Groupoid Cohomology and the Dixmier‐Douady Class

Suppose A is a continuous-trace C*-algebra with spectrum A. (Throughout, unless otherwise stated, all C*-algebras will be assumed to be separable and all topological spaces and groupoids will be assumed to be locally compact, Hausdorff, and second countable.) In [4], Dixmier and Douady showed how to associate an element 8(A) in the Cech cohomology group H (A ;Z) with A in such a way that two algebras, Ax and A2, are stably isomorphic if and only if 8(Ai) = 8(A2)The element 8(A) has therefore come to be known as the Dixmier-Douady class of A. It is well known that every class in H(A ; Z) can arise as a 8(A) and that 8(A) = 0 precisely when A is stably isomorphic to C0(A). (See [17, § 3] for more details and further references. As is customary, we identify H*(A ;Z) with H(A, ¥), where Sf denotes the sheaf of germs of continuous T-valued functions on A.) Suppose, now, that A is the C*-algebra of a locally compact groupoid G with Haar system {A"}ueC(o), that is, A = C*(G, A). In various contexts, in recent years, the problems of deciding when C*(G, A) has continuous trace and identifying its Dixmier-Douady class has arisen (cf., for example, [10,11,12,14,15,6,7]). In [11], we showed that if R is a principal groupoid, then C*(R, A) is a continuous-trace algebra if and only if the usual action of R on its unit space X is proper. (A principal groupoid is essentially an equivalence relation on its unit space. Consequently, we have made the consistent notational convention of denoting them by R. Further, when discussing a relation R on a space X, we shall simply refer to the unit space of R as X.) In this event, 8(C*(R, A)) = 0, because as we also showed in [11], (C*(R, A)) is homeomorphic to the quotient space of /^-equivalence classes X/R with the quotient topology, and C*(R, A) is stably isomorphic to CQ(X/R). On the other hand, for certain principal groupoids R, it is possible to find a groupoid 2-cocycle a such that the twisted groupoid C*-algebra determined by a (and A) C*(R, a, A) in the sense of Renault [18] has continuous trace with non-zero Dixmier-Douady class [14]. Indeed, as Raeburn and Taylor show, there is a very tight relation between Cech cohomology and groupoid cohomology in the particular setting they considered. They showed that given a locally compact space Y, and an element 8 G H(Y, y), one can build a relation R on a space X such that X/R is homeomorphic to Y and one can construct an explicit a e H(R, T) from 8 so that the Dixmier-Douady class of C*(R, a, A) is 8 for any choice of A. This, then, begs the question of what