Twists over étale groupoids and twisted vector bundles (Q2814399)

Let \(\mathcal{G}\) denote a locally compact Hausdorff groupoid. Twists \(\tau\) over \(\mathcal{G}\) then are equivalence classes of specific central extension \(\mathcal{R}\) of \(\mathcal{G}\) by the torus group \(\mathbb{T}\), i.e., the group of complex scalars of norm \(1\). Twists of this kind typically arise from projective Hilbert space bundles over the space \(\mathcal{G}^{(0)}\) of units of \(\mathcal{G}\), and these twists yield the relevant twisting data, needed to define corresponding twisted \(K\)-theory groups for \(\mathcal{G}\). In the present context, the twisted \(K\)-theory group of \(\mathcal{G}\) with a twist \(\tau\) as above by definition is the \(K\)-group of the associated twisted groupoid \(C^*\)-algebra, typically denoted by \(\mathcal{C}^*(\mathcal{G};\mathcal{R})\), whose definition goes back to \textit{J. Renault} [A groupoid approach to \(C^*\)-algebras. Berlin-Heidelberg-New York: Springer-Verlag (1980; Zbl 0433.46049)]. One central question is whether or not for a given twist \(\tau\) as above, there is a so-called twisted vector bundle, i.e., a vector bundle \(E\) over the unit space \(\mathcal{G}^{(0)}\) such that \(\mathcal{R}\) acts by linear isomorphisms on \(E\) over \(\mathcal{G}^{(0)}\) and with the property that the restriction of the action to the fibers of \(\mathcal{R}\) is always given by scalar multiplication. Twisted vector bundles yield cycles in \(K_0(\mathcal{C}^*(\mathcal{G}; \mathcal{R}))\), however, the natural question whether or not all elements of \(K_0(\mathcal{C}^*(\mathcal{G}; \mathcal{R}))\) can be represented by twisted vector bundles has been answered by \textit{N. C. Phillips} [Equivariant \(K\)-theory for proper actions. Harlow: Longman Scientific \& Technical; New York: Wiley (1989; Zbl 0684.55018)] who showed that this does not even hold in the untwisted setting.NEWLINENEWLINEOne necessary condition for the existence of twisted vector bundles for a given twist \(\tau\) is that the equivalence class of \(\tau\) in the Brauer group \(\mathrm{Br}(\mathcal{G})\) of \(\mathcal{G}\) is a torsion element; the relevant Brauer group \(\mathrm{Br}(\mathcal{G})\) here is the one defined in [\textit{A. Kumjian} et al., Am. J. Math. 120, No. 5, 901--954 (1998; Zbl 0916.46050)]. In the present paper, the authors give a criterion for when a torsion element in this Brauer group can be represented by a twist for which there exists a twisted vector bundle. Their investigation uses the classifying space \(B\mathcal{G}\) of the groupoid, and the condition is phrased as a condition on this classifying space. A further result of the article concerns the set of elements of the Brauer group which can be represented by twists which have an associated twisted vector bundles. Here, they show that this set of elements indeed is a subgroup of the Brauer group \(\mathrm{Br}(\mathcal{G})\). This fits well with a conjecture stated in [\textit{J.-L. Tu} et al., Ann. Sci. Éc. Norm. Supér. (4) 37, No. 6, 841--910 (2004; Zbl 1069.19006)] which says that the set of elements of the Brauer group which can be represented by twists with an associated twisted vector bundle coincides with the torsion subgroup of the Brauer group \(\mathrm{Br}(\mathcal{G})\). The conjecture in [Tu et al., loc. cit.], however, was only stated for proper Lie groupoids \(\mathcal{G}\) which act cocompactly on the space of units \(\mathcal{G}^{(0)}\).