The notion of amenability has its origins in the beginnings of modern measure theory: Does a finitely additive set function exist which is invariant under a certain group action? Since the 1940s, amenability has become an important concept in abstract harmonic analysis (or rather, more generally, in the theory of semitopological semigroups). In 1972, B.E. Johnson showed that the amenability of a locally compact group G can be characterized in terms of the Hochschild cohomology of its group algebra L^1(G): this initiated the theory of amenable Banach algebras. Since then, amenability has penetrated other branches of mathematics, such as von Neumann algebras, operator spaces, and even differential geometry. Lectures on Amenability introduces second year graduate students to this fascinating area of modern mathematics and leads them to a level from where they can go on to read original papers on the subject. Numerous exercises are interspersed in the text.

The notion of amenability has its origins in the beginnings of modern measure theory: Does a finitely additive set function exist which is invariant under a certain group action? Since the 1940s, amenability has become an important concept in abstract harmonic analysis (or rather, more generally, in the theory of semitopological semigroups). In 1972, B.E. Johnson showed that the amenability of a locally compact group G can be characterized in terms of the Hochschild cohomology of its group algebra L^1(G): this initiated the theory of amenable Banach algebras. Since then, amenability has penetrated other branches of mathematics, such as von Neumann algebras, operator spaces, and even differential geometry. Lectures on Amenability introduces second year graduate students to this fascinating area of modern mathematics and leads them to a level from where they can go on to read original papers on the subject. Numerous exercises are interspersed in the text.

The main topic of the book is amenable groups, i.e., groups on which there exist invariant finitely additive measures. It was discovered that the existence or non-existence of amenability is responsible for many interesting phenomena such as, e.g., the Banach-Tarski Paradox about breaking a sphere into two spheres of the same radius. Since then, amenability has been actively studied and a number of different approaches resulted in many examples of amenable and non-amenable groups. In the book, the author puts together main approaches to study amenability. A novel feature of the book is that the exposition of the material starts with examples which introduce a method rather than illustrating it. This allows the reader to quickly move on to meaningful material without learning and remembering a lot of additional definitions and preparatory results; those are presented after analyzing the main examples. The techniques that are used for proving amenability in this book are mainly a combination of analytic and probabilistic tools with geometric group theory.

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The subject of amenability has its roots in the work of Lebesgue at the turn of the century. In the 1940s, the subject began to shift from finitely additive measures to means. This shift is of fundamental importance, for it makes the substantial resources of functional analysis and abstract harmonic analysis available to the study of amenability. The ubiquity of amenability ideas and the depth of the mathematics involved points to the fundamental importance of the subject. This book presents a comprehensive and coherent account of amenability as it has been developed in the large and varied literature during this century. The book has a broad appeal, for it presents an account of the subject based on harmonic and functional analysis. In addition, the analytic techniques should be of considerable interest to analysts in all areas. In addition, the book contains applications of amenability to a number of areas: combinatorial group theory, semigroup theory, statistics, differential geometry, Lie groups, ergodic theory, cohomology, and operator algebras. The main objectives of the book are to provide an introduction to the subject as a whole and to go into many of its topics in some depth. The book begins with an informal, nontechnical account of amenability from its origins in the work of Lebesgue. The initial chapters establish the basic theory of amenability and provide a detailed treatment of invariant, finitely additive measures (i.e., invariant means) on locally compact groups. The author then discusses amenability for Lie groups, "almost invariant" properties of certain subsets of an amenable group, amenability and ergodic theorems, polynomial growth, and invariant mean cardinalities. Also included are detailed discussions of the two most important achievements in amenability in the 1980s: the solutions to von Neumann's conjecture and the Banach-Ruziewicz Problem. The main prerequisites for this book are a sound understanding of undergraduate-level mathematics and a knowledge of abstract harmonic analysis and functional analysis. The book is suitable for use in graduate courses, and the lists of problems in each chapter may be useful as student exercises.

This volume provides readers with a detailed introduction to the amenability of Banach algebras and locally compact groups. By encompassing important foundational material, contemporary research, and recent advancements, this monograph offers a state-of-the-art reference. It will appeal to anyone interested in questions of amenability, including those familiar with the author’s previous volume Lectures on Amenability. Cornerstone topics are covered first: namely, the theory of amenability, its historical context, and key properties of amenable groups. This introduction leads to the amenability of Banach algebras, which is the main focus of the book. Dual Banach algebras are given an in-depth exploration, as are Banach spaces, Banach homological algebra, and more. By covering amenability’s many applications, the author offers a simultaneously expansive and detailed treatment. Additionally, there are numerous exercises and notes at the end of every chapter that further elaborate on the chapter’s contents. Because it covers both the basics and cutting edge research, Amenable Banach Algebras will be indispensable to both graduate students and researchers working in functional analysis, harmonic analysis, topological groups, and Banach algebras. Instructors seeking to design an advanced course around this subject will appreciate the student-friendly elements; a prerequisite of functional analysis, abstract harmonic analysis, and Banach algebra theory is assumed.

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