连通可解群的$C^*$ -代数的迹
Traces of $C^*$-algebras of connected solvable groups
摘要：
本文给出了任意连通的、第二可数的、局部紧的、可解群$G$的$C^*$ -代数$C^*(G)$的迹态单形的一个显式描述。我们证明了$C^*(G)$的每一个迹态都是从阿贝尔化群的$C^*$ -代数的迹态提升而来的，并且除非$G$是阿贝尔群，否则$C^*(G)$的所有迹态的核的交集都是真理想.因此，连通可解非交换李群的$C^*$ -代数不能嵌入到单有单位元的AF-代数中。
英文摘要：
We give an explicit description of the tracial state simplex of the $C^*$-algebra $C^*(G)$ of an arbitrary connected, second countable, locally compact, solvable group $G$. We show that every tracial state of $C^*(G)$ lifts from a tracial state of the $C^*$-algebra of the abelianized group, and the intersection of the kernels of all the tracial states of $C^*(G)$ is a proper ideal unless $G$ is abelian. As a consequence, the $C^*$-algebra of a connected solvable nonabelian Lie group cannot embed into a simple unital AF-algebra.
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