AF-embeddability for Lie groups with $T_1$ primitive ideal spaces

Abstract

We study simply connected Lie groups $G$ for which the hull-kernel topology of the primitive ideal space $\text{Prim}(G)$ of the group $C^{\ast }$-algebra $C^{\ast }(G)$ is $T_1$, that is, the finite subsets of $\text{Prim}(G)$ are closed. Thus, we prove that $C^{\ast }(G)$ is AF-embeddable. To this end, we show that if $G$ is solvable and its action on the centre of $[G,G]$ has at least one imaginary weight, then $\text{Prim}(G)$ has no nonempty quasi-compact open subsets. We prove in addition that connected locally compact groups with $T_1$ ideal spaces are strongly quasi-diagonal.

23 pages, accepted for publication in J. London Math. Soc

References in corpus (2)

• Every simple higher dimensional noncommutative torus is an AT algebra
• Connective C*-algebras

Cited by in corpus (2)

• Square-integrable representations and the coadjoint action of solvable Lie groups
• On stably finiteness for $C^{\ast }$-algebras of exponential solvable Lie groups