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Some Structural Results for Measured Equivalence Relations and Their Associated von Neumann Algebras


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 (pmp) equivalence relations $\mathcal{R}$. We show that $L(\mathcal{R})$ is prime whenever $\mathcal{R}$ is non-amenable, ergodic, and admits an unbounded 1-cocycle into a mixing orthogonal representation weakly contained in the regular representation. This is accomplished by constructing the \emph{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 then study extensions of pmp equivalence relations $\mathcal{R}$ following the joint work \cite{BHI15} with Lewis Bowen and Adrian Ioana. By extending the techniques of Gaboriau and Lyons \cite{GL07}, we prove that if $\mathcal{R}$ is non-amenable and ergodic, it has an extension $\tilde{\mathcal{R}}$ which contains the orbits of a free ergodic pmp action of the free group $\mathbb F_2$. This allows us to prove that any such $\mathcal{R}$ admits uncountably many ergodic extensions which are pairwise not (stably) von Neumann equivalent. We further deduce that any non-amenable unimodular locally compact second countable group admits uncountably many free ergodic pmp actions which are pairwise not von Neumann equivalent (hence, pairwise not orbit equivalent).

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