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 ( R ) for countable probability measure preserving equivalence relations R . We show that L ( R ) is prime whenever 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 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 ( R 1 ) ⊗ ‾ L ( R 2 ) ⊗ ‾ ⋯ ⊗ ‾ L ( R k ) . As a corollary, we get a unique factorization result in the equivalence relation setting for products of the form R 1 × R 2 × ⋯ × R k . We finish with an application to the measure equivalence of groups.