call the group G an almost direct product of the groups Gi . Examples are given by compact connected groups 36.1, and also by semi-simple groups: 35.1 Theorem. A locally compact connected group of finite dimension is semisimple if, and only if, it is the almost direct product of a finite family (Si )1≤i≤n of almost simple (closed connected) subgroups Si . Proof. This follows from the corresponding theorem on Lie groups [4], III, §9, 2 no. 8, Prop. 26 via the Approximation Theorem 32.1 (a) and 34.5. 35.2 Theorem. Let G be a locally compact connected group. (a) If G is almost simple, then every proper closed normal subgroup is contained in the center Z of G, and Z is totally disconnected. In particular, G/Z is a simple Lie group with ind G/Z = ind G < ∞. (b) If G is semi-simple and of finite dimension, then every closed connected normal subgroup is of the form Si1 . . . Sik , where the Sij are some of the almost simple factors from 35.1.
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Proof. Let N be a proper closed normal subgroup of G. The connected component N1 is a proper closed connected normal subgroup of G. If G is almost simple, we infer that N1 = {1}. Via conjugation, the connected group G acts trivially on the totally disconnected group N. Therefore N is contained in Z. Applying this reasoning to the case where N = Z, we obtain that Z is totally disconnected. The rest of assertion (a) follows from 33.2 and 32.1. Assertion (b) follows from 34.5 and the corresponding theorem on Lie groups [4], I, §6, no. 2, Cor. 1; III, §6, no. 6, Prop. 14. 2 Our next observation makes precise the intuition that an almost simple group either has large compact subgroups, or large solvable subgroups. 35.3 Theorem. Let G be a locally compact connected almost simple group. Then there exist a compact subgroup C and closed connected subgroups T and D of G such that the following hold. (a) The group C is compact and semi-simple, T is a subgroup of dimension at most 1 that centralizes C, and D is solvable. (b) G = T CD, and ind G ≤ ind C + ind D + 1. (c) The group D is a simply connected, compact-free linear Lie group. (d) The center Z of G is contained in T C, and T C/Z is a maximal compact subgroup of G/Z, while CZ/Z is the commutator group of T C/Z. Proof. The centralizer of the commutator group of a maximal compact subgroup of a simple Lie group has dimension at most 1. The assertions follow immediately from the Iwasawa decomposition for simple real Lie groups [14], VI, 5.3 by an application of 34.5 and 32.1. 2 35.4 Theorem. Let G be a locally compact group, and assume that A is a closed connected Abelian normal subgroup such that ind A < ∞. Then there exists a minimal closed connected Abelian normal subgroup M ≤ A, and 0 < ind M ≤ ind A. Moreover: (a) Either the group M is compact, or it is isomorphic with Rm , where m = ind M. (b) If M is compact, then M lies in the center of the connected component G1 . Proof. The set A of closed connected Abelian normal subgroups of G that are contained in A is partially ordered by inclusion. Since ind X = ind Y for X, Y ∈ A implies that X = Y by 33.5 (b), there are only chains of finite length in A. The maximal compact subgroup C of a minimal element of A is a closed connected characteristic subgroup of M, hence either M = C or C = {1} by minimality. In the latter case, M ∼ 2 = Rm by 23.11. Assertion (b) is immediate from 25.4.
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From 35.4, we infer that the class of locally compact connected groups of finite dimension splits into the class of semi-simple groups, and the class of groups with a minimal closed connected Abelian normal subgroup M. The action of G on M via conjugation is well understood: 35.5 Theorem. Let G be a locally compact group, and assume that there exists a minimal closed connected Abelian normal subgroup M ∼ = Rm . (a) The group G acts (via conjugation) R-linearly and irreducibly on M. (b) The factor group L = G/ CG (M) is a linear Lie group, in fact, a closed subgroup of GL(m, R). The commutator group S of L is also closed in GL(m, R), and we have that L ∼ = SZ, where S is either trivial or semi-simple, and Z is the connected component of the center of L. Moreover, Z is isomorphic to a closed connected subgroup of the multiplicative group C∗ . (c) For every one-parameter subgroup R of M, we have that ind G/ CG (R) ≤ ind M. Proof. The action via conjugation yields a continuous homomorphism from G to GL(m, R), cf. 24.6. Every invariant subspace V of M ∼ = Rm is a closed connected normal subgroup of G. Minimality of M implies that V = M, or V is trivial. This proves assertion (a). The factor group G/ CG (M) is a Lie group [18], VIII.1.1, which acts effectively on M ∼ = Rm . This action is a continuous homomorphism of Lie groups. From [4], II.6.2, Cor. 1(ii) we infer that the image L of G/ CG (M) in GL(m, R) is an analytic subgroup. Moreover, we know that L is irreducible on Rm . According to [10], the group L is closed in GL(m, R). Hence we may identify L and G/ CG (M), cf. 33.11. The commutator group S of L is closed, see [18], XVIII.4.5. From [64], 3.16.2 we infer that the radical of the group L is contained in the center Z of L, whence L = SZ. According to Schur’s Lemma [33], p. 118, p. 257, the centralizer of L in EndR (M) is a (not necessarily commutative) field. Since this field is also a finite-dimensional algebra over R, we infer that it is isomorphic to R, C, or H, cf. 26.9. Thus Z generates a commutative subfield of H, hence Z ≤ C∗ . This completes the proof of assertion (b). Assertion (c) follows readily from 33.10 (b), since by linearity CG (R) = CG (r) 2 for every non-trivial element r of R. 35.6 Theorem. Let A be a locally compact connected Abelian group, and write A = RC, where R ∼ = Ra and C is compact and connected. (a) The group of automorphisms of A is isomorphic to the semi-direct product Aut(C) Hom(Ra , C) GL(a, R), the connected component Aut(A)1 is isomorphic to Hom(C ∗ , Ra ) GL(a, R).
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(b) If ind C = c < ∞, then Aut(A)1 is a linear Lie group; in fact, there is a monomorphism ι : Aut(A) → GL(c, Q) Hom(Qc , Ra ) GL(a, R), where Q and GL(c, Q) carry the discrete topologies. Proof. The group Aut(A) leaves invariant the (unique) maximal compact subgroup C of A. Together with the remarks in 25.8, this gives the first part of the assertion. From Hom(R, C) ∼ = Hom(C ∗ , Ra ) ≤ Hom(Q ⊗ C ∗ , Ra ) we infer that there exists a monomorphism from Aut(A) to the group L := Aut(C) Hom(Qc , Ra )
GL(a, R). Now assume that ind C < ∞. According to [15], 24.28, we have that ind(Q ⊗ C ∗ ) = rank C ∗ = ind C. Hence L is a (linear) Lie group. Since L has no small subgroups, the same holds for Aut(A). Hence Aut(A)1 is a (connected) Lie group [39], Ch. III, 4.4, and the restriction of ι to Aut(A)1 is analytic, see 2 [18], VII, Th. 4.2 or [64], Sect. 2.11. 35.7 Corollary. Let G be a locally compact connected group, and assume that A is a closed connected normal Abelian subgroup of G. If ind A < ∞, then G/ CG (A) is an analytic subgroup of Rca GL(a, R), where C is a compact group of dimension c, and A ∼ = Ra × C. An important application is the following. 35.8 Theorem. Let G be a compact group, and assume that a is a natural number, and that C is a compact connected Abelian group. For every continuous homomorphism μ : G → Aut(Ra × C), the following hold: (a) Both Ra and C are invariant under Gμ . (b) There exists a positive definite symmetric bilinear form on Ra that is invariant under Gμ . Consequently, μ induces a completely reducible R-linear action of G on Ra . (c) If Gμ is connected, then Gμ acts trivially on C. Proof. Assertion (a) follows from 35.6 and the fact that Hom(C ∗ , R) is compactfree. The group Gμ induces a compact subgroup of GL(a, R). Using Weyl’s Trick 14.35, one finds a Gμ -invariant positive definite symmetric bilinear form q on Ra . If V is a Gμ -invariant subspace of Ra , then the orthogonal complement with respect to q is Gμ -invariant as well. This completes the proof of assertion (b). 2 The last assertion follows from 25.4. 35.9 Theorem. In every locally compact connected group G of finite dimension, there exists a maximal closed connected solvable √ √ normal subgroup (called √ the solvable radical G of G). Of course, G = G if G is solvable, and G is √ non-trivial if G is not semi-simple. The factor group G/ G is semi-simple (or trivial).
