**Mathematical Surveys and Monographs**

Volume: 160;
2010;
237 pp;
Hardcover

MSC: Primary 37; 03;

Print ISBN: 978-0-8218-4894-4

Product Code: SURV/160

List Price: $81.00

AMS Member Price: $64.80

MAA Member Price: $72.90

**Electronic ISBN: 978-1-4704-1387-3
Product Code: SURV/160.E**

List Price: $81.00

AMS Member Price: $64.80

MAA Member Price: $72.90

#### Supplemental Materials

# Global Aspects of Ergodic Group Actions

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*Alexander S. Kechris*

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 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.

#### Readership

Graduate students and research mathematicians interested in ergodic theory and descriptive set theory.

#### Reviews & Endorsements

This monograph is written with a great exposition as well as with precise proofs and references. We would like to emphasize that the author has undertaken the tremendous job of collecting and systematizing the material from a huge number of different sources. The choice of material (both classical and new) is excellent. Furthermore, this seems to be the only book that discusses in such detail the topological structure of group actions and their subsets. This monograph could undoubtedly serve as a starting point for any researcher in the field.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Global Aspects of Ergodic Group Actions

- Contents vii8 free
- Introduction ix10 free
- Chapter I. Measure preserving automorphisms 114 free
- Chapter II. The space of actions 6174
- Chapter III. Cocycles and cohomology 125138
- Appendix A. Realifications and complexifications 187200
- Appendix B. Tensor products of Hilbert spaces 189202
- Appendix C. Gaussian probability spaces 191204
- Appendix D. Wiener chaos decomposition 193206
- Appendix E. Extending representations to actions 197210
- Appendix F. Unitary representations of abelian groups 199212
- Appendix G. Induced representations and actions 203216
- Appendix H. The space of unitary representations 207220
- Appendix I. Semidirect products of groups 223236
- Bibliography 227240
- Index 233246 free