GROUP ACTIONS ON GRAPHS AND C*-CORRESPONDENCES

If G acts on a C*-correspondence H over the C*-algebra A (see Definition 3), then by the universal property G acts on the Cuntz-Pimsner algebra O-H and we study the crossed product O-H (sic) G and the fixed point algebra O-H(G). Using intertwiners, we define the Doplicher-Roberts algebra O-rho of a representation rho of a compact group G on H and prove that under certain conditions O(H)(G )is isomorphic to O-rho. The action of G commutes with the gauge action on O-H, therefore G acts also on the core algebras O-H(T), where T denotes the unit circle. We give applications for the action of a group G on the C*-correspondence H-E associated to a topological graph E. If G is finite and E is discrete and locally finite, we prove that the crossed product C*(E) (sic) G is isomorphic to the C*-algebra of a graph of C*-correspondences and stably isomorphic to a locally finite graph algebra. If C*(E) is simple and purely infinite and the action of G is outer, then C*(E)(G) and C*(E) (sic) G are also simple and purely infinite with the same K-theory groups. We illustrate with several examples.