The Brauer Semigroup of a groupoid and a symmetric imprimitivity theorem

TL;DR

This paper introduces a generalized equivariant Brauer semigroup for groupoids, extending existing concepts and establishing an isomorphism under groupoid equivalence that preserves crossed product classes, thus broadening the symmetric imprimitivity theorem.

Contribution

It defines a new equivariant Brauer semigroup for groupoids and proves an isomorphism that generalizes Raeburn's symmetric imprimitivity theorem.

Findings

The equivariant Brauer semigroup is well-defined for locally compact Hausdorff groupoids.

Groupoid equivalence induces an isomorphism of these semigroups.

The isomorphism preserves Morita equivalence classes of crossed products.

Abstract

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.