An equivalence theorem for reduced Fell bundle C*-algebras

TL;DR

This paper proves an equivalence theorem for reduced Fell bundle C*-algebras, showing how equivalences of Fell bundles induce isomorphisms between their reduced cross-sectional algebras, generalizing known results in groupoid crossed-products.

Contribution

It establishes a linking bundle construction that relates equivalent Fell bundles over groupoids to their reduced C*-algebras, extending previous imprimitivity theorems to the reduced setting.

Findings

Constructs a linking bundle L(E) over the linking groupoid L.

Shows the full cross-sectional algebra of L(E) contains those of B and C as full corners.

Generalizes Quigg and Spielberg's result to reduced crossed products.

Abstract

We show that if E is an equivalence of upper semicontinuous Fell bundles B and C over groupoids, then there is a linking bundle L(E) over the linking groupoid L such that the full cross-sectional algebra of L(E) contains those of B and C as complementary full corners, and likewise for reduced cross-sectional algebras. We show how our results generalise to groupoid crossed-products the fact, proved by Quigg and Spielberg, that Raeburn's symmetric imprimitivity theorem passes through the quotient map to reduced crossed products.