Inclusions of $C^*$-algebras arising from fixed-point algebras

Siegfried Echterhoff, Mikael Rørdam

Published 2021-08-19Version 1

We examine inclusions of $C^*$-algebras of the form $A^H \subseteq A \rtimes_{r} G$, where $G$ and $H$ are groups acting on a unital simple $C^*$-algebra $A$ by outer automorphisms and when $H$ is finite. We show that $A^H \subseteq A$ is $C^*$-irreducible, in the sense that all intermediate $C^*$-algebras are simple, if $H$ moreover is abelian. We further show that $A^H \subseteq A \rtimes_{r} G$ is $C^*$-irreducible when $H$ is abelian, if the two actions of $G$ and $H$ on $A$ commute, and the combined action of $G \times H$ on $A$ is outer. We illustrate these results with examples of outer group actions on the irrational rotation $C^*$-algebras. We exhibit, among other examples, $C^*$-irreducible inclusions of AF-algebras that have intermediate $C^*$-algerbras that are not AF-algebras, in fact, the irrational rotation $C^*$-algebra appears as an intermediate $C^*$-algebra.