Von Neumann Algebras of Equivalence Relations with Nontrivial One-Cohomology

Abstract

Using Popa's deformation/rigidity theory, we investigate prime decompositions of von Neumann algebras of the form $L(\mathcal{R})$ for countable probability measure preserving equivalence relations $\mathcal{R}$. We show that $L(\mathcal{R})$ is prime whenever $\mathcal{R}$ 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 $\textit{Gaussian extension }\tilde{\mathcal{R}}$ of $\mathcal{R}$ and subsequently an s-malleable deformation of the inclusion $L(\mathcal{R}) \subset L(\tilde{\mathcal{R}})$. 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 $L(\mathcal{R}_1) \otimes L(\mathcal{R}_2) \otimes \cdots \otimes L(\mathcal{R}_k)$. As a corollary, we get a unique factorization result in the equivalence relation setting for products of the form $\mathcal{R}_1 \times \mathcal{R}_2 \times \cdots \times \mathcal{R}_k$. We finish with an application to the measure equivalence of groups.