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Proof. Obviously, the radical is generated by the union of all closed connected normal solvable subgroups, cf. [31], Th. 15. 2 If G is a connected linear Lie group, or a simply connected Lie group, it is known [18],√XVIII.4, [64], 3.18.13 that there exists a closed subgroup S of G such √ that G = S G and ind(S ∩ G) = 0. Such a (necessarily semi-simple) subgroup is called a Levi-complement in G. Even for Lie groups, however, such an S does not exist in general (see [64], Ch. 3, Exercise 47 for an example). Apart from the fact that, in the Lie case, one has at least an analytic (possibly non-closed) Levi complement [64], 3.18.13, one also has some information about the general case: 35.10 Theorem. (a) Let L be a Lie group, and let S be a semi-simple analytic subgroup of L. Then there is an Abelian closed connected subgroup C ≤ CG (S) such that S = SC, and ind(S ∩ C) = 0. √ (b) Let G be a locally compact connected group of finite dimension, and let G be the solvable √ radical of G.√Then there exists a closed subgroup H of G such that G = H G and (H ∩ G)1 ≤ CG (H ). Proof. Without loss, we may assume that S is dense in L. The adjoint action of S on the Lie algebra l of L is completely reducible, hence there exists a complement c of the Lie algebra s of S such that [s, c] ≤ c ∩ [l, l]. According to [18], XVI.2.1, we have that [l, l] = [s, s] = s. This implies [s, c] ≤ s ∩ c = 0, and assertion (a) follows. If G is a Lie group, then assertion (b) can be obtained from assertion √ (a), applied to a Levi complement S of G, and the connected component of SC∩ G is contained in the radical C of SC. Applying 34.5, we obtain assertion (b) in general. 2 35.11 Remark. If G is an algebraic group, then the decomposition in 35.10 (b) is the so-called algebraic Levi decomposition into an almost semi-direct product of a reductive group and the unipotent radical, see [43], Ch. 6.
36 Compact Groups We have shown in Section 19 that every compact group models a projective limit of compact Lie groups (namely, closed subgroups of unitary groups of Hilbert spaces of finite dimension). By the following result, the structure theory of compact connected groups is, essentially, reduced to the theory of compact almost simple Lie groups and the theory of compact Abelian groups:
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36.1 Theorem. (a) Let G be a compact connected group. Then there exist a compact connected Abelian group C, a family (Si )i∈I of almost simple compact Lie groups Si , and a surjective homomorphism η : C × i∈I Si → G with ind ker η = 0. The image C η is the connected of the center of G, and the commutator
component η group G equals S . i∈I i (b) Conversely, of the form C × i∈I Si , as in a, is compact; hence
η every group also C × i∈I Si . Proof. [34], Th. 1, Th. 2, cf. [4], App. I, no. 3, Prop. 2, [31], Remark after Lemma 2 2.4, [66], §25. Note that, in general, the connected component C η of the center of G is not a complement, but merely a supplement of the ηcommutator group in G. Since the topology of the commutator group is well understood, a complement i∈I Si would be fine in order to show the more delicate topological features of G. The following result asserts the existence of a complement (which, in general, is not contained in the center of G). 36.2 Theorem. Every compact connected group is a semi-direct product of its commutator group and an Abelian compact connected group. Proof. [22], 2.4. A generalization to locally compact groups, involving rather technical assumptions, is given in [21], Th. 6. 2 For a compact connected group G, let η : C × i∈I Si → G be a quotient morphism as in 36.1. The possible factors Si are known from Lie Theory; see, e.g., [43], Ch. 5. In order to understand the structure of C, one employs the Pontryaginvan Kampen duality for (locally) compact Abelian groups. See Chapter F, and also [50], [51] for a treatment that stresses the functorial aspects of duality. The dual C ∗ is a discrete torsion-free Abelian group of rank c, and c equals the covering dimension of C if one of the two is finite [40], Th. 34, p. 108, [15], 24.28. Hence there are embeddings Z(c) → C ∗ and C ∗ → Q ⊗ C ∗ ∼ = Q(c) . Dualizing again, we obtain a convenient description of the class of compact connected Abelian groups (note that covering dimension and small inductive dimension coincide for the class of groups that we are interested in, see 36.7 below): 36.3 Theorem. Let C be a compact connected Abelian group. (a) If C has finite (covering) dimension c, then there are quotient morphisms σ : (Q∗ )c → C and τ : C → Tc , both with totally disconnected kernel. (b) If C has infinite (covering) dimension, then there exists a cardinal number c such that there are quotient morphisms σ : (Q∗ )c → C and τ : C → Tc , both with totally disconnected kernel.
