Locally Compact Groups (EMS Textbooks in Mathematics) by Markus Stroppel (PDF)

Description

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.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This is the only book I’ve come across that presents the Pontryagin duality theorem in the language of category theory. The Pontryagin duality theorem states that the double dual of a locally compact abelian group is naturally isomorphic to itself, and “naturally” has a precise meaning: there is a natural isomorphism from the identity functor to the double dual functor in the category of locally compact abelian groups.The book starts with topological spaces and proves the kind of point set topology results that are used when working with topological groups. Stroppel presents these results using the idea of a universal property whenever this is possible, such as for the product topology and quotient topology. One topic this book covers that is beyond the usual point set topology course is small inductive dimension. The chapter on topological groups proves the important open mapping theorem for topological groups, that any surjective continuous homomorphism from a locally compact sigma-compact topological group to a locally compact Hausdorff group is an open map. In the chapter on the Haar integral, we pay a brief visit to Hilbert modules.The chapter on categories of topological groups is good and can be read as an introduction to category theory. It includes the extremely important ideas of direct and projective limits. Stroppel defines precisely what a natural transformation is.Although this book uses heavy formalism, it is worked out carefully and a patient reader should be able to handle it without feeling confused and frustrated. I first opened this book when I was a master’s student and learned about natural transformations from it.

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