Traces of C*-algebras of connected solvable groups

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. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

The paper provides an explicit description of the tracial state simplex of the C*-algebra C*(G) of a connected, second countable, locally compact, solvable group G. It is shown that every tracial state of C*(G) can be derived from a tracial state of the C*-algebra of the abelianized group, and the intersection of the kernels of all tracial states forms a proper ideal unless G is abelian. Consequently, the C*-algebra of a connected solvable nonabelian Lie group cannot embed into a simple unital AF-algebra.