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Sometimes, one needs a more detailed description, as supplied by the following: 36.4
Remark. Dualizing the description 1of Q asn inductive limit of the system 1 Z (endowed with natural inclusions n Z → d Z), we obtain that the character n n∈N group Q∗ models the projective limit (see 17.14) of the system (Tn )n∈N , where Tn = T for each n, with quotient morphisms t → t d : Tnd → Tn . Every onedimensional compact connected group is a quotient of Q∗ . Within the boundaries that are set up by the fact that locally compact connected Abelian groups are divisible (see 23.27), we are free to prescribe the torsion subgroup of a one-dimensional compact connected group. In fact, let P be the set of all prime numbers, and let P ⊆ P be an arbitrary subset. In the multiplicative monoid of natural numbers, let NP be the submonoid generated by P (i.e., NP consists of all natural numbers whose prime decomposition uses only factors from P ). With this notation, we have: 36.5 Theorem. For every subset P ⊆ P, there exists a compact connected group C with ind C = 1 and the following properties: If c ∈ C has finite order n, then n ∈ NP . Conversely, for every n ∈ NP there exists some c ∈ C of order n. Proof. The limit SP of the subsystem (Tn )n∈NP of the projective system considered in 36.4 has the required property. 2 See [15], 10.12–10.15 for alternate descriptions of the ‘solenoids’ SP . 36.6 Examples. Of course, SP = Q∗ , and S∅ = T. The group S{p} is the dual of 1 the group ∞ n=0 p n Z, its torsion group has elements of orders that are not divisible by p. We conclude this chapter with an observation that relates 36.3 to the inductive dimension function. 36.7 Theorem. Small inductive dimension and covering dimension coincide for compact connected Abelian groups. Proof. Let A be a compact connected Abelian group, and let d denote its covering dimension. The dual group A∗ is discrete by 20.6 and torsion-free (since A is connected, see 23.26). Assume first that d is finite. According to [40], Th. 34, p. 108, we have the equality d = rank A∗ . For a maximal free subgroup F of A∗ we infer that F ∼ = Zd , and A∗ /F is a torsion group. Consequently, the annihilator F ⊥ is totally disconnected, and has inductive dimension 0 by 33.5 (a). Now Td ∼ = F∗ ∼ = ⊥ d A/(F ), and we conclude from 33.2 and 33.5 that ind A = ind T = d. If d is infinite, then rank A is infinite, and we infer that ind A is infinite, as well. 2
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37 Countable Bases, Metrizability In several instances, in particular when describing locally compact groups as projective limits (see 17.14) of Lie groups, we shall benefit from the following observation. 37.1 Theorem. Let G be a locally compact connected group of finite dimension. Then the topology of G has a countable neighborhood basis. In particular, the topology is separable and metrizable. Proof. In view of the Theorem of Malcev and Iwasawa 32.5 (c), it suffices to consider the case where G is compact. According to 36.2 below, the group G is the semi-direct product of its commutator group G and some Abelian compact connected group A, both of finite dimension. Since G is a Lie group by 36.1, there remains to show that A has a countable basis. The weight of A (i.e., the minimal cardinality of a neighborhood basis for A) equals the cardinality of the character group A∗ , see [15], 24.15. Since A is connected, we have that A∗ is torsion-free, see 23.26 and 23.24 (b). Thus A∗ is isomorphic to a subgroup of Qr , where r is the torsion-free rank of A∗ . From [15], 24.28 we infer that r = ind A is finite. (Note that covering dimension, as used in [15], coincides with small inductive dimension for A by 36.7.) Thus A∗ is countable, and A possesses a countable neighborhood basis. The assertion that G is metrizable follows from the existence of a countable neighborhood basis, see [15], 8.3. 2 For the conclusion of 37.1, neither the assumption that G is connected nor the assumption of finite dimension can be dispensed with. 37.2 Examples. If D is an uncountable discrete Abelian group, then the dual group D ∗ is a compact group of weight |D| and infinite dimension. For a concrete example, take D = ZN . Then D ∗ = TN is connected (of infinite dimension). The group D itself is an example of a disconnected group of uncountable weight (and dimension 0). 37.3 Corollary. Every locally compact connected group of finite dimension models the projective limit of a sequence of finite coverings of a Lie group. Proof. Assume that G satisfies the assumptions. According to 32.1 (b), there exists a totally disconnected compact normal subgroup N of G such that G/N is a Lie group. Since G has a countable basis, we find a descending sequence Un of relatively compact neighborhoods of 1 with trivial intersection. Now 32.1 (b) asserts the existence of a descending sequence Nn of compact normal subgroups such that Nn ⊆ Un and G/(Nn ) is a Lie group for every n. Since Nn is totally disconnected, we obtain for every k ≤ n that Nn /(Nk ) is finite, viz. G/(Nn ) is a finite covering of G/(Nk ). 2
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38 Non-Lie Groups of Finite Dimension In this section, we construct some examples of non-Lie groups, and solve the problem whether or not a given simple Lie group is the quotient of some almost simple non-Lie group. 38.1 Lemma. Assume that (J, ) is a directed set, let (πij : Gj → Gi )ij be a projective system of locally compact groups, and assume that G models the projective limit (see 17.14). If πij has compact kernel for all i, j such that i j , then the projective limit is a locally compact group. Proof. Let G be the projective limit. For every i ∈ J , the natural map πi : G → Gi has compact kernel, since this kernel is the projective limit of compact groups. Hence πi is a perfect map by 32.8, and the preimage of a compact neighborhood in Gi is a compact neighborhood in G. 2 38.2 Lemma. Assume that (J, ) is a directed set, and let (πij : Gj → Gi )ij be a projective system of locally compact groups such that ker πij is finite for all i j . Let G model the projective limit. If J has a smallest element, then every projection πi : G → Gi has compact totally disconnected kernel. Proof. Assume that a is the smallest element of J . For every i ∈ J , let Ki denote the kernel of πia . The kernel of πa models the projective limit K of the system
i from the fact πij |K Kj : Kj → Ki ij . Since ker πi ≤ ker πa , the assertion follows that K is a closed subgroup of the compact totally disconnected group i∈I Ki . 2 38.3 Theorem. Let G be a locally compact connected group of finite dimension. (a) If G is not a Lie group, and N is a compact totally disconnected normal subgroup such that G/N is a Lie group, then there exists an infinite sequence πn : Ln+1 → Ln of cn -fold coverings of connected Lie groups such that L0 = G/N and 1 < cn < ∞ for every n, and G models the projective limit of the system (Ln )n∈N . (b) Conversely, let L be a connected Lie group, and let πn : Ln+1 → Ln be an infinite sequence of cn -fold coverings of connected Lie groups such that L0 = L and 1 < cn < ∞ for every n. Then there exists a locally compact connected non-Lie group G with a compact totally disconnected normal subgroup N such that G/N ∼ = L. Proof. Assertion (a) is an immediate consequence of 37.3, recall that a covering of a Lie group is a Lie group again. In the situation of (b), consider the projective system πn : Ln+1 → Ln . By 38.1, the limit is modeled by a locally compact group G. The projective limit N of the
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kernels of the natural maps G → Gi is modeled by a compact infinite group, and N is totally disconnected by 38.2. Hence G is not a Lie group. 2 38.4 Remarks. (a) Our technical assumption in 38.2 that I has a smallest element seems to be adequate for the application in 38.3 (b). On the bottom of page 260 in [28], one finds an example of an infinite-dimensional projective limit of a system of one-dimensional groups which shows that 38.2 does not hold without some assumption of the sort. (b) Theorem 38.3 (b) could also be derived from [28], 2.2, 3.3. Roughly speaking, the method of K. H. Hofmann, T. S. Wu and J. S. Yang [28] consists of a dimension-preserving compactification of the center of a given group. 38.5 Remark. The fundamental group of a semi-simple compact Lie group is finite; see, e.g., [64], Th. 4.11.6. This implies that, for a connected Lie group L, the existence of a sequence of coverings as in Theorem 38.3 is equivalent to the existence of a central torus in a maximal compact subgroup of L. The simple Lie groups with this property are sometimes called hermitian groups, they give rise to non-compact irreducible hermitian symmetric spaces [14], VIII.6.1. In the terminology of [62], C,p R,2 the corresponding simple Lie algebras are the real forms Al (1 ≤ p ≤ l+1 2 ), Bl R,2 H (l ≥ 2), CR (l ≥ 4), DH 2p (p ≥ 3), D2p+1 (p ≥ 2), E6(−14) , E7(−25) . l (l ≥ 3), Dl In [43], these algebras are denoted as sup,l+1−p (including sup,p ), so2,l−1 , sp2l (R), so2,l−2 , u∗2p (H), u∗2p+1 (H), EIII, and EVII, respectively. Consequently, we know the locally compact almost simple non-Lie groups. 38.6 Theorem. Let G be a locally compact connected almost simple group. Then G is not a Lie group if, and only if, the center Z of G is totally disconnected but not discrete. In this case, the factor group G/Z is a hermitian group (cf. 38.5), and G is the projective limit of a sequence of finite coverings of G/Z. Of course, a similar result holds for semi-simple non-Lie groups: at least one of the almost simple factors in 35.1 is not a Lie group.
