On transitive algebras containing a standard finite von Neumann subalgebra

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., LFn and (M2(C), 12 Tr) ∗ (M2(C), 12 Tr), are studied. Brown measures of certain operators in (M2(C), 12 Tr) ∗ (M2(C), 12 Tr) are explicitly computed. © 2007 Elsevier Inc. All rights reserved