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Crossed Products of von Neumann Algebras by Equivalence Relations and Their Subalgebras
Igor Fulman, University of Iowa, Iowa City, IA

Memoirs of the American Mathematical Society
1997; 107 pp; softcover
Volume: 126
ISBN-10: 0-8218-0557-6
ISBN-13: 978-0-8218-0557-2
List Price: US$44
Individual Members: US$26.40
Institutional Members: US$35.20
Order Code: MEMO/126/602
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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.

Table of Contents

  • Introduction
  • Preliminaries
  • Unitary realization of \(\alpha _{(y,x)}\)
  • Construction of \(\tilde M^\nabla\)
  • Coordinate representation of elements of \(\tilde M\)
  • The expectation \(E\)
  • Coordinates in \(\tilde M^\nabla\)
  • The expectation \(E'\)
  • Tomita-Takesaki theory for \(\tilde M\) and \(\tilde M^\nabla\)
  • \(I(M)\) -automorphisms of \(\tilde M\)
  • Flows of automorphisms
  • The Feldman-Moore-type structure theorem
  • Isomorphisms of crossed products
  • Bimodules and subalgebras of \(\tilde M\)
  • Spectral theorem for bimodules
  • Analytic algebra of a flow of automorphisms
  • Properties of \(\tilde M\)
  • Hyperfiniteness and dilations
  • The construction of Yamanouchi
  • Examples and particular cases
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