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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
DOI: https://doi.org/10.1090/S0002-9947-1977-0578730-2
MathSciNet review: 0578730
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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}$.


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DOI: https://doi.org/10.1090/S0002-9947-1977-0578730-2
Article copyright: © Copyright 1977 American Mathematical Society

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