A Course in Abstract Harmonic Analysis

by Gerald B. Folland

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.

Product Details

ISBN-13:	9780849384905
Publisher:	Taylor & Francis
Publication date:	09/05/2000
Series:	Textbooks in Mathematics Series , #18
Edition description:	New Edition
Pages:	288
Product dimensions:	6.25(w) x 9.25(h) x 0.75(d)

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