Locally Compact Groups (EMS Textbooks in Mathematics)

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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

vi

Preface

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

v

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 . . . . . . . . . . . . . .

26 26 43 50 53 66 72

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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

25 26 27

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

287

Index of Symbols

291

Subject Index

297

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

3

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 ) :=


5

( 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

6

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.


1.15 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}.

8

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 .


9

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

10

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 .


11

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.

12

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).


13

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.

14

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).


15

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.

16

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


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


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.


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.


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


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

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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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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


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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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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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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):

40


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 .


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


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


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


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


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


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 = 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


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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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.

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(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.

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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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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

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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

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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

6. Quotients

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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.

7. Solvable and Nilpotent Groups

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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

12. Existence and Uniqueness of Haar Integrals

115

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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D The Haar Integral

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

118

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 .


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

120

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.


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

122

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 .

13. The Module Function

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.

124

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


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.

126

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,

127


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

128

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.

14. Applications to Linear Representations

129

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

15. Categories

145

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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147

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


151

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.


153

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.

154


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


155

,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

156


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

158


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.


159

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.

160


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 FF 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.

161


(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.

162


−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


163

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 .

164


(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

18. Projective Limits of Topological Groups

165

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}.

166


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.

18. Projective Limits of Topological Groups

167

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 .

168


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 .

···

19. Compact Groups

169

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.

170


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.

19. Compact Groups

171

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.

172


(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.

19. Compact Groups

173

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 .

20. Characters and Character Groups

175

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

176

F Locally Compact Abelian Groups

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


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.

178


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


179

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 .

180


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∗ .

21. Compactly Generated Abelian Lie Groups

181

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 _

∗

182


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


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.

184


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.


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.

186


Let κ : Ra+b → L0 = Ra × Tb × {(0, 0)} be the natural map. /A | | π || | | _

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