Structure of Relatively Biexact Group von Neumann Algebras

Abstract

Using computations in the bidual of \({\mathbb {B}}(L^2M)\) 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 \(L\Gamma \) where \(\Gamma \) is an infinite group that is biexact relative to a finite family of subgroups \(\{\Lambda _i\}_{i\in I}\) such that each \(\Lambda _i\) is almost malnormal in \(\Gamma \). 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.