On Relative Biexactness of Amalgamated Free Product von Neumann Algebras

Abstract

Given weakly exact tracial von Neumann algebras $M_1,M_2$ with a common injective amalgam $B$, we prove that the amalgamated free product $M_1{\overline{\ast }}_B{M}_2$ is biexact relative to $\{M_1,M_2\}$. In the case where $M_1$ and $M_2$ are injective, we further show that $M_1{\overline{\ast }}_B{M}_2$ is biexact relative to the amalgam $B$, and if $B$ is mixing in each of $M_1$ and $M_2$, $M_1{\overline{\ast }}_B{M}_2$ itself is biexact. As applications, we derive structural decomposition results and subalgebra absorption theorems for amalgamated free product von Neumann algebras, extending those previously known in the group case.

30 pages; added new decomposition results and a section of upgrading theorems