AF‐embeddability for Lie groups with T1 primitive ideal spaces

Abstract

We study simply connected Lie groups G for which the hull-kernel topology of the primitive ideal space Prim(G) of the group C * -algebra C * (G) is T1, that is, the finite subsets of Prim(G) are closed. Thus, we prove that C * (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 Prim(G) has no nonempty quasi-compact open subsets. We prove in addition that connected locally compact groups with T1 ideal spaces are strongly quasi-diagonal.