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

Abstract

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