C ~*-动力系统谱的收敛强度和重数
Strength of convergence and multiplicities in the spectrum of a C*-dynamical system
摘要：
我们考虑可分的$C^*$ -动力系统$(A,G,α)$，其中群$G$在$C^*$ -代数$A$的谱$\hat A$上的诱导作用是自由的.本文研究了伴随交叉积$C^*$ -代数$A\rtimes_αG$的表示论如何依赖于$A$的表示论以及$G$对$\hat A$作用的性质.我们的主要工具包括计算与$A\rtimes_αG$的不可约表示相关联的多重数的上界和下界。我们应用我们的技巧给出了$A\rtimes_αG$是（i）连续迹$G$-代数，（ii）Fell $C^*$ -代数和（iii）有界迹$C^*$-代数的充要条件.当$G$是顺从的时，我们也给出了交叉积$C^*$ -代数$A\rtimes_αG$是（iv）阈$C^*$ -代数和（v）I型$C^*$ -代数的充要条件.（i），（iii）-（v）中的结果推广了某些较早的特殊情况，其中假定$A$具有相应的性质.
英文摘要：
We consider separable $C^*$-dynamical systems $(A,G,α)$ 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 $A\rtimes_αG$ 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 $A\rtimes_αG$. We apply our techniques to give necessary and sufficient conditions, in terms of $A$ and the action of $G$ on $\hat A$, for $A\rtimes_αG$ 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 $A\rtimes_αG$ 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.
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