On Relative Biexactness of Amalgamated Free Product von Neumann Algebras

Given weakly exact tracial von Neumann algebras M₁, M₂ with a common injective amalgam B, we prove that the amalgamated free product M₁*̄BM₂ is biexact relative to ₁,M₂\. In the case where M₁ and M₂ are injective, we further show that M₁*̄BM₂ is biexact relative to the amalgam B, and if B is mixing in each of M₁ and M₂, M₁*̄BM₂ 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.