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

TL;DR

This paper investigates the structure of inclusions of $C^*$-algebras arising from group actions, characterizing when such inclusions are $C^*$-irreducible and classifying intermediate algebras, with examples involving irrational rotation algebras.

Contribution

It provides a characterization of $C^*$-irreducibility for inclusions from fixed-point and crossed product algebras, and offers a Galois type classification for certain cases.

Findings

$A^H times_{r} G$ is $C^*$-irreducible iff $G$ and $H$ have trivial intersection in outer automorphisms.

Intermediate $C^*$-algebras can be classified when $H$ is abelian and actions commute.

Examples include $C^*$-irreducible inclusions of AF-algebras with non-AF intermediate algebras.

Abstract

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 $H$ is finite. It follows from a theorem of Izumi that $A^H \subseteq A$ is $C^*$-irreducible, in the sense that all intermediate $C^*$-algebras are simple. We show that $A^H \subseteq A \rtimes_{r} G$ is $C^*$-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 $C^*$-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 $C^*$-algebras. We exhibit, among other examples, $C^*$-irreducible inclusions of AF-algebras that have intermediate $C^*$-algebras that are not AF-algebras, in fact, the irrational rotation $C^*$-algebra appears as an intermediate $C^*$-algebra.