Von Neumann algebras of equivalence relations with nontrivial one-cohomology

摘要

Using Popa's deformation/rigidity theory, we investigate prime decompositions of von Neumann algebras of the form L(R) for countable probability measure preserving equivalence relations R. We show that L(R) is prime whenever R is nonamenable, ergodic, and admits an unbounded 1-cocycle into a mixing orthogonal representation weakly contained in the regular representation. This is accomplished by constructing the Gaussian extension (R) over tilde of R. and subsequently an s-malleable deformation of the inclusion L(72,) subset of L((R) over tilde). We go on to note a general obstruction to unique prime factorization, and avoiding it, we prove a unique prime factorization result for products of the form L(R-1) (circle times) over bar L(R-2) (circle times) over bar ... (circle times) over barL(R-k). As a corollary, we get a unique factorization result in the equivalence relation setting for products of the form R-1 x R-2 x ... x R-k. We finish with an application to the measure equivalence of groups. (C) 2015 Elsevier Inc. All rights reserved.