Graded Brauer groups of a groupoid with involution

We define a group BrRˆ(G) containing, in a sense, the graded complex and orthogonal Brauer groups of a locally compact groupoid GG equipped with an involution. When the involution is trivial, we show that the new group naturally provides a generalisation of Donovan–Karoubiʼs graded orthogonal Brauer group GBrO  . More generally, it is shown to be a direct summand of the well-known graded complex Brauer group. In addition, we prove that BrRˆ(G) identifies with a direct sum of a Real cohomology group and the abelian group ExtRˆ(G,S1) of Real graded S1S1-central extensions. A cohomological picture is then given.