Locally compact groups play an important role in many areas of mathematics as well as in physics. The class of locally compact groups admits a strong structure theory, which allows to reduce many problems to groups constructed in various ways from the additive group of real numbers, the classical linear groups and from finite groups. The book gives a systematic and detailed introduction to the highlights of that theory. In the beginning, a review of fundamental tools from topology and the elementary theory of topological groups and transformation groups is presented. Completions, Haar integral, applications to linear representations culminating in the Peter-Weyl Theorem are treated. Pontryagin duality for locally compact Abelian groups forms a central topic of the book. Applications are given, including results about the structure of locally compact Abelian groups, and a structure theory for locally compact rings leading to the classification of locally compact fields. Topological semigroups are discussed in a separate chapter, with special attention to their relations to groups. The last chapter reviews results related to Hilbert's Fifth Problem, with the focus on structural results for non-Abelian connected locally compact groups that can be derived using approximation by Lie groups. The book is self-contained and is addressed to advanced undergraduate or graduate students in mathematics or physics. It can be used for one-semester courses on topological groups, on locally compact Abelian groups, or on topological algebra. Suggestions on course design are given in the preface. Each chapter is accompanied by a set of exercises that have been tested in classes.
Author(s): Markus Stroppel
Publisher: European Mathematical Society
Year: 2006
Language: English
Pages: 313
Preface......Page 6
Maps and Topologies......Page 12
Connectedness and Topological Dimension......Page 30
Basic Definitions and Results......Page 37
Subgroups......Page 54
Linear Groups over Topological Rings......Page 61
Quotients......Page 63
Solvable and Nilpotent Groups......Page 77
Completion......Page 83
The Compact-Open Topology......Page 102
Transformation Groups......Page 110
Sets, Groups, and Rings of Homomorphisms......Page 116
Existence and Uniqueness of Haar Integrals......Page 124
The Module Function......Page 134
Applications to Linear Representations......Page 139
Categories......Page 154
Products in Categories of Topological Groups......Page 158
Direct Limits and Projective Limits......Page 167
Projective Limits of Topological Groups......Page 176
Compact Groups......Page 180
Characters and Character Groups......Page 185
Compactly Generated Abelian Lie Groups......Page 192
Pontryagin's Duality Theorem......Page 202
Applications of the Duality Theorem......Page 205
Maximal Compact Subgroups and Vector Subgroups......Page 214
Automorphism Groups of Locally Compact Abelian Groups......Page 218
Locally Compact Rings and Fields......Page 223
Homogeneous Locally Compact Groups......Page 241
Topological Semigroups......Page 253
Embedding Cancellative Directed Semigroups into Groups......Page 261
Compact Semigroups......Page 265
Groups with Continuous Multiplication......Page 270
The Approximation Theorem......Page 272
Dimension of Locally Compact Groups......Page 275
The Rough Structure......Page 279
Notions of Simplicity......Page 283
Compact Groups......Page 287
Countable Bases, Metrizability......Page 290
Non-Lie Groups of Finite Dimension......Page 291
Arcwise Connected Subgroups......Page 292
Algebraic Groups......Page 296
Bibliography......Page 298
Index of Symbols......Page 302
Subject Index......Page 308
