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

Siegfried Echterhoff¹  ; Mikael Rørdam² 

¹Westfälische Wilhelm-Universität Münster, Germany
²University of Copenhagen, Denmark

Groups, geometry, and dynamics, Tome 18 (2024) no. 1, pp. 127-145

Résumé

We examine inclusions of C∗-algebras of the form AH⊆A⋊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 AH⊆A is C∗-irreducible, in the sense that all intermediate C∗-algebras are simple. We show that AH⊆A⋊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.

Détail

DOI : 10.4171/ggd/743

Classification : 46-XX
Mots-clés : Irreducible inclusion of C∗-algebras, crossed product, fixed-point algebra, irrational rotation algebra