39 Arcwise Connected Subgroups In the theory of Lie groups, arcwise connectedness plays an important role. In fact, according to a theorem of H.Yamabe [13], the arcwise connected subgroups of a Lie group are in one-to-one correspondence with the subalgebras of the corresponding Lie algebra. Our aim in this section is to extend this to the case of locally compact
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groups of finite dimension. To this end, we shall refine the topology of the arc component, and show that we obtain a Lie group topology. 39.1 Definition. Let G be a topological group, and let U be a neighborhood basis at 1. For W ∈ U, let UW = {U ∈ U | U ⊆ W }, of course UW is again a neighborhood basis at 1. For every U ∈ U, we denote by U arc the arc component of 1 arc | U ∈ U }. in U . For W ∈ U, let Uarc W W = {U Easy verification shows that the system {V g | V ∈ Uarc W , g ∈ G} forms a basis for a group topology on G. For every W ∈ U, we obtain the same group topology on G, this topology shall be denoted by TUloc arc . The following proposition clearly implies that the topology TUloc arc is locally arcwise connected. 39.2 Proposition. (a) The topology TUloc arc is finer than the original one on G. (b) A function α : [0, 1] → G is continuous with respect to the original topology if, and only if, it is continuous with respect to TUloc arc . Proof. For U ∈ U and g ∈ U , we find V ∈ U such that V g ⊆ U . Now V arc g ⊆ U , and we infer that U ∈ TUloc arc . The ‘if’-part of assertion (b) follows immediately from (a). So assume that α is continuous with respect to the original topology, let r ∈ [0, 1] and U ∈ U. By continuity, there is a connected neighborhood I of r such that I α ⊆ U r α . Now continuity of α with respect to TUloc arc follows from the fact that I α ⊆ U arc r α . 2 39.3 Corollary. (a) With respect to TUloc arc , the arc component is again arcwise connected. Thus the arc component Garc of G coincides with the arc component of G with respect to TUloc arc . (b) Algebraically, Garc is generated by U arc for every U ∈ U. While Garc is understood to be endowed with the induced original topology, we shall write Gloc arc for the topological group Garc with the topology induced from TUloc arc . According to 39.2 (b), the inclusion Garc → G yields a continuous injection ι : Gloc arc → G. 39.4 Theorem. Assume that G is a locally compact group of finite dimension, and let U be a neighborhood basis at 1. Then the following hold: (a) If W ∈ U is the direct product of a compact totally disconnected normal subgroup C of G and some local Lie group ⊆ G, then Garc is algebraically generated by the connected component 1 . In particular, G1 ≤ Garc C.
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(b) The factor group G/C is a Lie group, in fact, the natural map π : G → G/C restricts to a topological isomorphism of onto a neighborhood of 1 in G/C. (c) Gloc arc is a connected Lie group, and ιπ : Gloc arc → (G/C)1 is a covering. (d) The arc component Garc is dense in G1 . (e) The sets Hom(R, G) and Hom(R, Garc ) coincide. The map α → αι : Hom(R, Gloc arc ) → Hom(R, G) is a bijection. Proof. For every U ∈ U, the connected component G1 is contained in the subgroup U that is algebraically generated by U . In particular, G1 ≤ W = C ; recall that C is a normal subgroup of G. The connected component 1 is arcwise connected, therefore 1 = W arc . This implies that 1 is open in Gloc arc , whence Gloc arc = 1 . This proves assertion (a). From the fact that W is the direct product of C and , we conclude that π | is injective. The quotient map π is open, hence π = W π is open in G/C. Therefore, the group G/C is locally isomorphic to , and (b) is proved. Since V := ( 1 )π is open in G/C, we obtain that (G/C)1 is generated by V . Hence ιπ : Gloc arc → (G/C)1 is surjective, and assertion (d) holds. An application of 34.5 (c) to the closure of Garc and the restriction of π to G1 yields assertion (d). Finally, assertion (e) is an immediate consequence of 39.2 (b). 2 39.5 Remarks. (a) From K. Iwasawa’s local product theorem [12], Th. B we know that in every locally compact group there exists a neighborhood W with the properties that are required in 39.4 (a). (b) In view of 39.4 (e), we define the Lie algebra of G as Hom(R, G), cf. [35], II.11.9, p. 140. We then have the exponential map exp : Hom(R, G) → G : α → 1α . For every subalgebra s of Hom(R, G), it seems reasonable to define the corresponding arcwise connected subgroup that is generated by exp s. This is in contrast with R. Lashof’s definition [37], 4.20, while our definition of the Lie algebra essentially amounts to the same as R. Lashof’s. (c) A source for further information on G might be the quotient morphism η : Gloc arc × C → G = Garc C : (x, c) → x ι c. Note that η is a local isomorphism, and therefore a quotient map.
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(d) As an immediate consequence of the local product theorem, we have that a locally compact group of finite dimension is a Lie group if, and only if, it is locally connected. However, it is not clear a priori that Gloc arc is locally compact. We collect some consequences of 39.4. 39.6 Theorem. Let G be a locally compact connected group of finite dimension. (a) Let M be a compact normal subgroup such that ind M = 0 and G/M is a Lie group. For the natural map πM : G → G/M, we have that ιπM : Gloc arc → G/M is a covering. In particular, ind G = ind G/M = ind Gloc arc . (b) The group G is a Lie group if, and only if, the composite ιπM is a finite covering. (c) If H is a connected Lie group, and α : H → G is a continuous homomorphism, then α factors through ι. (d) The group G is a Lie group if, and only if, the morphism ι is surjective. Proof. The kernel K = Garc ∩M of ιπ is closed in Gloc arc and totally disconnected. Since Gloc arc is a Lie group, we infer that K is discrete. Since Gloc arc and G/M are connected Lie groups of the same dimension, we conclude that ιπ is surjective, hence assertion (a) holds. If G is a Lie group, then G = Garc = Gloc arc . Being totally disconnected, the subgroup M is discrete and compact, hence finite. Thus πM is a finite covering. Now assume that ιπM has finite kernel K = Garc ∩ M. Let U be a neighborhood of 1 in G such that U ∩ K = {1}. According to 32.1 (b), there exists an normal totally disconnected compact subgroup N such that N ⊆ U and G/N is a Lie group. For every such N , we obtain that ιπN is an isomorphism. If N is non-trivial, we may pass to a neighborhood V of 1 in U such that N is not contained in V . Then we find a normal compact subgroup N ⊆ N ∩ V , and obtain a proper covering G/N → G/N, in contradiction to the fact that ιπN is an isomorphism. This implies that N = {1}, and G is a Lie group. Thus assertion (b) is proved. In the situation of (c), it suffices to show that α is continuous with respect to TUloc arc ; in fact H α is arcwise connected, hence contained in Garc . For every U ∈ U, we find a neighborhood V of 1 in H such that V α ⊆ U . Since H is locally arcwise connected, we may assume that V is arcwise connected. This implies that V α ⊆ U arc , whence α is continuous with respect to TUloc arc . In order to prove (d), assume first that ι is surjective. Then ι is a homeomorphism by the Open Mapping Theorem 6.19, hence G is a Lie group. The proof of (d) is completed by the observation that every connected Lie group is arcwise connected. 2 We remark that 39.6 (d) is a result of M. Goto, see [13].
