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 $\AA$ be a transitive algebra containing $\M'$. In this paper we prove that if $\AA$ is 2-fold transitive, then $\AA$ is strongly dense in $\B(\H)$. This implies that if a transitive algebra containing a standard finite von Neumann algebra (in the sense of [Ha1]) is 2-fold transitive, then $\AA$ is strongly dense in $\B(\H)$. Non-selfadjoint algebras related to free products of finite von Neumann algebras, e.g., $\L{\mathbb{F}_n}$ and $(M_2(\cc), {1/2}Tr)*(M_2(\cc), {1/2}Tr)$, are studied. Brown measures of certain operators in $(M_2(\cc), {1/2}Tr)*(M_2(\cc), {1/2}Tr)$ are explicitly computed.