On maximal injective subalgebras of tensor products of von Neumann algebras

Let M_i be a von Neumann algebra, and B_i be a maximal injective von Neumann subalgebra of M_i, i=1,2. If M_1 has separable predual and the center of B_1 is atomic, e.g., B_1 is a factor, then B_1\tensor B_2 is a maximal injective von Neuamnn subalgebra of M_1\tensor M_2. This partly answers a question of PopaComment: 15 pages, a simple proof of the main theorem (Theorem 4.1 in the new version) suggested by referee is included in the new version. Typos are correcte

On spectra and Brown's spectral measures of elements in free products of matrix algebras

We compute spectra and Brown measures of some non self-adjoint operators in (M_2(\cc), {1/2}Tr)*(M_2(\cc), {1/2}Tr), the reduced free product von Neumann algebra of M_2(\cc) with M_2(\cc). Examples include $AB$ and $A+B$, where A and B are matrices in (M_2(\cc), {1/2}Tr)*1 and 1*(M_2(\cc), {1/2}Tr), respectively. We prove that AB is an R-diagonal operator (in the sense of Nica and Speicher \cite{N-S1}) if and only if Tr(A)=Tr(B)=0. We show that if X=AB or X=A+B and A,B are not scalar matrices, then the Brown measure of X is not concentrated on a single point. By a theorem of Haagerup and Schultz \cite{H-S1}, we obtain that if X=AB or X=A+B and $X\neq \lambda 1$, then X has a nontrivial hyperinvariant subspace affiliated with (M_2(\cc), {1/2}Tr)*(M_2(\cc), {1/2}Tr).Comment: final version. to appear on Math. Sca

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.Comment: 24 pages, to appear on Journal of Functional Analysi