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The Brauer group of a locally compact groupoid. (English) Zbl 0916.46050

In the superb introduction it is argued that the analysis of the Brauer group Br(G) of a groupoid G extends both the Mackey approach to projective representations of locally compact groups and that of Dixmier and Douady to continuous trace \(\text{C}^*\)-algebras. Moreover, it suggests perspectives of extending and unifying various related cohomology theories. The groupoid G in question, having the unit space \(\text{ G}^{(0)}\), is 2-nd countable, Hausdorff, locally compact and with Haar system. Br(G) consists of classes \([\mathcal A,\alpha]\), modulo the suitable Morita equivalence, of systems \((\mathcal A,\alpha)\) where \(\mathcal A\) is an elementary \(\text{ C}^*\)-bundle over \(\text{ G}^{(0)}\) satisfying Fell’s condition and \(\alpha\) is a G-action on \(\mathcal A\) by *-isomorphisms. It is endowed with a group structure. If \(\text{ G}=X\times H\) is a transformation groupoid of a group H, then \(\text{Br} (G) = Br_H(X)\), the equivariant Brauer group, cf. [D. Crocker, A. Kumjian, I. Raeburn, D. P. Williams, J. Funct. Anal., 146, No. 1, 151-184 (1997; Zbl 0873.22003)]. Established is the isomorphism of Brauer groups for equivalent groupoids, as defined in [P. S. Muhly, J. N. Renault, D. P. Williams, J. Oper. Theory 17, 3-22 (1987; Zbl 0645.46040], and the generalized twist group \({\mathcal E}(\text{G})\) is isomorphic to the subgroup \(\text{ Br}_0(G)\subset Br(G)\) of \([\mathcal A,\alpha]\) with trivial Dixmier-Douadi invariant. This study of twists over G (cf. [P. S. Muhly and D. P. Williams, Proc. Lond. Math. Soc., III. Ser. 71, No. 1, 109-134 (1995; Zbl 0902.46045)]), aimed at replacing the sheaf cohomology of [A. Kumjian, J. Oper. Theory 20, 207-240 (1988; Zbl 0692.46066)], is combined with viewing the directed collection \({\mathcal P}\)(G) (of all 2-nd countable principal G-spaces \(X\), whose imprimitivity groupoid \(\text{G}^X\) has a Haar system) as a generalization of all open covers of \(\text{G}^{(0)}\), directed by refinement, to prove that the inductive limit Ext(G,\(\mathbb{T}\)) of \({\mathcal E}({\text G}^X)\), \(X \in {\mathcal P}\text{(G)}\), is isomorphic to Br(G). Illustrative examples are given as well as a viewpoint based on Renault’s concept of a dual groupoid of a \(\text{C}^*\)-algebra.

MSC:

46L55 Noncommutative dynamical systems
46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)
54H15 Transformation groups and semigroups (topological aspects)
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