Transformation group C∗-algebras with continuous trace

Abstract

We obtain several results characterizing when transformation group C ∗ -algebras have continuous trace. These results can be stated most succinctly when ( G, Ω ) is second countable, and the stability groups are contained in a fixed abelian subgroup. In this case, C ∗ (G, Ω) has continuous trace if and only if the stability groups vary continuously on Ω and compact subsets of Ω are wandering in an appropriate sense. In general, we must assume that the stability groups vary continuously, and if ( G, Ω ) is not second countable, that the natural maps of G S x onto G · x are homeomorphisms for each x . Then C ∗ (G, Ω) has continuous trace if and only if compact subsets of Ω are wandering and an additional C ∗ -algebra, constructed from the stability groups and Ω, has continuous trace.