Noncommutative topological boundaries and amenable invariant random intermediate subalgebras

TL;DR

This paper introduces a noncommutative topological boundary concept for tracial von Neumann algebras and uses it to characterize invariant amenable subalgebras and subequivalence relations under group actions.

Contribution

It generalizes previous results by defining a noncommutative boundary and applying it to classify invariant amenable subalgebras and subequivalence relations.

Findings

Invariant amenable subalgebras are contained in Rad(Γ) ⋉ A.

The results extend to invariant subequivalence relations of group actions.

Provides a new framework for understanding boundaries in noncommutative settings.

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 the main results of [AHO23], 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 the invariant subequivalence relations of $\mathcal{R}_{\Gamma \curvearrowright X}$.