A Course in Abstract Harmonic Analysis  book cover
2nd Edition

A Course in Abstract Harmonic Analysis

ISBN 9781498727136
Published September 25, 2015 by Chapman & Hall
334 Pages 3 B/W Illustrations

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Book Description

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
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
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

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

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Gerald B. Folland received his Ph.D in mathematics from Princeton University, New Jersey, USA in 1971. After two years at the Courant Institute of Mathematical Sciences, New York, USA, he joined the faculty of the University of Washington, Seattle, USA, where he is now professor emeritus of mathematics. He has written a number of research and expository articles on harmonic analysis and its applications, and he is the author of eleven textbooks and research monographs.


Praise for the Previous Edition

"This delightful book fills a long-standing gap in the literature on abstract harmonic analysis. … To the reviewer's knowledge, no one existing book contains all of the topics that are treated in this one. … [The author's] respect for the subject shows on every hand…through 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