Traces of $C^{\ast }$-algebras of connected solvable groups

Abstract

We give an explicit description of the tracial state simplex of the $C^{\ast }$-algebra $C^{\ast }(G)$ of an arbitrary connected, second countable, locally compact, solvable group $G$. We show that every tracial state of $C^{\ast }(G)$ lifts from a tracial state of the $C^{\ast }$-algebra of the abelianized group, and the intersection of the kernels of all the tracial states of $C^{\ast }(G)$ is a proper ideal unless $G$ is abelian. As a consequence, the $C^{\ast }$-algebra of a connected solvable nonabelian Lie group cannot embed into a simple unital AF-algebra.

arXiv comment9 pages