Continuous-trace groupoid crossed products

Fulman, Igor; Muhly, Paul S.; Williams, Dana P.

doi:10.1090/s0002-9939-03-07158-2

Cited by 5 publications

(5 citation statements)

References 13 publications

(15 reference statements)

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“…In this event, the twisted K-theory, as we defined it, coincides with twisted K-theory defined in [3,23]. Indeed the C * -algebra C * (M, σ) is Morita equivalent to the continuous trace C * -algebra defined by the correspondant Dixmier-Douady class (see for instance Theorem 1 in [18]).…”

Section: 21mentioning

confidence: 78%

Twisted longitudinal index theorem for foliations and wrong way functoriality

Rouse

Wang²

2011

Advances in Mathematics

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Section: 21mentioning

confidence: 78%

Twisted longitudinal index theorem for foliations and wrong way functoriality

Rouse

Wang²

2011

Advances in Mathematics

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“…In this event, the twisted K-theory, as we defined it, coincides with twisted K-theory defined in [2,21]. Indeed the C * -algebra C * (M, α) is Morita equivalent to the continuous trace C * -algebra defined by the corresponding Dixmier-Douady class (see for instance Theorem 1 in [18]).…”

Section: (Twisting On Fiber Product Groupoid) Let Nmentioning

confidence: 80%

Geometric Baum-Connes assembly map for twisted Differentiable Stacks

Rouse¹,

Wang²

2016

Ann. Sci. École Norm. Sup.

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We construct the geometric Baum-Connes assembly map for twisted Lie groupoids, that means for Lie groupoids together with a given groupoid equivariant P U (H)−principle bundle. The construction is based on the use of geometric deformation groupoids, these objects allow in particular to give a geometric construction of the associated pushforward maps and to establish the functoriality. The main results in this paper are to define the geometric twisted K-homology groups and to construct the assembly map. Even in the untwisted case the fact that the geometric K-homology groups and the geometric assembly map are well defined for Lie groupoids is new, as it was only sketched by Connes in his book for general Lie groupoids without any restrictive hypothesis, in particular for non Hausdorff Lie groupoids.We also prove the Morita invariance of the assembly map, giving thus a precise meaning to the geometric assembly map for twisted differentiable stacks. We discuss the relation of the assembly map with the associated assembly map of the S 1 -central extension. The relation with the analytic assembly map is treated, as well as some cases in which we have an isomorphism. One important tool is the twisted Thom isomorphism in the groupoid equivariant case which we establish in the appendix. RésuméNous construisons le morphisme d'assemblage géométrique de Baum-Connes pour des groupoïdes de Lie tordus,à savoir des groupoïdes de Lie avec un P U (H)-fibré principal equivariant. La construction est basé dans l'utilisation des groupoïdes de déformation, ces objets permettent en particulier de donner une construction géométrique des morphismes shriek associés et d'établir la fonctorialité. Les résultats principaux de cet article sont la définition des groupes de K-homologie géométrique tordue et la construction du morphisme d'assemblage. Même dans le cas non tordu le fait que les groupes de K-homologie géométrique et le morphisme d'assemblage (géométrique) pour des groupoïdes de Lie sont bien définis est nouveau, en effet, ceci aété esquissé par Connes dans son livre pour des groupoïdes de Lie générales sans aucune restriction, en particulier pour des groupoïdes non séparés.Nous montrons aussi l'invariance par Morita du morphisme d'assemblage, donnant ainsi un sens précis au morphisme d'assemblage géométrique de Baum-Connes pour des Champs différentiables tordus. Nous discutons la relation de notre morphisme d'assemblage avec le morphisme associéà la S 1 -extension central. La relation avec le morphisme analytique est traité, ainsi que quelques cas où il y a isomorphisme. Un outil important est le morphisme de Thom tordu dans le cas equivariant par rapportà un groupoïde que nousétablissons dans l'appendice.

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“…First, it will be helpful to recast [3, Theorem 1] in terms of Brauer groups as defined in [4]. 3 To start off, we only need G to be a proper and not necessarily locally trivial.…”

Section: The Dixmier-douady Classmentioning

confidence: 99%

“…Thus the pair (srf, a) is exactly what is needed to define an element in the Brauer group Br(G) as defined in [4,Definition 2.14]. As a special case of [3,Theorem 1], it follows that the groupoid crossed product C*(G; srf) has continuoustrace if and only if G is a proper principal groupoid. Thus if C*(G; &/) has continuous trace we can, and do, assume that that G = £2 * £2 = {(a>, co') e Q x £2 : p(co) = pip)')) for a continuous open surjection p : Q -> Y.…”

Section: Introductionmentioning

confidence: 99%

“…It follows from [5, Proposition 2.2] and [4, Theorem 4.1] that [sf, a] *-> [srf G * n] is an isomorphism of Br(G) onto Br(G (0) /G), where ^G " " = s t (#/)/G and where s t (srf) is the pull-back of si to G via the source map s (see, for example,[4, page 914]). Furthermore,[3, Theorem 1] implies that C 0 (G (0) /G, srf Gm ) is Morita equivalent to C*(G\ &/). This proves the first assertion.On the other hand,[4, Theorem 4.1] implies that the inverse is given by…”

mentioning

confidence: 99%

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