Equivalence and disintegration theorems for Fell bundles and their C^*-algebras

Abstract

We study the C * -algebras of Fell bundles. In particular, we prove the analogue of Renault's disintegration theorem for groupoids. As in the groupoid case, this result is the key step in proving a deep equivalence theorem for the C * -algebras of Fell bundles.

A stabilization theorem for Fell bundles over groupoids

“…The vectorvalued Tietze Extension Theorem [27,Proposition A.5] implies that the set {f (x) : f ∈ Γ c (G; B) } is dense in V (x). The Hofmann-Fell theorem (see, for example, [10,Theorem II.13.18], [14], [15], and [17, Theorem 1.2]) implies that there is a unique topology such that V is an upper semicontinuous Banach bundle over G (0) and such that Γ(G (0) ; V ) contains {f : f ∈ Γ c (G; B) }.…”

“…We g)) B(g) by Lemma 3.3, and multiplication induces an imprimitivity-bimodule isomorphism between B(g) ⊗ A(s(g)) B(h) and B(gh) (see Lemma 1.2 of [27]). Moreover, V (r(gh)) = V (r(g)).…”

“…If B and C are equivalent Fell bundles then C * (G; B) and C * (H; C) are Morita equivalent [27,Theorem 6.4] and so are C * red (G; B) and C * red (H; C) [20,43]. An important example for this note is the Fell bundle associated with a groupoid dynamical system.…”

“…Following [27] (see also [10,11,20]), a Fell bundle over G is an uppersemicontinuous Banach bundle p : B → G endowed with a partially defined multiplication and an involution that respect p such that the fibres A(x) over x ∈ G (0) are C * -algebras and each fibre B(g) is an A(r(g))-A(s(g)) imprimitivity bimodule with respect to the inner products and actions induced by the multiplication in B. Note that for x ∈ G (0) we will write both A(x) and B(x) for the fiber over x to distinguish its dual role as a C * -algebra and as the trivial imprimitivity bimodule over A(x).…”

“…Muhly sketched in [25] how one can extend Kumjian's techniques to non-étale groupoids, but still only for continuous bundles. Here we elaborate on a variation of Muhly's approach and in doing so prove a generalization of his result (Theorem 3.7): given a second-countable saturated upper-semicontinuous Fell bundle p : B → G over a second-countable Hausdorff locally compact groupoid we show that there is a groupoid dynamical system (in the sense of [28]) such that the Fell bundle associated with the groupoid dynamical system is equivalent with B (see [27]). Therefore the full and reduced C * -algebras of B are Morita equivalent to the full and reduced C * -algebras, respectively, of the corresponding groupoid dynamical system.…”