Quasidiagonality of $C^*$-algebras of solvable Lie groups

TL;DR

This paper characterizes when the $C^*$-algebras of certain solvable Lie groups are quasidiagonal, focusing on groups of the form ${ m R}^m times { m R}$ and type I groups, with implications for amenable groups.

Contribution

It provides a complete characterization of quasidiagonality for $C^*$-algebras of specific solvable Lie groups and identifies examples of amenable groups with non-quasidiagonal $C^*$-algebras.

Findings

Characterization of quasidiagonality for $C^*$-algebras of ${ m R}^m times { m R}$ groups.

Determination of strongly quasidiagonal $C^*$-algebras for connected simply connected solvable Lie groups of type I.

Examples of amenable Lie groups with non-quasidiagonal $C^*$-algebras.

Abstract

We characterize the solvable Lie groups of the form ${\mathbb R}^m\rtimes {\mathbb R}$, whose $C^*$-algebras are quasidiagonal. Using this result, we determine the connected simply connected solvable Lie groups of type~I whose $C^*$-algebras are strongly quasidiagonal. As a by-product, we give also examples of amenable Lie groups with non-quasidiagonal $C^*$-algebras.