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( R) for countable probability measure preserving (pmp) equivalence relations R . We show that L(R) is prime whenever 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 Gaussian extension R of R } and subsequently an s-malleable deformation of the inclusion L(R) ⊂ L( 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(R1)⊗ L(R2) ⊗ ··· ⊗ L(Rk). As a corollary, we get a unique factorization result in the equivalence relation setting for products of the form R1 x R 2 x ··· x R k .;We then study extensions of pmp equivalence relations R following the joint work [BHI15] with Lewis Bowen and Adrian Ioana. By extending the techniques of Gaboriau and Lyons [GL07], we prove that if R is non-amenable and ergodic, it has an extension R˜ which contains the orbits of a free ergodic pmp action of the free group F2 . This allows us to prove that any such 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).