A new look at crossed product correspondences and associated C⁎C⁎-algebras

When a locally compact group acts on a C⁎C⁎-correspondence, it also acts on the associated Cuntz–Pimsner algebra in a natural way. Hao and Ng have shown that when the group is amenable the Cuntz–Pimsner algebra of the crossed product correspondence is isomorphic to the crossed product of the Cuntz–Pimsner algebra. In this paper, we have a closer look at this isomorphism in the case where the group is not necessarily amenable. We also consider what happens at the level of Toeplitz algebras.