The Brauer semigroup of a groupoid and a symmetric imprimitivity theorem

Abstract

In this paper we define a monoid called the Brauer semigroup for a locally compact Hausdorff groupoid E E whose elements consist of Morita equivalence classes of E E -dynamical systems. This construction generalizes both the equivariant Brauer semigroup for transformation groups and the Brauer group for a groupoid. We show that groupoid equivalence induces an isomorphism of Brauer semigroups and that this isomorphism preserves the Morita equivalence classes of the respective crossed products, thus generalizing Raeburn’s symmetric imprimitivity theorem.