Let the groupoid G with unit space G⁰ act via a representation ρ on a C*-correspondence H over the C₀(G⁰)-algebra A. By the universal property, G acts on the Cuntz-Pimsner algebra O_H which becomes a C₀(G⁰)-algebra. The action of G commutes with the gauge action on O_H, therefore G acts also on the core algebra O_H^T.

We study the crossed product O_H x G and the fixed point algebra O_H^G and obtain similar results as in [5], where G was a group. Under certain conditions, we prove that O_H x G ≅ O_{H x G}, where H x G is the crossed product C*-correspondence and that O_H^G ≅ O_ρ, where O_ρ 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) x G is stably isomorphic to a graph algebra. We illustrate with some examples.

The author would like to thank Alex Kumjian and Leonard Huang for helpful and illuminating discussions.