Definition 0.1   A representation of a topological $*$ -algebra is defined as
a continuous $*$ -morphism from to $B(\H)$ , where is a topological groupoid, (usually a locally compact groupoid, ), $\H $ is a Hilbert space, and $B(\H)$ is the $C^*$ -algebra of bounded linear operators on the Hilbert space $\H$ ; of course, one considers the inductive limit (strong) topology to be defined on , and then also an operator weak topology to be defined on $B(\H)$ .