representation of a $C_c(𝖦)$ *- topological algebra

Definition 0.1.

A representation of a $C_c\mathit{}\mathrm{(}\mathrm{G}\mathrm{)}$ (http://planetmath.org/C_cG) topological $\mathrm{*}$–algebra is defined as
a continuous $\mathrm{*}$–morphism from $C_c\mathit{}\mathrm{(}\mathrm{G}\mathrm{)}$ to $B\mathit{}\mathrm{(}\mathrm{H}\mathrm{)}$, where $\mathrm{G}$ is a topologicalgroupoid, (usually a locally compact groupoid, ${\mathrm{G}}_{l\mathit{}c}$), $\mathrm{H}$ is a Hilbert space,and $B(\mathscr{H})$ is the $C^{*}$-algebra of bounded linear operators on the Hilbert space $\mathscr{H}$;of course, one considers the inductive limit (strong) topology to be defined on $C_c(𝖦)$,and then also an operator weak topology to be defined on $B(\mathscr{H})$.