Discrete C*-coactions and C*-algebraic bundles

Abstract

Discrete C'-coactions are shown to be equivalent to discrete C*-algebraic bundles. Simplicity, primeness, liminality, postliminality, and nuclearity are related to the fixed point algebra and the cocrossed product. Ergodic, and more generally homogeneous, C -coactions are characterized.1991 Mathematics subject classification (Amer. Math. Soc): 46L55.

Aperiodicity, topological freeness and pure outerness: from group actions to Fell bundles

“…We are going to construct a canonical Morita globalisation using a variant of Takesaki-Takai duality. This construction is already studied by Quigg [49] in the language of coactions. If B comes from a partial action α, then this Morita globalisation agrees with the Morita enveloping action of α analysed in [3].…”

“…Examples of such properties are: being liminal, antiliminal, Type I 0 , Type I or nuclear (see [49]). We warn the reader that the "ideal property" is not "ideal" in this sense.…”

“…We use that every Fell bundle is Morita equivalent to a Fell bundle that comes from a partial action that globalises to an action by automorphisms. Such a Morita globalisation for Fell bundles over discrete groups is constructed by Abadie [3] and by Quigg [49] using Takai Duality (see Section 7). In order to show that this construction preserves the non-triviality properties we are interested in, we introduce the concepts of a Morita covering and a dense Morita covering (see Section 6).…”

“…Section 6 studies permanence of non-triviality conditions for single Hilbert bimodules under Morita restrictions and dense Morita coverings. Section 7 studies a canonical Morita globalisation for a Fell bundle that goes back to Abadie [3] and Quigg [49]. In Section 8, we use Morita globalisations and Morita coverings to generalise and extend relationships between non-triviality conditions from single automorphisms to single Hilbert bimodules.…”

Fell bundles over groupoids

“…Remark. If the groupoid is a (discrete) group this result may be obtained by combining [Q2,Cor. 2.7] and [Kt,Th.…”