Noncommutative topological boundaries and amenable invariant intermediate subalgebras

Abstract: As an analogue of the topological boundary of discrete groups $\Gamma$, we define the noncommutative topological boundary of tracial von Neumann algebras $(M,\tau)$ and apply it to generalize a recent result by Amrutam-Hartman-Oppelmayer, showing that for a trace preserving action $\Gamma \curvearrowright (A,\tau_A)$ on an amenable tracial von Neumann algebra, any $\Gamma$-invariant amenable intermediate subalgebra between $A$ and $\Gamma\ltimes A$ is necessarily a subalgebra of $\mathrm{Rad}(\Gamma) \ltimes A$. By taking $(A,\tau_A)=L^\infty(X,\nu_X)$ for a free pmp action $\Gamma \curvearrowright (X,\nu_X)$, we obtain a similar result for invariant subequivalence relations of $\mathcal{R}_{\Gamma \curvearrowright X}$.

geometric topologynumber theoryoperator algebrasrepresentation theory

Audience: researchers in the topic

( paper | video )

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