Global aspects of ergodic group actions / Alexander S. Kechris.

Format:

Book

Author/Creator:

Kechris, A. S., 1946- author.

Series:

Mathematical surveys and monographs ; no. 160.

Mathematical surveys and monographs ; volume 160

Language:

English

Subjects (All):

Measure-preserving transformations.

Ergodic theory.

Automorphisms.

Physical Description:

1 online resource (258 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2010]

Language Note:

English

Summary:

The subject of this book is the study of ergodic, measure preserving actions of countable discrete groups on standard probability spaces. It explores a direction that emphasizes a global point of view, concentrating on the structure of the space of measure preserving actions of a given group and its associated cocycle spaces. These are equipped with canonical topological actions that give rise to the usual concepts of conjugacy of actions and cohomology of cocycles. Structural properties of discrete groups such as amenability, Kazhdan's property (T) and the Haagerup Approximation Property play a significant role in this theory as they have important connections to the global structure of these spaces. One of the main topics discussed in this book is the analysis of the complexity of the classification problems of conjugacy and orbit equivalence of actions, as well as of cohomology of cocycles. This involves ideas from topological dynamics, descriptive set theory, harmonic analysis, and the theory of unitary group representations. Also included in this title is a study of properties of the automorphism group of a standard probability space and some of its important subgroups, such as the full and automorphism groups of measure preserving equivalence relations and connections with the theory of costs. The book contains nine appendices that present necessary background material in functional analysis, measure theory, and group representations, thus making the book accessible to a wider audience.

Contents:

""Contents""; ""Introduction""; ""Chapter I. Measure preserving automorphisms""; ""Chapter II. The space of actions""; ""Chapter III. Cocycles and cohomology""; ""Appendix A. Realifications and complexifications""; ""Appendix B. Tensor products of Hilbert spaces""; ""Appendix C. Gaussian probability spaces""; ""Appendix D. Wiener chaos decomposition""; ""Appendix E. Extending representations to actions""; ""Appendix F. Unitary representations of abelian groups""; ""Appendix G. Induced representations and actions""; ""Appendix H. The space of unitary representations""

""Appendix I. Semidirect products of groups""""Bibliography""; ""Index""

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

Description based on print version record.

ISBN:

1-4704-1387-6