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- W2775842518 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 ${mathcal O}_{mathcal H}rtimes Gcong {mathcal O}_{mathcal Hrtimes G}$, where $mathcal Hrtimes G$ is the crossed product $C^*$-correspondence and that ${mathcal O}_{mathcal H}^Gcong{mathcal O}_rho$, 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." @default.
- W2775842518 created "2018-01-05" @default.
- W2775842518 creator A5083741780 @default.
- W2775842518 date "2017-12-28" @default.
- W2775842518 modified "2023-09-27" @default.
- W2775842518 title "Groupoid actions on $C^*$-correspondences" @default.
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