Groupoid actions on C*-correspondences

Let the groupoid G with unit space G(0) act via a representation rho on a C*-correspondence H over the C-0(G(0))-algebra A. By the universal property, G acts on the Cuntz-Pimsner algebra OH which becomes a C-0(G(0))-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 (sic) 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 (sic) G congruent to O-H proportional to G, where H (sic) G is the crossed product C*-correspondence and that OGH congruent to O rho, where 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) (sic) G is stably isomorphic to a graph algebra. We illustrate with some examples.