Traces of $C^*$-algebras of connected solvable groups

TL;DR

This paper explicitly describes the tracial states of the $C^*$-algebra of connected solvable groups, revealing structural properties and limitations on embeddings into AF-algebras.

Contribution

It provides an explicit characterization of the tracial state space of $C^*(G)$ for connected solvable groups and shows how these states relate to the abelianized group.

Findings

Every tracial state lifts from the abelianized group

The intersection of kernels of all tracial states is proper unless G is abelian

The $C^*$-algebra of a nonabelian solvable Lie group cannot embed into a simple unital AF-algebra

Abstract

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.