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Suppose G, a discrete group acting on the compact operators $\mathcal{K}$ by automorphism of C*-algebra $\mathcal{K}$. Can we view the crossed product as a groupoid C*-algebra of some groupoid?

This question occurred to me when I was thinking of possible formulation of Baum-Connes conjecture for projective representation of the group.

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  • $\begingroup$ Since all automorphisms of $\mathcal K$ are inner, the crossed product ${\mathcal K} \rtimes G$ is $*$-isomorphic to ${\mathcal K} \otimes \mathrm{C}^*_\lambda(G)$. The latter is a groupid $\mathrm{C}^*$-algebra. $\endgroup$ – Narutaka OZAWA Jun 30 '14 at 21:47
  • $\begingroup$ @NarutakaOZAWA if you look at this paper arxiv.org/pdf/math/0609784v2.pdf (page 5, second paragraph from top), this action by $G$ comes from a 2-cocycle from 2nd cohomology group of $G$, and the crossed product $\mathcal{K} \rtimes G$ highly depends on that cocycle. So can you please clarify the 1st statement? $\endgroup$ – SiOn Jul 1 '14 at 20:32
  • $\begingroup$ You are right. It was isomorphic to ${\mathcal K}\otimes\mathrm{C}^*_\lambda(G,\bar{\omega})$. It isn't clear if it's a groupoid $\mathrm{C}^*$-algebra. $\endgroup$ – Narutaka OZAWA Jul 1 '14 at 22:03
  • $\begingroup$ yes, even not clear for finite groups(for trivial group its obvious only)! $\endgroup$ – SiOn Jul 1 '14 at 23:03
  • $\begingroup$ Depending on what you want to do, describing a C*-algebra as a twisted groupoid C"-algebra is often good enough. So @Ozawa's last comment could be what you need. $\endgroup$ – Ruy Oct 13 '16 at 21:48
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Have you seen Packer-Raeburn Stabilization trick? ( Transactions of the AMS 334(2), p. 685-718, 1992). Specially theorem 3.4 therein stablishes that the algebra is morita equivalent to a Twisted Group C*- algebra, summarizing the answer by N. Ozawa and the correction above.

By considering the trick of viewing the algebra $K\rtimes G$ as a locally trivial bundle with a $G$-action over a contractible $G$- space $X$ ( For instance, the total space of the universal $G$-Bundle $EG$, but also the space $\underline{E}G$ works), you can see the $C*$ algebra in question is morita equivalent to the algebr a of sections of the Fell bundle over the transformation groupoid $X\rtimes G$. THis has ben documented in Kumjian's monograph http://www.theta.ro/jot/archive/1988-020-002/1988-020-002-002.pdf

Other references on this kind of tricks include http://jlms.oxfordjournals.org/content/50/1/170.abstract , and part 5 here: https://arxiv.org/abs/1501.06050

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