Inclusions of $C^{\ast }$-algebras arising from fixed-point algebras

Abstract

We examine inclusions of $C^{\ast }$-algebras of the form $A^H\subseteq A{⋊}_{r}G$, where $G$ and $H$ are groups acting on a unital simple $C^{\ast }$-algebra $A$ by outer automorphisms and $H$ is finite. It follows from a theorem of Izumi that $A^H\subseteq A$ is $C^{\ast }$-irreducible, in the sense that all intermediate $C^{\ast }$-algebras are simple. We show that $A^H\subseteq A{⋊}_{r}G$ is $C^{\ast }$-irreducible for all $G$ and $H$ as above if and only if $G$ and $H$ have trivial intersection in the outer automorphisms of $A$, and we give a Galois type classification of all intermediate $C^{\ast }$-algebras in the case when $H$ is abelian and the two actions of $G$ and $H$ on $A$ commute. We illustrate these results with examples of outer group actions on the irrational rotation $C^{\ast }$-algebras. We exhibit, among other examples, $C^{\ast }$-irreducible inclusions of AF-algebras that have intermediate $C^{\ast }$-algebras that are not AF-algebras; in fact, the irrational rotation $C^{\ast }$-algebra appears as an intermediate $C^{\ast }$-algebra.