Strength of convergence and multiplicities in the spectrum of a C*-dynamical system

Abstract

We consider separable $C^{\ast }$-dynamical systems $(A,G,\alpha )$ for which the induced action of the group $G$ on the spectrum $\hat{A}$ of the $C^{\ast }$-algebra $A$ is free. We study how the representation theory of the associated crossed-product $C^{\ast }$-algebra $A{⋊}_{\alpha }G$ depends on the representation theory of $A$ and the properties of the action of $G$ on $\hat{A}$. Our main tools involve computations of upper and lower bounds on multiplicity numbers associated to irreducible representations of $A{⋊}_{\alpha }G$. We apply our techniques to give necessary and sufficient conditions, in terms of $A$ and the action of $G$ on $\hat{A}$, for $A{⋊}_{\alpha }G$ to be (i) a continuous-trace $C^{\ast }$-algebra, (ii) a Fell $C^{\ast }$-algebra and (iii) a bounded-trace $C^{\ast }$-algebra. When $G$ is amenable, we also give necessary and sufficient conditions for the crossed-product $C^{\ast }$-algebra $A{⋊}_{\alpha }G$ to be (iv) a liminal $C^{\ast }$-algebra and (v) a Type I $C^{\ast }$-algebra. The results in (i), (iii)--(v) extend some earlier special cases in which $A$ was assumed to have the corresponding property.

Publication version, to appear in Proc. London Math. Soc

References in corpus (2)

• The transformation groups whose C^*-algebras are Fell algebras
• Inducing Primitive Ideals

Cited by in corpus (1)

• The equivalence relations of local homeomorphisms and Fell algebras