Structure of Relatively Biexact Group von Neumann Algebras

Using computations in the bidual of \({\mathbb {B}}(L^2M)\) $\mathbb{B}(L^{2}M)$ 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 \(L\Gamma \) $L\mathrm{\Gamma }$ where \(\Gamma \) $\mathrm{\Gamma }$ is an infinite group that is biexact relative to a finite family of subgroups \(\{\Lambda _i\}_{i\in I}\) ${\{{\mathrm{\Lambda }}_i\}}_{i\in I}$ such that each \(\Lambda _i\) ${\mathrm{\Lambda }}_i$ is almost malnormal in \(\Gamma \) $\mathrm{\Gamma }$ . This generalizes the result of Ding et al. (Properly proximal von Neumann algebras, 2022. arXiv:2204.00517) 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.