Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Ergodic equivalence relations, cohomology, and von Neumann algebras. II


Authors: Jacob Feldman and Calvin C. Moore
Journal: Trans. Amer. Math. Soc. 234 (1977), 325-359
MSC: Primary 22D40; Secondary 28A65, 46L10
MathSciNet review: 0578730
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let R be a Borel equivalence relation with countable equivalence classes, on the standard Borel space $ (X,\mathcal{A},\mu )$. Let $ \sigma $ be a 2-cohomology class on R with values in the torus $ \mathbb{T}$. We construct a factor von Neumann algebra $ {\mathbf{M}}(R,\sigma )$, generalizing the group-measure space construction of Murray and von Neumann [1] and previous generalizations by W. Krieger [1] and G. Zeller-Meier [1].

Very roughly, $ {\mathbf{M}}(R,\sigma )$ is a sort of twisted matrix algebra whose elements are matrices $ ({a_{x,y}})$, where $ (x,y) \in R$. The main result, Theorem 1, is the axiomatization of such factors; any factor M with a regular MASA subalgebra A, and possessing a conditional expectation from M onto A, and isomorphic to $ {\mathbf{M}}(R,\sigma )$ in such a manner that A becomes the ``diagonal matrices"; $ (R,\sigma )$ is uniquely determined by M and A. A number of results are proved, linking invariants and automorphisms of the system (M, A) with those of $ (R,\sigma )$. These generalize results of Singer [1] and of Connes [1]. Finally, several results are given which contain fragmentary information about what happens with a single M but two different subalgebras $ {{\mathbf{A}}_1},{{\mathbf{A}}_2}$.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 22D40, 28A65, 46L10

Retrieve articles in all journals with MSC: 22D40, 28A65, 46L10


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1977-0578730-2
PII: S 0002-9947(1977)0578730-2
Article copyright: © Copyright 1977 American Mathematical Society