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285
39.7 Examples. (a) Dense one-parameter subgroups (or, more generally, dense analytical subgroups) are familiar from Lie theory; most prominent, perhaps, is the ‘dense wind’ R → T2 . In the realm of Lie groups, the closure of a non-closed analytical subgroup has larger dimension than the subgroup itself. (b) The dual of the monomorphism Q → R, where Q carries the discrete topology, yields a monomorphism R → Q∗ with dense image. Note that ind Q∗ = 1 = ind R. Equidimensional immersions are typical for non-Lie groups; see [28], and 39.6.
40 Algebraic Groups In this last section, we briefly indicate how certain results from the theory of complex algebraic groups yield results on the rough structure of locally compact groups of finite dimension. Let G be a locally compact group. If ind G < ∞, and A, B ∈ Struc(G) such that A < B, then ind A < ind B by 33.5 (b). Consequently, every chain in Struc(G) has a maximal and a minimal element. This corresponds to the fact that analytic (arcwise connected) subgroups of a Lie group are in one-to-one correspondence to the subalgebras of the Lie algebra, where the dimension function is obviously injective on every chain. Upper bounds for the dimension of subgroups of a given locally compact group G yield lower bounds for the dimension of separable metric spaces that admit a non-trivial action of G, cf. 33.10. In order to gain information about the maximal elements in Struc(G), we shall try to employ information from the theory of algebraic groups. The maximal algebraic subgroups of a complex algebraic group are understood quite well. E.g., one has the following result, cf. [29], 30.4. 40.1 Theorem. Let G be a reductive complex algebraic group. Then every maximal algebraic subgroup of G either is parabolic or has reductive Zariski-component. 2 Parabolic subgroups are those that contain a Borel subgroup. Every parabolic subgroup is a conjugate of a so called standard parabolic subgroup, and these are easy to describe. In fact, they are in one-to-one correspondence to the subsets of a basis for the lattice of roots of G relative to a maximal torus. Cf. [29], 30.1. The maximal reductive subgroups of reductive complex algebraic groups are known; see, e.g., [3]. There arises the question as to what extent these results are applicable in order to describe the maximal closed subgroups of a given locally compact group, or even
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a Lie group. First of all, we note that an important class of Lie groups consists in fact of algebraic groups, cf. [43], Ch. 3, Th. 5. 40.2 Theorem. Let G be a connected complex linear Lie group, and assume that G equals its commutator group. Then G admits a unique complex algebraic structure. In particular, every complex semi-simple linear Lie group is complex algebraic. While every algebraic subgroup of a complex algebraic group G is closed in the Lie topology, the converse does not hold in general. However, the structure of the algebraic closure H alg of a connected analytic subgroup H of G (i.e., the smallest algebraic subgroup that contains H ) is to some extent controlled by the structure of H . In particular, the commutator group of H alg equals that of H , cf. [19], VIII.3.1. This implies the following. 40.3 Theorem. Let G be a complex semi-simple linear Lie group. Then every maximal closed connected subgroup is algebraic. Via complexification, we obtain an estimate for the possible dimensions of maximal closed subgroups of real semi-simple Lie groups (and thus of locally compact semi-simple groups). 40.4 Theorem. Let G be a semi-simple (real) Lie group. If H is a proper subgroup, then ind H ≤ mG , where mG denotes the maximal (complex) dimension of proper subgroups of the complexification of G. Since, e.g., the parabolic subgroups have no counterpart in compact real forms, the estimate in 40.4 may be quite rough. However, it is attained in the case of split real forms. Note that maximal closed subgroups of compact semisimple Lie groups correspond to reductive subgroups of the complexification, see [3]. 40.5 Example. Consider a complex simple Lie group of type G2 . Then a reductive subgroup is either semi-simple of type A2 , A1 × A1 , A1 , or a product of A1 with a one-dimensional centralizer, or Abelian of dimension at most two. The maximal parabolic subgroups are semi-direct products of a Levi factor of type A1 and a solvable radical of dimension 6. Consequently, if G is a locally compact almost simple group such that the factor group modulo the center is a real form of G2 , then the maximal elements in Struc(G) have dimension at most 9. Note that, if G is the compact real form, then every subgroup is reductive, and the maximal elements in Struc(G) have dimension at most 8. Since ind G = 14, we infer that if G acts non-trivially on a separable metric space X, then ind X ≥ 5, and ind X ≥ 6 if G is compact.
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Index of Symbols
Sets, Maps, Topology ind X Nx S top V uni F fil T0 , T1 , T2 , T3 , T4 ← Tϕ Tx T |Y V|Y X α∈A Xα α∈A Xα α∈A (Xα , Xα ) X/ϕ Xa Y◦ Y
X R◦R R◦R◦R
small inductive dimension, 2.16 := {N ⊆ X | ∃T ∈ Tx : T ⊆ N } neighborhood filter, 1.1 topology generated by (subbasis) S, 1.1 uniform structure generated by (uniformity) V, 8.1 filter generated by (filterbasis) F , 1.22 separation properties, 1.14 := {x ∈ X | x ϕ ∈ T }, 1.4 := {T ∈ T | x ∈ T }, 1.1 := {Y ∩ T | T ∈ T } induced topology, 1.1 := {V ∩ Y 2 | V ∈ V} induced uniformity, 8.9 cartesian product, 1.8 product topology, 1.10 product of topological spaces, 1.10 ← := {T ⊆ Y | T ϕ ∈ X} quotient topology, 1.34 connected component of a, 2.4 interior of Y , 1.1 closure of Y , 1.1 := {(x, x) | x ∈ X} diagonal, 1.16, 8.1 := {(x, z) ∈ X2 | ∃y ∈ X : (x, y) ∈ R (y, z)}, 8.1 := {(w, z) ∈ X2 | ∃x, y ∈ X : (w, x), (x, y), (y, z) ∈ R}, 8.5
