Intermediate subalgebras for reduced crossed products of discrete groups

TL;DR

This paper investigates the structure of intermediate subalgebras in reduced crossed product C*-algebras arising from group actions, establishing a Galois correspondence under certain conditions.

Contribution

It introduces the concept of pointwise residual proper outerness as a key condition for a Galois correspondence in reduced crossed products, generalizing freeness.

Findings

Established a sufficient condition for Galois correspondence in reduced crossed products.

Proved the condition is almost always necessary under the approximation property.

Connected noncommutative outerness with the structure of intermediate subalgebras.

Abstract

Let $\alpha : \Gamma \curvearrowright A$ be an action of a discrete group $\Gamma$ on a unital C*-algebra $A$ by *-automorphisms and let $A \rtimes_{\alpha,\lambda} \Gamma$ denote the corresponding reduced crossed product C*-algebra. Assuming that $\Gamma$ satisfies the approximation property, we establish a sufficient and (almost always) necessary condition on the action $\alpha$ for the existence of a Galois correspondence between intermediate C*-algebras for the inclusion $A \subseteq A \rtimes_{\alpha,\lambda} \Gamma$ and partial subactions of $\alpha$. This condition, which we refer to as pointwise residual proper outerness, is a natural noncommutative generalization of freeness.