The Brauer Semigroup of a groupoid and a symmetric imprimitivity theorem

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