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.
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