Von Neumann algebras of equivalence relations with nontrivial one-cohomology

Using Popa's deformation/rigidity theory, we investigate prime decompositions of von Neumann algebras of the form L(R)L(R) for countable probability measure preserving equivalence relations RR. We show that L(R)L(R) is prime whenever RR 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˜of  RR 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×R2×⋯×RkR1×R2×⋯×Rk. We finish with an application to the measure equivalence of groups.