R
:= {(y, x) | (x, y) ∈ R}, 8.1 := {z ∈ X | ∃y ∈ Y : (y, z) ∈ V }, 8.3 := {x}V , 8.3 := {C V | V ∈ V, C ∈ C}, 8.29 set of homeomorphisms, 1.4 := Homeo(X, X) group of homeomorphisms, 1.4, 9.15, 9.16
↔
YV xV
CV
Homeo(X, Y ) Homeo(X)
Topological Groups, Rings, Fields, Semigroups AF (C, +) (C× , ·) char R frobp
algebraic closure of the commutative field F , 26.19 additive group of complex numbers, 3.5 multiplicative group of complex numbers, 3.5 characteristic of the ring R, 26.13 Frobenius endomorphism, 26.16
292
Index of Symbols
D(p, d) Fq F [[t]] F ((t)) Fσ ((t%) H N P Qp (R, +) (R× , ·) T Z(n) Z(p∞ ) Zp R n×n GL(n, R) SL(n, R) O(n, R) SO(n, R) U(n, C) SU(n, C) CG (X) NG (X) Fix(ϕ) Fix( ) xG Gx compr G (Y ) Aut(G) Mor(G, H ) Mor(G, G) ker ϕ zG A [ϕ]R B S
:= {(qi )i∈d ∈ (Qp )d | supi∈d |qi |p < ∞} minimal divisible extension of (Zp )d , 27.7 finite field with q elements, 26.19, Exercise 26.11, Exercise 26.12 ring F [[t]] of formal power series over F , 26.3 field of formal Laurent series over F , 26.3 field of skew Laurent polynomials over F , 26.4, Exercise 26.7 field of Hamilton’s quaternions, Exercise 3.2, cp. Exercise 26.8 nonnegative integers (including 0) set of prime numbers field of p-adic numbers, 26.5, 8.58 additive group of real numbers, 3.5 multiplicative group of real numbers, 3.5 := R/Z circle group, Exercise 6.1, 20.1 := Z/nZ additive group, orring 3.30 := Z + pzn | z ∈ Z, n ∈ N Prüfer group, 17.1, Exercise 18.8, 4.19, 4.24 ring of p-adic integers, 17.2, Exercises 18.6–18.11, 8.49, 8.58 ring of all n × n matrices with entries in R, 3.5 group of all invertible n × n matrices with entries in R, 3.5 := {S ∈ GL(n, R) | det S = 1} special linear group, 3.5 := {M ∈ Rn×n | MM = 1} (real) orthogonal group, 3.8 := O(n, R) ∩ SL(n, R) special orthogonal group, 3.8 := {M ∈ Cn×n | MM ∗ = 1} unitary group, 3.8 := U(n, C) ∩ SL(n, C) special unitary group, 3.8 := {g ∈ G | ∀x ∈ X : xg = gx} centralizer of X in G, 3.37, 3.39 := {g ∈ G | ∀x ∈ X : g −1 xg ∈ X} normalizer of X in G, 3.37, 3.39 {x ∈ X | x ϕ = x}, 1.16 := := ϕ∈ Fix(ϕ), 1.16 orbit of x under G, 10.9 stabilizer of x, 10.9 := {g ∈ G | ∀y ∈ Y : (y, g)ω ∈ Y } compression semigroup, 28.1 group of automorphisms of G, 3.32 set of all continuous group homomorphisms from G to H , 11.1 as topological ring, 11.7 kernel of homomorphism, 3.25 endomorphism of commutative group G, 3.31, 23.22, 23.23, 27.4 matrix description of group homomorphism, 11.14
Index of Symbols
X G/H πH A⊕B C δ N A∨B A∧B [A, B] 1 (B) NA 1 CA (B) Struc(G) Cp(G) CpFree(G) d(G) D(G) dn+1 (G) Dn+1 (G) zn (G) Zn (G) z√n (G) G Comp(G) Tors(A)
Ap φ(d) RU LU R(G,T ) L(G,T ) S(G,T )
subgroup generated by X, 4.1 homogeneous space 6.1, 6.3 natural surjection G → G/H , 6.1 interior direct product, 6.24 semidirect product, 10.11 smallest closed subgroup that contains both A and B, 34.2 biggest connected subgroup that is contained in A and B, 34.2 closure of the subgroup generated by {a −1 b−1 ab | a ∈ A, b ∈ B}, 34.1 connected component of the normalizer of B, taken in A, 34.2 connected component of the centralizer of B, taken in A, 34.2 the rough structure of G, 34.2 the set of compact connected subgroups of G, 34.2 the set of compact-free closed connected subgroups of G, 34.2 derived group, 7.1 := d(G) Hausdorff derived group, 7.1 := d(dn (G)) derived series, 7.1 := D(Dn (G)) Hausdorff derived series, 7.1 descending (or lower) central series, 7.10 Hausdorff descending (or lower) central series, 7.10 ascending (or upper) central series, 7.10 solvable radical of G, 35.9 := {C ≤ G | C is compact}, 23.15 := n∈N{0} ker nA , torsion subgroup of the Abelian group A, 23.22 := n≥0 ker pn A , p-primary part, 4.18 := |Z(d)× |, Euler’s function, 6.32 := {(x, y) ∈ G2 | yx −1 ∈ U } member of the right uniformity, 8.2 := {(x, y) ∈ G2 | x −1 y ∈ U } member of the left uniformity, 8.2 := {RU | U ∈ T1 } right uniformity, 8.2 := {LU | U ∈ T1 } left uniformity, 8.2 := {RU ∩ LV | U, V ∈ T1 } bilateral uniformity, 8.2
Spaces of Functions C(X, Y ) $C, U % Sc-o Tc-o , Tp-o
293
space of all continuous maps from X to Y , 9.1 := {ϕ ∈ C(X, Y ) | C ϕ ⊆ U } subbasic element for Tc-o , 9.1 := {$C, U % | X ⊇ C is compact, Y ⊇ U is open} subbasis for Tc-o , 9.1 compact-open topology, 9.1, point-open topology, 9.18
294
Index of Symbols
BεC (ϕ) R(X) B(X) Cc (X), CcC (X) ϕ≥0 V+ C + (X) B + (X) Cc+ (X) ϕa (·|·) L2 (G) S⊥ H End(H ) conv(X) Eμ
:= {ψ ∈ C(Y, Z) | ∀x ∈ C : d(x ϕ , x ψ ) < ε} subbasis element for the topology of compact convergence, 9.20 space of all functions of finite support, 12.1 := {ϕ ∈ C(X, R) | supx∈X |x ϕ | < ∞} space of all bounded continuous functions, 12.1 spaces of continuous functions with compact support, 12.1, 14.1 positive (nonnegative) function, 12.1 := {ϕ ∈ V | ∀x ∈ X : x ϕ ≥ 0} the positive cone in V , 12.1 the positive cone in C(X, R), 12.1 := C + (X) ∩ B(X) the positive cone in B(X), 12.1 := C + (X) ∩ Cc (X) the positive cone in Cc (X), 12.1 translate of ϕ (by applying a −1 to the argument), 10.6 scalar product, 14.5 completion of CcC (G), 14.8 := {y ∈ V | ∀x ∈ S : (x|y) = 0} orthogonal space, 14.9 space of all continuous linear forms on H , 14.10 semigroup of continuous linear endomorphisms, 14.11 convex hull of X, 14.13 := {x ∈ H | x ϕ = μ · x} eigenspace, 14.20
Notions Related to Haar Integrals ϕ
Bψ (ϕ : ψ)
special set of sequences, 12.7 ϕ := inf k∈N bk | (bk )k∈N ∈ Bψ , 12.7
p(ϕ, ψ) pψ PV 5evϕ G ϕ dλ rμ,λ mod α mod
:= (ϕ:ψ) (η:ψ) , 12.13 := (p(ϕ, ψ))ϕ∈Cc+(G) , 12.13 := {pψ | 0 = ψ ∈ Cc+ (G) , supp ψ ⊆ V }, 12.15 evaluation at 5ϕ, 12.17 5 := ϕ dλ = G g ϕ dg Haar integral, 12.24 positive real factor relating Haar integrals, 13.2 value of the module function at α, 13.2 module function Aut(G) → ]0, ∞[, 13.2
Categorical Notions ob C Mor C Mor C (X, Y ) Set
object class of the category C, 15.1 class of all morphisms in the category C, 15.1 set of morphisms from X to Y in C, 15.1 category of sets and maps, 15.2
