Fell bundles over groupoids

We study the C*-algebras associated to Fell bundles over groupoids and give a notion of equivalence for Fell bundles which guarantees that the associated C*-algebras are strong Morita equivalent. As a corollary we show that any saturated Fell bundle is equivalent to a semi-direct product arising from the action of the groupoid on a C*-bundle. A C*-algebraic bundle (see [F2, §11]) over a locally compact group may be thought of as a continuous version of a group grading in a C*-algebra; one may regard the associated C*-algebra as a fairly general sort of crossed product of the fiber algebra over the neutral element by the group (in [LPRS] it is shown that the C*-algebra is endowed with a coaction by the group). There is a natural extension of this definition to groupoids (see [Yg]) which when specialized to trivial groupoids (i.e. topological spaces) yields the more usual notion of C*-algebra bundle. This object is referred to below as a Fell bundle. Closely related notions have appeared in the literature: in recent work [Yn], Yamanouchi studies the analogous notion in the von Neumann algebra setting under the name integrable coaction. Fell bundles are reminiscent of the C*-categories discussed in [GLR]. They are also presaged in [Re2, Def. 5.3] Def. 5.3 (the object is used to construct a strong Morita equivalence bimodule). Since each fiber of a saturated Fell bundle may be regarded as a strong Morita equivalence bimodule, the theory of Fell bundles provides a natural locus for proving theorems related to strong Morita equivalence of the kind which appear in [MRW] and [Re2]. In §1 we fix notation and review some well-known facts concerning groupoids, Banach bundles, Hilbert modules, and equivalence bimodules. The notion of Fell bundle is defined in §2 and some examples are discussed, including the semi-direct product which results from a groupoid acting on a C*-algebra bundle fibered over the unit space. In §3 the associated C*-algebra is constructed in the case that the groupoid is r-discrete; the norm is defined by an analog of the left regular representation (the resulting C*-algebra may be regarded as the reduced C*-algebra associated to the Fell bundle). Finally, a Morita equivalence theorem of the sort discussed above is proved (Th. 4.2). This is used to show that there is an action of a groupoid on a C*-algebra bundle obtained from a saturated Fell bundle so that the C*-algebra associated to the Fell bundle is strong Morita equivalent to that of the semi-direct product (Cor. 4.5). See [Kt, Th. 8], [PR, Cor. 3.7], [Q1, Th. 3.1], [Q2, Cor. 2.7], and [Yn, Th. 7.8] for related results. §1 Preliminaries 1.1 Given a groupoid Γ, let Γ denote the unit space and r, s : Γ → Γ denote the range and source maps (respectively); let Γ denote the collection of composable pairs; write the inverse map γ 7→ γ∗. We tacitly assume that all groupoids under discussion are locally compact and Hausdorff, that the structure maps are continuous, and that they admit left Haar systems (see [Re1]). Let ∆ denote the transitive equivalence relation with unit space ∆ = {0, 1}; write ∆ = {0, 1, ∂, ∂∗} where s(∂) = 0 and r(∂) = 1. A groupoid Γ is said to be trivial if Γ = Γ. 1.2 Given a Banach bundle, p : E → X, let Cc(E) denote the collection of compactly supported continuous sections of E and C0(E) denote the collection of continuous sections of E vanishing at ∞. Note that C0(E) may be regarded as the completion of Cc(E) in the supremum norm. For x ∈ X, let Ex denote the fiber over x, p−1(x). We shall tacitly assume that the total space, E, of any Banach bundle under consideration is second countable and that the base space, X, is locally compact and Hausdorff; it follows that the base space is second countable and that both the fiber, Ex, and the associated Banach space, C0(E), are separable (see [F2, Prop. 10.10]). By a result Date: January 1997 (minor corrections July 2006). See [Mu] for a more recent version of the results of §4. 1991 Mathematics Subject Classification. Primary 46L55; Secondary 46L40, 46L45. Proc. Amer. Math Soc. 126 (1998) 1115–1125.