On Relative Biexactness of Amalgamated Free Product von Neumann Algebras

Abstract: We show that if $M_{1}, M_{2}$ are weakly exact tracial von Neumann algebras admitting a common injective amalgam $B$, then the amalgamated free product $M_{1}\overline{*}{B}M{2}$ is biexact relative to ${M_{1},M_{2}}$. When each $ M_i $ is injective, we also show that the amalgmated free product is biexact relative to the amalgam $B$.