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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
    Commented Jun 30, 2014 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
    Commented Jul 1, 2014 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
    Commented Jul 1, 2014 at 22:03
  • $\begingroup$ yes, even not clear for finite groups(for trivial group its obvious only)! $\endgroup$
    – SiOn
    Commented Jul 1, 2014 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
    Commented Oct 13, 2016 at 21:48

1 Answer 1

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Have you seen Packer-Raeburn Stabilization trick?

Judith A. Packer and Iain Raeburn, On the structure of twisted group $C^*$-algebras, Trans. Amer. Math. Soc. 334 no 2 (1992), 685-718, doi:10.1090/S0002-9947-1992-1078249-7

Specially Theorem 3.4 therein establishes 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 algebra of sections of the Fell bundle over the transformation groupoid $X\rtimes G$. This has been documented in Kumjian's monograph

Alex Kumjian, On equivariant sheaf cohomology and elementary $C^*$-bundles, J. Operator Theory 20 (1988), no. 2, 207–240 (journal pdf)

Other references on this kind of tricks include

Siegfried Echterhoff, Morita Equivalent Twisted Actions and a New Version of the Packer-Raeburn Stabilization Trick, Journal of the London Mathematical Society 50 Issue 1 (1994) 170–186, doi:10.1112/jlms/50.1.170

and part 5 here:

Noé Bárcenas, Twisted geometric K-homology for proper actions of discrete groups, Journal of Topology and Analysis (2018) doi:10.1142/S1793525319500729, arXiv:1501.06050.

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  • $\begingroup$ The first link above doesn't have a Theorem 3.4 (Thanks to Michael Murray for pointing this out_! It's possible the paper Nicolas meant was: Judith A. Packer and Iain Raeburn, Twisted crossed products of C∗-algebras, Math. Proc. Cambridge Philos. Soc. 106 (1989), no. 2, 293–311, MR1002543, doi:10.1017/S0305004100078129, which does have what looks to be a relevant Theorem 3.4 $\endgroup$
    – David Roberts ♦
    Commented Feb 26, 2020 at 4:20

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