Electronic ISBN:  9781470401870 
Product Code:  MEMO/126/602.E 
107 pp 
List Price:  $47.00 
MAA Member Price:  $42.30 
AMS Member Price:  $28.20 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 126; 1997MSC: Primary 47; 46;
In this book, the author introduces and studies the construction of the crossed product of a von Neumann algebra \(M = \int _X M(x)d\mu (x)\) by an equivalence relation on \(X\) with countable cosets. This construction is the generalization of the construction of the crossed product of an abelian von Neumann algebra by an equivalence relation introduced by J. Feldman and C. C. Moore. Many properties of this construction are proved in the general case. In addition, the generalizations of the Spectral Theorem on Bimodules and of the theorem on dilations are proved.
ReadershipGraduate students and research mathematicians interested in operator algebras.

Table of Contents

Chapters

1. Introduction

2. Preliminaries

3. Unitary realization of $\alpha _{(y,x)}$

4. Construction of $\tilde {M}^\nabla $

5. Coordinate representation of elements of $M$

6. The expectation $E$

7. Coordinates in $\tilde {M}^\nabla $

8. The expectation $E’$

9. TomitaTakesaki theory for $\tilde {M}$ and $\tilde {M}^\nabla $

10. $I(M)$automorphisms of $\tilde {M}$

11. Flows of automorphisms

12. The FeldmanMooretype structure theorem

13. Isomorphisms of crossed products

14. Bimodules and subalgebras of $\tilde {M}$

15. Spectral theorem for bimodules

16. Analytic algebra of a flow of automorphisms

17. Properties of $\tilde {M}$

18. Hyperfiniteness and dilations

19. The construction of Yamanouchi

20. Examples and particular cases


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In this book, the author introduces and studies the construction of the crossed product of a von Neumann algebra \(M = \int _X M(x)d\mu (x)\) by an equivalence relation on \(X\) with countable cosets. This construction is the generalization of the construction of the crossed product of an abelian von Neumann algebra by an equivalence relation introduced by J. Feldman and C. C. Moore. Many properties of this construction are proved in the general case. In addition, the generalizations of the Spectral Theorem on Bimodules and of the theorem on dilations are proved.
Graduate students and research mathematicians interested in operator algebras.

Chapters

1. Introduction

2. Preliminaries

3. Unitary realization of $\alpha _{(y,x)}$

4. Construction of $\tilde {M}^\nabla $

5. Coordinate representation of elements of $M$

6. The expectation $E$

7. Coordinates in $\tilde {M}^\nabla $

8. The expectation $E’$

9. TomitaTakesaki theory for $\tilde {M}$ and $\tilde {M}^\nabla $

10. $I(M)$automorphisms of $\tilde {M}$

11. Flows of automorphisms

12. The FeldmanMooretype structure theorem

13. Isomorphisms of crossed products

14. Bimodules and subalgebras of $\tilde {M}$

15. Spectral theorem for bimodules

16. Analytic algebra of a flow of automorphisms

17. Properties of $\tilde {M}$

18. Hyperfiniteness and dilations

19. The construction of Yamanouchi

20. Examples and particular cases