Perturbations of crossed product C*-algebras by amenable groups

TL;DR

This paper investigates the stability of crossed product C*-algebras by amenable groups under small perturbations, establishing conditions under which such algebras remain isomorphic or equal.

Contribution

It proves that sufficiently close intermediate C*-subalgebras of a crossed product by an amenable group are isomorphic or identical, extending stability results in operator algebras.

Findings

Close intermediate C*-subalgebras are isomorphic.

If the inclusion is irreducible, the subalgebras are equal.

Stability of crossed products under perturbations.

Abstract

We study uniform perturbations of crossed product C$^*$-algebras by amenable groups. Given a unital inclusion of C$^*$-algebras $C\subseteq D$ and sufficiently close separable intermediate C$^*$-subalgebras $A$, $B$ for this inclusion with a conditional expectation from $D$ onto $B$, if $A=C\rtimes G$ with $G$ discrete amenable, then $A$ and $B$ are isomorphic. Furthermore, if $C\subseteq D$ is irreducible, then $A=B$.