For any hyperbolic group G, every amenable subalgebra Q of L(G) that intersects L(H) diffusely for an infinite maximal amenable subgroup H is contained in L(H); the result extends to acylindrically hyperbolic groups.
The separable case of Kadison's problem on orthonormal bases of unitaries for type $\mathrm{II}_1$ factors
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abstract
In 1967, Kadison asked ``does every type $\mathrm{II}_1$ factor have an orthonormal (with respect to the trace) basis consisting of unitaries?'' Using a noncommutative Lyapunov theorem of Akemann and Weaver, we prove that if $M$ is a separable diffuse finite von Neumann algebra with a normal faithful trace $\tau$, then $L^2(M,\tau)$ admits an orthonormal basis consisting of self-adjoint unitaries in $M$. Consequently, we affirm the separable case of the Kadison problem.
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Amenable absorption in von Neumann algebras of hyperbolic groups
For any hyperbolic group G, every amenable subalgebra Q of L(G) that intersects L(H) diffusely for an infinite maximal amenable subgroup H is contained in L(H); the result extends to acylindrically hyperbolic groups.