Groupoid actions on $C^{\ast }$-correspondences

Let the groupoid $G$ with unit space $G^0$ act via a representation $\rho$ on a $C^{\ast }$-correspondence $\mathcal{H}$ over the $C_0(G^0)$-algebra $A$. By the universal property, $G$ acts on the Cuntz-Pimsner algebra ${\mathcal O}{\mathcal H}$whichbecomesa$C_0(G^0)$-algebra\mathrm{.}Theactionof$G$ commutes with the gauge action on ${\mathcal{O}}_{\mathcal{H}}$, therefore $G$ acts also on the core algebra ${\mathcal{O}}_{\mathcal{H}}^{\mathbb{T}}$. We study the crossed product ${\mathcal{O}}_{\mathcal{H}}⋊G$ and the fixed point algebra ${\mathcal{O}}_{\mathcal{H}}^G$ and obtain similar results as in \cite{D}, where $G$ was a group. Under certain conditions, we prove that ${\mathcal{O}}_{\mathcal{H}}⋊G\cong {\mathcal{O}}_{\mathcal{H}⋊G}$, where $\mathcal{H}⋊G$ is the crossed product $C^{\ast }$-correspondence and that ${\mathcal{O}}_{\mathcal{H}}^G\cong {\mathcal{O}}_{\rho }$, where ${\mathcal O}\rho$ is the Doplicher-Roberts algebra defined using intertwiners. The motivation of this paper comes from groupoid actions on graphs. Suppose $G$ with compact isotropy acts on a discrete locally finite graph $E$ with no sources. Since $C^{\ast }(G)$ is strongly Morita equivalent to a commutative $C^{\ast }$-algebra, we prove that the crossed product $C^{\ast }(E)⋊G$ is stably isomorphic to a graph algebra. We illustrate with some examples.