Traces of $C^*$-algebras of connected solvable groups
Ingrid Beltita, Daniel Beltita

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 -algebra of an arbitrary connected, second countable, locally compact, solvable group . We show that every tracial state of lifts from a tracial state of the -algebra of the abelianized group, and the intersection of the kernels of all the tracial states of is a proper ideal unless is abelian. As a consequence, the -algebra of a connected solvable nonabelian Lie group cannot embed into a simple unital AF-algebra.
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