Groupoid actions on \(C^*\)-correspondences

Abstract

Let the groupoid \(G\) with unit space \(G^0\) act via a representation \(\rho\) on a \(C^*\)-correspondence \({\mathcal H}\) over the \(C_0(G^0)\)-algebra \(A\). By the universal property, \(G\) acts on the Cuntz-Pimsner algebra \({\mathcal O}_{\mathcal H}\) which becomes a \(C_0(G^0)\)-algebra. The action of \(G\) 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 \({\mathcal O}_{\mathcal H}\rtimes G\) 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 \({\mathcal O}_{\mathcal H}\rtimes G\cong {\mathcal O}_{\mathcal H\rtimes G}\), where \(\mathcal H\rtimes G\) is the crossed product \(C^*\)-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 \(C^*(G)\) is strongly Morita equivalent to a commutative \(C^*\)-algebra, we prove that the crossed product \(C^*(E)\rtimes G\) is stably isomorphic to a graph algebra. We illustrate with some examples.