Index of Symbols
Top TG TA HG HA DG DA LCG LCA CG CA TR DR HR CR / / // ◦
◦
/
/
/ 2, /
category of topological spaces and continuous maps, 15.2 category of topological groups and continuous homomorphisms, 15.2 category of Abelian topological groups, 15.2 category of Hausdorff groups, 15.2 category of Abelian Hausdorff groups, 15.2 category of discrete groups, 15.2 category of discrete Abelian groups, 15.2 category of locally compact Hausdorff groups, 15.2 category of locally compact Abelian Hausdorff groups, 15.2 category of compact Hausdorff groups, 15.2 category of compact Abelian Hausdorff groups, 15.2 category of topological rings, 15.2 category of discrete rings, 15.2 category of Hausdorff rings, 15.2 category of compact Hausdorff rings, 15.2 monic, 15.9 epic, 15.9 open map, 15.9 embedding, 15.9 open embedding, 15.9 quotient morphism, 15.9 morphism whose existence is not yet clear, 15.9
Notions from Duality A∗ ϕ∗ a, α εA ε X ⊥A X⊥ X ⊥⊥ R∗ T∗ Z∗ Z(n)∗ Zp ∗ Q∗
character group of A, 20.1 adjoint of ϕ, 20.1 := a α application of character, 20.1 canonical embedding A → A∗∗ , 20.9 := (εA )A∈LCA natural transformation, 20.11 := {α ∈ A∗ | X, α = 0} annihilator of X, 20.19 annihilator of/ X, 20.19 0 := a ∈ A a, X⊥ = 0 double annihilator of X, 20.19 ∼ = R dual of R, 21.5 ∼ = Z dual of T, 21.6 ∼ = T dual of Z, 21.7 ∼ = Z(n) dual of Z(n), 21.7 ∼ = Z(p∞ ), dual of the group of p-adic integers, Exercise 23.2 dual of the discrete group Q, Exercise 23.7
295
Subject Index
A Abelian group, 3.1 action, 10.1, 10.3 action by linear isometries, 14.21 action by linear transformations, 14.21 action by unitary transformations, 14.21 additive notation, 3.1 adjoint, 20.1 algebraic closure, 26.19, Exercise 26.4, Exercise 26.5, Exercise 26.6, Exercise 26.10, almost direct product, 33.7, 33.8, 35.1 almost semi-direct product, 33.7, 33.9 almost simple, 35.1 (a), 35.3 (a) almost simple non-Lie groups, 38.6 annihilator, 20.19 annihilator mechanism, 23.5 anti-homomorphism, 3.25 approximation of compact Hausdorff groups by Lie groups, 19.8 approximation of locally compact Abelian Hausdorff groups by Lie groups, 21.16 approximation of locally compact groups by Lie groups, 32.1 arc, 2.15 arc component, 2.15 arcwise connected, 2.15 Arzela–Ascoli Theorem, 9.24 ascending central series, 7.10 automorphism, 15.3 automorphism group, 9.17 automorphism of a topological group, 3.32 B Banach space, 14.5
basis, 1.1 bilateral uniformity, 8.2 bonding morphisms, 17.5, 17.12 bounded continuous functions, 12.1 C canonical projection, 16.1, 1.8 cartesian product, 1.8, 3.35 category, 15.1 Cauchy sequence, 1.30 Cauchy filter(basis), 8.20 center of a field, 3.7 centralizer, 3.37, 3.39 character, 20.1 character group, 20.1 characteristic (of a ring), 26.13 circle group, 3.8 closed map , 1.4 closed set, 1.1 closure, 1.1 co-finite topology, 1.24 commutative diagram, 15.9 commutative group, 3.1 compact fields, 26.18 compact group, 3.3 compact Hausdorff groups as projective limits of unitary groups, 19.4 compact Abelian Hausdorff groups as projective limits, 19.5 compact operator, 14.17, 14.20 compact rings, 26.10 compact space, 1.18 compact-open topology, 9.1 compact-free group, 23.15, 23.18 compactly generated group, 4.1, 6.11, 6.12
298
Subject Index
complemented (lattice of closed connected subgroups), 34.6 complete group, 8.46 complete metric space, 1.30 complete regularity, 1.25 completion of a group, 8.45 completion of a field (need not be a field), 8.59 completion of a normed vector space, 14.7 completion of a ring, 8.53 completion of a uniform space, 8.28, 8.31 complex Haar integral, 14.2 compression semigroup, 28.1 cone (over a directed system), 17.5 cone (over a projective system), 17.12 connected, 2.1 connected component, 2.4 connected locally compactAbelian group, 23.18, 23.27 connected locally compact fields, 26.9 connected locally compact rings, 26.8 constant, 17.5, 17.12 continuity of composition, 9.4 continuity of evaluation, 9.6 continuous action, 9.8 continuous at a point, 1.5 continuous map, 1.4 contravariant functor, 15.1 converge, (filterbasis) 1.22 convex, 14.13 convex hull, 14.13, 14.16 covariant functor, 15.1 cyclic group, 4.1, 6.32, 6.33 cyclic subgroups, 6.26 cyclotomic polynomial, 26.17 D dense subset, 1.1 derived group, 7.1 derived length, 7.1
derived series, 7.1 descending central series, 7.10 diagonal, 1.16, 8.1 direct limit, 17.8 directed set, 17.3, 17.4 directed system, 17.5 discrete topology, 1.2 divisible group, 4.14, 4.21, 4.24 23.24, 23.27, 27.2 (b) double annihilator, 20.19, 23.6 double dual, 20.9 dual (group), 20.1 dual of an extension, 23.4, 23.7 E embedding of topological spaces, 1.1 embedding of uniform spaces, 8.28 endomorphism, 15.3 entourage, 8.1 epic, 15.4, 23.2 equicontinuous, 9.22 Euler’s function, 6.32 exact sequence, 3.27 exact sequence of topological groups, 3.34, 23.7 exponent, 27.2 (b) extension, 3.34, 23.4 extension properties, 6.7, 6.15, 23.13, 24.4 F factor group, 3.29 factor ring, 3.29 faithful module, 14.22 field, 3.2 field of p-adic numbers, 26.5 field of formal Laurent series, 26.3 finite field, Exercise 26.11, Exercise 26.12 filterbasis, 1.22 first countable space, 1.3
Subject Index
forgetful functor, 16.5 formal Laurent series, 26.3 formal power series, 26.3 free Abelian group, 6.27, 6.28, 6.29, 6.30 Frobenius endomorphism, 26.16 functions of compact support, 12.1 functions of finite support, 12.1 functor, 15.1 G group, 3.1 H Haar integral, 12.3, 12.20, 12.23 Hahn–Banach–Mazur Theorem, 14.14 Halder’s Lemma, 2.22 Hamilton’s quaternions, Exercise 3.2 Hasse diagram, 3.34 Hausdorff derived group, 7.1 Hausdorff derived length, 7.1, 7.7 Hausdorff derived series, 7.1, 7.7 Hausdorff descending central series, 7.10 Hausdorff lower central series, 7.10 Hausdorff nilpotency class, 7.10 Hausdorff space, 1.14 Hausdorff-nilpotent group, 7.10 Hausdorff-solvable group, 7.1, 7.7 Hilbert G-module, 14.22 Hilbert space, 14.5 homeomorphism, 1.4 homogeneous, 1.7 homogeneous group, 27.1, 27.26 homogeneous space, 6.3 homomorphism, 3.25 I ideal, 3.28 indiscrete topology, 1.2 induced topology (by a metric), 1.2
