A Course in Abstract Harmonic Analysis: (Textbooks in Mathematics)

A concise introduction to Fourier analysis on groups and unitary representation theory. The emphasis of the book is on important ideas, rather than on topics of only technical or specialized interest. After a brief review of the relevant parts of Banach algebra theory and spectral theory, the book covers the basic facts about locally compact groups, and Haar measure and unitary representations - including the Gelfand-Raikov existence among them. It is intended for mathematicians.

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

Review :
"This delightful book fills a long-standing gap in the literature on abstract harmonic analysisTo the reviewer's knowledge, no one existing book contains al of the topics that are treated in this one[The author's] respect for the subject shows on every handthrough his careful writing style, which is concise, yet simple and elegant. The reviewer would encourage anyone with an interest in harmonic analysis to have this book in his or her personal library. ..a fine book that the reviewer would have been proud to write." -Robert S. Doran in Mathematical Reviews, Issue 98c