Coactions of Hopf C*-algebras on Cuntz-Pimsner algebras

Original · EN

Unifying two notions of an action and coaction of a locally compact group on a C*-correspondence we introduce a coaction (σ,δ) of a Hopf C*-algebra S on a C*-correspondence (X,A). We show that this coaction naturally induces a coaction ζ of S on the associated Cuntz-Pimsner algebra Oₓ under the weak δ-invariancy for the ideal Jₓ. When the Hopf C*-algebra S is defined by a well-behaved multiplicative unitary, we construct a C*-correspondence (XσS,AδS) from (σ,δ) and show that it has a representation on the reduced crossed product OₓζS by the induced coaction ζ. This representation is used to prove an isomorphism between the C*-algebra OₓζS and the Cuntz-Pimsner algebra OXσS under the covariance assumption which is guaranteed in particular if the ideal JXσS of AδS is generated by the canonical image of Jₓ in M(AδS) or the left action on X by A is injective. Under this covariance assumption, our results extend the isomorphism result known for actions of amenable groups to arbitrary locally compact groups. The Cuntz-Pimsner covariance condition which was assumed for the same isomorphism result concerning group coactions is shown to be redundant.