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Crossed Products of von Neumann Algebras by Equivalence Relations and Their Subalgebras

Igor Fulman University of Iowa, Iowa City, IA
Available Formats:
Electronic ISBN: 978-1-4704-0187-0
Product Code: MEMO/126/602.E
List Price: $47.00 MAA Member Price:$42.30
AMS Member Price: $28.20 Click above image for expanded view Crossed Products of von Neumann Algebras by Equivalence Relations and Their Subalgebras Igor Fulman University of Iowa, Iowa City, IA Available Formats:  Electronic ISBN: 978-1-4704-0187-0 Product Code: MEMO/126/602.E  List Price:$47.00 MAA Member Price: $42.30 AMS Member Price:$28.20
• Book Details

Memoirs of the American Mathematical Society
Volume: 1261997; 107 pp
MSC: 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.

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. Tomita-Takesaki theory for $\tilde {M}$ and $\tilde {M}^\nabla$
• 10. $I(M)$-automorphisms of $\tilde {M}$
• 11. Flows of automorphisms
• 12. The Feldman-Moore-type 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
• Requests

Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 1261997; 107 pp
MSC: 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.

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. Tomita-Takesaki theory for $\tilde {M}$ and $\tilde {M}^\nabla$
• 10. $I(M)$-automorphisms of $\tilde {M}$
• 11. Flows of automorphisms
• 12. The Feldman-Moore-type 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
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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