On transitive algebras containing a standard finite von Neumann subalgebra

Abstract

Let M be a finite von Neumann algebra acting on a Hilbert space H and A be a transitive algebra containing M ′ . In this paper we prove that if A is 2-fold transitive, then A is strongly dense in B ( H ) . This implies that if a transitive algebra containing a standard finite von Neumann algebra (in the sense of [U. Haagerup, The standard form of von Neumann algebras, Math. Scand. 37 (1975) 271–283]) is 2-fold transitive, then A is strongly dense in B ( H ) . Non-selfadjoint algebras related to free products of finite von Neumann algebras, e.g., L F n and ( M 2 ( C ) , 1 2 Tr ) ∗ ( M 2 ( C ) , 1 2 Tr ) , are studied. Brown measures of certain operators in ( M 2 ( C ) , 1 2 Tr ) ∗ ( M 2 ( C ) , 1 2 Tr ) are explicitly computed.