Bounded trace C*-algebras and integrable actions

Abstract

Let G be a second countable group, A be a separable C * -algebra with bounded trace and α a strongly continuous action of G on A. Suppose that the action of G on A induced by α is free and the G-orbits are locally closed. We show that the crossed product A × α G has bounded trace if and only if G acts integrably (in the sense of Rieffel and an Huef) on A. In the course of this, we show that the extent of non-properness of an integrable action gives rise to a lower bound for the size of the (finite) upper multiplicities of the irreducible representations of the crossed product. Mathematics Subject Classification (1991): 46L55

Principal groupoid $C^*$-algebras with bounded trace

“…It is evident from [1,2] that integrability and k-times convergence in the orbit space of a transformation group are closely related. Moreover, Lemma 2.6 of [8] says that, if a principal groupoid fails to be proper and the orbit space G 0 /G is Hausdorff, then there exists a sequence that converges 2-times in G 0 /G in the sense of Definition 3.6.…”

“…(a) Condition (2) in Definition 3.6 is needed so that the composition in (3) makes sense. (b) Definition 3.6 does not require that x n → z, but as in the transformationgroup case ([2, Definition 2.2]), this can be arranged by changing the sequence which converges k-times: replace x n by r(γ (1) n ) and replace γ …”

“…The sequence {x n } converges k-times in G/G 0 to z for every k ∈ N \ {0}, so it certainly converges l times. So there exist l sequences {γ (1) n }, {γ (2) n }, . .…”

“…Moreover, by construction (see Proposition 3.11), we may take γ (1) n = x n . Temporarily fix n. Set g (1) n := g (1) , and for 2 ≤ j ≤ l set …”

“…This theorem is thus very much in the spirit of [8,Theorem 2.4], [4,Theorem 7.9], [4,Theorem 4.1] (see also [3,Corollary 5.9]) and [4,Theorem 5.3], which characterize when principal-groupoid C * -algebras are, respectively, continuous-trace, Fell, CCR and GCR C * -algebras. The key technical tools used to prove Theorem 4.4 are, first, a homeomorphism of the spectrum of C * (G) onto the orbit space [4, Proposition 5.1] and, second, a generalisation to groupoids of the notion of k-times convergence in the orbit space of a transformation group from [1].…”