On the structure of relatively biexact group von Neumann algebras

Using computations in the bidual of $\mathbb{B}(L^{2}M)$ we develop a new technique at the von Neumann algebra level to upgrade relative proper proximality to full proper proximality. This is used to structurally classify subalgebras of $L\mathrm{\Gamma }$ where $\mathrm{\Gamma }$ is an infinite group that is biexact relative to a finite family of subgroups $\{{\mathrm{\Lambda }}_i{\}}_{i\in I}$ such that each ${\mathrm{\Lambda }}_i$ is almost malnormal in $\mathrm{\Gamma }$. This generalizes the result of \cite{DKEP21} which classifies subalgebras of von Neumann algebras of biexact groups. By developing a combination with techniques from Popa’s deformation-rigidity theory we obtain a new structural absorption theorem for free products and a generalized Kurosh type theorem in the setting of properly proximal von Neumann algebras.