On Relative Biexactness of Amalgamated Free Product von Neumann Algebras

Given weakly exact tracial von Neumann algebras $M_1,M_2$ with a common injective amalgam $B$, we prove that the amalgamated free product $M_1{\bar{\ast }}_B{M}_2$ is biexact relative to $\{M_1,M_2\}$. In the case where $M_1$ and $M_2$ are injective, we further show that $M_1{\bar{\ast }}_B{M}_2$ is biexact relative to the amalgam $B$, and if $B$ is mixing in each of $M_1$ and $M_2$, $M_1{\bar{\ast }}_B{M}_2$ 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.