299
induced topology (by a uniformity), 8.3 induced topology (on a subspace), 1.1 induced uniformity, 8.9 induced uniform structure, 8.9 inductive dimension, 2.16 interior, 1.1 interior direct product, 6.24, 6.23 Intermediate Value Theorem, Exercise 2.2 invariant integral, 12.3 isometric linear representation, 14.21 isomorphism, 15.3 isomorphism of topological groups, 3.32 Isomorphism Theorems, 6.16, 6.17 J jointly continuous, 9.7 K kernel of a homomorphism, 3.25 L left module, 3.2 left uniformity, 8.2 left uniform structure, 8.2 left vector space, 3.2 Levi complement, 35.10 Levi decomposition, 35.11 linear representation, 14.21 local homomorphism, 21.11 locally compact group, 3.3 locally compact space, 1.18 locally compact vector space, 26.40 locally euclidean, 1.7 locally homogeneous, 1.7 local ring, 26.27, 26.33 local section, 33.2 lower central series, 7.10
300
Subject Index
M
O
Malcev–Iwasawa Theorem, 32.5 matrix description of homomorphisms, 11.14, 25.8 maximal compact subgroup, 23.11, 24.12 maximal vector subgroup, 24.1, 24.2, 24.11 meager, 1.28 metrizable space, 1.3, 37.1 minimal closed connected Abelian normal subgroup, 35.4, 35.5 minimal divisible extension of (Zp )d , 27.7, 27.11 model a direct limit, 17.8 model a projective limit, 17.14 modified compact-open topology, 9.14 module function, 13.2, 26.34 module, 3.2 monic, 15.4, 23.2 morphism of directed systems, 17.5 morphism of projective systems, 17.12 multiplication by scalars, 3.2 multiplicative notation, 3.1
open action, 10.10 open map, 1.4 Open Mapping Theorem, 6.19 open set, 1.1 orbit, 10.9 orthogonal group, 3.8 P
p-adic integers, 17.2, Exercise 17.8, 8.49 p-adic numbers, 26.5, 8.58 p-adic topology, Exercise 18.6 ff p-adic valuation, 26.31 path, 2.12 path component, 2.12 pathwise connected, 2.12 perfect map, 32.8 permutation representation, 10.1, 10.3 Peter–Weyl Theorem, 14.33, 19.1 p-group, 4.18 point-open topology (= product topology), 9.18, 9.19 Pontryagin Duality Theorem, 22.6 positive cone, 12.1 positive functions, 12.1 N positive linear form, 12.1 natural map, 3.29 positive linear map, 12.1 natural transformation, 20.11 positive operator, 14.18, 14.20 neighborhood, 1.1 pre-compact, 1.30 neighborhood basis, 1.1, 37.1 pre-Hilbert space, 14.5 neighborhood filter, 1.1 pre-order, 17.3 nilpotency class, 7.10 preserve products, 16.5 nilpotent group, 7.10 prime field, 26.13 non-archimedean absolute value, 26.30, pro-finite group, 19.9 26.34 product (in a category), 16.1 non-archimedean valuation, 26.30, 26.34 product topology, 1.10 no small subgroups, 6.13, 6.14, 21.18 product uniformity, 8.8 normalizer, 3.37, 3.39, 3.40 projective limit, 17.14 normal space, 1.14 projective system, 17.12 normed vector, space 14.5 proper morphism, 20.14, 23.2, 23.7 nowhere dense, 1.28 Prüfer group, 17.1, 17.9, 4.19, 4.24
Subject Index
Q quaternions, Exercise 3.2 Exercise 26.8 quotient map, 1.34 quotient topology, 1.34
301
special unitary group, 3.8 split extension, 3.34 Splitting Theorem, 23.11 stabilizer, 10.9 subbasis, 1.1 subcategory, 15.1 subgroup, 3.1 R subgroup lattice, 3.34 radical, 35.5, 35.9, 35.10, 35.11 submodule of a Hilbert module, 14.22 Raikov completion, 8.44, 8.45 subspace, 1.1 rank (of free Abelian group), 6.27 suited neighborhood, 9.22 regular space, 1.14 representation as a topological transfor- support, 12.1 supremum norm, 12.1 mation group, 10.1, 10.3 right module, 3.2 T right uniformity, 8.2 right uniform structure, 8.2 topological dimension, 2.16 right vector space, 3.2 topological field, 3.2 ring, 3.2 topological group, 3.1 ring of formal power series, 26.3 topological G-module, 14.22 ring of p-adic integers, 17.2, topological Hilbert G-module, 14.22 Exercise 17.8 topological isomorphism (of groups), 3.32 S topological module, 3.2 topological prime field, 26.13 scalar product, 14.5 topological ring, 3.2 Schur’s Lemma, 35.5 topological semigroup, 3.2 second countable space, 1.3 semidirect product, 10.11, 10.12, 10.13 topological space, 1.1 topological transformation group, 10.1, semigroup, 3.2 10.3 semi-simple, 35.1 topological vector space, 3.2, 26.40 separable space, 1.3, 37.1 topology, 1.1 separation properties, 1.14 separation properties, for homogeneous topology generated by subbasis, 1.1 topology induced by a metric, 1.2 spaces 6.6 separation properties, for uniform spaces topology induced on subset, 1.1 topology of compact convergence, 9.20 8.7 topology of uniform convergence, 12.1 short exact sequence, 3.27 torsion-free group, 23.22, 23.24 σ -compact space, 1.18 torsion group, 23.22, 23.29, 23.30 small inductive dimension, 2.16, 36.7 totally bounded, 1.30 solvable group, 7.1 totally disconnected, 2.4 solvable radical, 35.9 special orthogonal group, 3.8 Tychonoff’s Theorem, 1.33
302
Subject Index
U ultrametric inequality, 26.30 uniform embedding, 8.28 uniformity, 8.1 uniformly continuous, 8.10 uniform space, 8.1 uniform structure, 8.1 uniform topology, 8.3 unimodular group, 13.6, 13.7, 13.8 unitary group, 14.5, 3.8 unitary representations, 14.21 unitary transformation, 14.5 universal property of cartesian product, 1.9 universal property of direct limits, 17.8 universal property of product topology, 1.12 universal property of products, 16.1 universal property of products of topological groups, 3.35 universal property of projective limits, 17.14
upper central series, 7.10 usual topology on R, Rn , 1.2 V valuation ideal, 26.30, 26.33 valuation ring, 26.30, 26.33 vector group, 24.1, 24.6, 25.5, 25.6 vector space, 3.2, 14.5, 14.7, 26.40, vector subgroup, 24.1, 24.4 W Wedderburn’s Theorem, 26.17 Weil completion, 8.44, 8.45 Weil’s Lemma, 6.26 weight, 37.1, 37.2 Weyl’s Trick, 14.35 Z Zorn’s Lemma, 4.21
