A Course in Abstract Harmonic Analysis

✍ Scribed by Gerald B. Folland

• Publisher: Chapman and Hall/CRC
• Year: 2015
• Tongue: English
• Leaves: 318
• Series: Textbooks in Mathematics
• Edition: 2
• Category: Library

✦ Synopsis

A Course in Abstract Harmonic Analysis is an introduction to that part of analysis on locally compact groups that can be done with minimal assumptions on the nature of the group. As a generalization of classical Fourier analysis, this abstract theory creates a foundation for a great deal of modern analysis, and it contains a number of elegant results and techniques that are of interest in their own right.

This book develops the abstract theory along with a well-chosen selection of concrete examples that exemplify the results and show the breadth of their applicability. After a preliminary chapter containing the necessary background material on Banach algebras and spectral theory, the text sets out the general theory of locally compact groups and their unitary representations, followed by a development of the more specific theory of analysis on Abelian groups and compact groups. There is an extensive chapter on the theory of induced representations and its applications, and the book concludes with a more informal exposition on the theory of representations of non-Abelian, non-compact groups.

Featuring extensive updates and new examples, the Second Edition:

• Adds a short section on von Neumann algebras
• Includes Mark Kac’s simple proof of a restricted form of Wiener’s theorem
• Explains the relation between SU(2) and SO(3) in terms of quaternions, an elegant method that brings SO(4) into the picture with little effort
• Discusses representations of the discrete Heisenberg group and its central quotients, illustrating the Mackey machine for regular semi-direct products and the pathological phenomena for nonregular ones

A Course in Abstract Harmonic Analysis, Second Edition serves as an entrée to advanced mathematics, presenting the essentials of harmonic analysis on locally compact groups in a concise and accessible form.

✦ Table of Contents

Banach Algebras and Spectral Theory
Banach Algebras: Basic Concepts
Gelfand Theory
Nonunital Banach Algebras
The Spectral Theorem
Spectral Theory of ∗-Representations
Von Neumann Algebras
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
Fourier Analysis 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: The Classics
More Examples, Good and Bad
Notes and References

Further Topics in Representation Theory
The Group C* Algebra
The Structure of the Dual Space
Tensor Products of Representations
Direct Integral Decompositions
The Plancherel Theorem
Examples

Appendices
A Hilbert Space Miscellany
Trace-Class and Hilbert-Schmidt Operators
Tensor Products of Hilbert Spaces
Vector-Valued Integrals