On Relative Biexactness of Amalgamated Free Product von Neumann Algebras

TL;DR

This paper proves that certain amalgamated free product von Neumann algebras are biexact relative to their components, leading to new structural and subalgebra results extending known group case theorems.

Contribution

It establishes biexactness properties of amalgamated free product von Neumann algebras under various conditions, extending previous results from the group case.

Findings

Amalgamated free product von Neumann algebras are biexact relative to their components.

Biexactness is shown when the component algebras are injective and the amalgam is mixing.

Structural decomposition and subalgebra absorption results are derived for these 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 $M_{1}\overline{*}_{B}M_{2}$ is biexact relative to $\{M_{1},M_{2}\}$. In the case where $ M_1 $ and $M_2$ are injective, we further show that $M_{1}\overline{*}_{B}M_{2}$ is biexact relative to the amalgam $B$, and if $B$ is mixing in each of $M_1$ and $M_2$, $M_{1}\overline{*}_{B}M_{2}$ 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.