Enveloping actions and Takai duality for partial actions

We show that any partial action on a topological space X is the restriction of a suitable global action, called enveloping action, that is essentially unique. In the case of C ∗ -algebras, we prove that any partial action has a unique enveloping action up to Morita equivalence, and that the correspo...

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Bibliographic Details
Published inJournal of functional analysis Vol. 197; no. 1; pp. 14 - 67
Main Author Abadie, Fernando
Format Journal Article
LanguageEnglish
Published Elsevier Inc 10.01.2003
Subjects
Crossed products by partial actions
Fell bundles
Takai duality
Fell bundles
Crossed products by partial actions
Takai duality
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Summary:We show that any partial action on a topological space X is the restriction of a suitable global action, called enveloping action, that is essentially unique. In the case of C ∗ -algebras, we prove that any partial action has a unique enveloping action up to Morita equivalence, and that the corresponding reduced crossed products are Morita equivalent. The study of the enveloping action up to Morita equivalence reveals the form that Takai duality takes for partial actions. By applying our constructions, we prove that the reduced crossed product of the reduced cross-sectional algebra of a Fell bundle by the dual coaction is liminal, postliminal, or nuclear, if and only if so is the unit fiber of the bundle. We also give a non-commutative generalization of the well-known fact that the integral curves of a vector field on a compact manifold are defined on all of R .
ISSN:0022-1236
1096-0783
DOI:10.1016/S0022-1236(02)00032-0