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Some Structural Results for Measured Equivalence Relations and Their Associated von Neumann Algebras
 Hoff, Daniel Joseph
 Advisor(s): Ioana, Adrian
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 (pmp) equivalence relations $\mathcal{R}$. We show that $L(\mathcal{R})$ is prime whenever $\mathcal{R}$ is nonamenable, ergodic, and admits an unbounded 1cocycle 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 smalleable 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 nonamenable 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 nonamenable 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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