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More About This Textbook
Overview
Abstract theory remains an indispensable foundation for the study of concrete cases. It shows what the general picture should look like and provides results that are useful again and again. Despite this, however, there are few, if any introductory texts that present a unified picture of the general abstract theory.
A Course in Abstract Harmonic Analysis offers a concise, readable introduction to Fourier analysis on groups and unitary representation theory. After a brief review of the relevant parts of Banach algebra theory and spectral theory, the book proceeds to the basic facts about locally compact groups, Haar measure, and unitary representations, including the Gelfand-Raikov existence theorem. The author devotes two chapters to analysis on Abelian groups and compact groups, then explores induced representations, featuring the imprimitivity theorem and its applications. The book concludes with an informal discussion of some further aspects of the representation theory of non-compact, non-Abelian groups.
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Table of Contents
Banach Algebras and Spectral Theory Banach Algebras: Basic Concepts Gelfand Theory Nonunital Banach Algebras The Spectral Theorem Spectral Theory of *-Representations Notes and References Locally Compact Groups Topological Groups Haar Measure Interlude: Some Technicalities The Modular Function Convolutions Homogeneous Spaces Notes and References Basic Representation Theory Unitary Representations Representations of a Group and its Group Algebra Functions of Positive Type Notes and References Analysis on Locally Compact Abelian Groups The Dual Group The Fourier Transform The Pontrjagin Duality Theorem Representations of Locally Compact Abelian Groups Closed Ideals in L1(G)
Spectral Synthesis The Bohr Compactification Notes and References Analysis on Compact Groups Representations of Compact Groups The Peter-Weyl Theorem The Fourier Transform on Compact Groups Examples Notes and References Induced Representations The Inducing Construction The Frobenius Reciprocity Theorem Pseudomeasures and Induction in Stages Systems of Imprimitivity The Imprimitivity Theorem Introduction to the Mackey Machine Examples Notes and References Further Topics in Representation Theory The Group C* Algebra The Structure of the Dual Space Tensor Products Direct Integral Decompositions The Plancherel Theorem Examples Appendix 1. Hilbert Space Miscellany Appendix 2. Trace-Class and Hilbert-Schmidt Operators Appendix 3. Vector-Valued Integrals Bibliography Index