Locally Compact Groups

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 one 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. This textbook gives a systematic and detailed introduction to the highlights of that theory. In its second edition, it includes three new chapters, treating applications of totally disconnected groups, and more recent developments in the structure theory of totally disconnected locally compact groups.

In the beginning, a review of fundamental tools from topology and the elementary theory of topological groups and transformation groups is developed. 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. While the first edition of this book concluded with a chapter on Hilbert's Fifth Problem, this second edition contains three further chapters on aspects of totally disconnected locally compact groups. These focus on automorphism groups of trees, Galois groups of field extensions of infinite degree, and new techniques due to George Willis.

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.

For the first edition of this book, please click here.