A course in abstract harmonic analysis. 2nd updated edition.

*(English)*Zbl 1342.43001
Textbooks in Mathematics. Boca Raton, FL: CRC Press (ISBN 978-1-4987-2713-6/hbk; 978-1-4987-2715-0/ebook). xiii, 305 p. (2016).

In the book under review, “harmonic analysis” is understood as those parts of analysis in which the action of a locally compact group plays an essential role: more specifically, the theory of unitary representations of locally compact groups, and the analysis of functions on such groups and their homogeneous spaces.

Written for readers with some knowledge of real analysis and basic functional analysis, the text offers a thorough treatment of classical results that constitute the core of the theory for locally compact groups. The presentation is particularly clear, making the book accessible and appropriate for an introduction to the topic. The more technical parts of the theory are abundantly illustrated by thoughtfully chosen and carefully treated examples.

In addition, each chapter contains a short section titled Notes and references, that offers a panorama of further topics, including some in Lie theory.

Here is an overview of the content of each chapter.

The first chapter contains background material on Banach and C*-algebras, as well as spectral theory.

Chapter 2 presents the basics of analysis on locally compact groups and their homogeneous spaces, including Haar measures, convolutions and quasi-invariant measures.

The third chapter is devoted to generalities about (unitary) representations of locally compact groups and their convolution algebras, including the Gelfand-Raikov Theorem and a discussion of certain properties in terms of functions of positive type.

Fourier analysis is discussed in the cases of abelian and compact groups, in Chapters 4 and 5 respectively, describing how the classical Fourier transform on \(\mathbb{R}\) generalizes in the representation-theoretic context for these types of groups.

Chapter 6 presents the theory of induced representations. The precise and accessible treatment of this somewhat technical yet fundamental topic is one of the highlights of the book. The notion of imprimitivity is introduced, as well as elements of the Mackey machine.

The final chapter of the book surveys various aspects of the representation theory of non-abelian non-compact groups. They include group C*-algebras, which allow to consider the dual space of a group as a non-commutative space in the sense of A. Connes, as well as tensor products of representations, direct integral decompositions and the Plancherel theorem. Most proofs are omitted from the treatment of these more advanced topics. Nevertheless the important results are stated with precision and illustrated by concrete examples. In particular, the dual spaces of Heisenberg groups, \(ax+b\) group and \(\mathrm{SL}(2,\mathbb{R})\) are described.

Four appendices contain background material about Hilbert spaces and their tensor products, trace-class and Hilbert-Schmidt operators and vector-valued integrals.

The author states in the introduction how beautiful he deems the subject matter. Thanks to his insightful exposition, his elegant style and the delicate balance between theory and examples, this book is very likely to convey this impression to the readers and efficiently prepare them to move on towards current developments in the harmonic analysis on groups.

For a review of the first edition, see [Zbl 0857.43001].

Written for readers with some knowledge of real analysis and basic functional analysis, the text offers a thorough treatment of classical results that constitute the core of the theory for locally compact groups. The presentation is particularly clear, making the book accessible and appropriate for an introduction to the topic. The more technical parts of the theory are abundantly illustrated by thoughtfully chosen and carefully treated examples.

In addition, each chapter contains a short section titled Notes and references, that offers a panorama of further topics, including some in Lie theory.

Here is an overview of the content of each chapter.

The first chapter contains background material on Banach and C*-algebras, as well as spectral theory.

Chapter 2 presents the basics of analysis on locally compact groups and their homogeneous spaces, including Haar measures, convolutions and quasi-invariant measures.

The third chapter is devoted to generalities about (unitary) representations of locally compact groups and their convolution algebras, including the Gelfand-Raikov Theorem and a discussion of certain properties in terms of functions of positive type.

Fourier analysis is discussed in the cases of abelian and compact groups, in Chapters 4 and 5 respectively, describing how the classical Fourier transform on \(\mathbb{R}\) generalizes in the representation-theoretic context for these types of groups.

Chapter 6 presents the theory of induced representations. The precise and accessible treatment of this somewhat technical yet fundamental topic is one of the highlights of the book. The notion of imprimitivity is introduced, as well as elements of the Mackey machine.

The final chapter of the book surveys various aspects of the representation theory of non-abelian non-compact groups. They include group C*-algebras, which allow to consider the dual space of a group as a non-commutative space in the sense of A. Connes, as well as tensor products of representations, direct integral decompositions and the Plancherel theorem. Most proofs are omitted from the treatment of these more advanced topics. Nevertheless the important results are stated with precision and illustrated by concrete examples. In particular, the dual spaces of Heisenberg groups, \(ax+b\) group and \(\mathrm{SL}(2,\mathbb{R})\) are described.

Four appendices contain background material about Hilbert spaces and their tensor products, trace-class and Hilbert-Schmidt operators and vector-valued integrals.

The author states in the introduction how beautiful he deems the subject matter. Thanks to his insightful exposition, his elegant style and the delicate balance between theory and examples, this book is very likely to convey this impression to the readers and efficiently prepare them to move on towards current developments in the harmonic analysis on groups.

For a review of the first edition, see [Zbl 0857.43001].

Reviewer: Pierre Clare (Hanover)

##### MSC:

43-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to abstract harmonic analysis |