Coactions of Hopf $C^{\ast }$-algebras on Cuntz-Pimsner algebras

Abstract

Unifying two notions of an action and coaction of a locally compact group on a $C^{\ast }$-cor\-re\-spond\-ence we introduce a coaction $(\sigma ,\delta )$ of a Hopf $C^{\ast }$-algebra $S$ on a $C^{\ast }$-cor\-re\-spond\-ence $(X,A)$. We show that this coaction naturally induces a coaction $\zeta$ of $S$ on the associated Cuntz-Pimsner algebra ${\mathcal{O}}_X$ under the weak $\delta$-invariancy for the ideal $J_X$. When the Hopf $C^{\ast }$-algebra $S$ is defined by a well-behaved multiplicative unitary, we construct a $C^{\ast }$-cor\-re\-spond\-ence $(X{⋊}_{\sigma }\hat{S},A{⋊}_{\delta }\hat{S})$ from $(\sigma ,\delta )$ and show that it has a representation on the reduced crossed product ${\mathcal{O}}_X{⋊}_{\zeta }\hat{S}$ by the induced coaction $\zeta$. This representation is used to prove an isomorphism between the $C^{\ast }$-algebra ${\mathcal{O}}_X{⋊}_{\zeta }\hat{S}$ and the Cuntz-Pimsner algebra ${\mathcal{O}}_{X{⋊}_{\sigma }\hat{S}}$ under the covariance assumption which is guaranteed in particular if the ideal $J_{X{⋊}_{\sigma }\hat{S}}$ of $A{⋊}_{\delta }\hat{S}$ is generated by the canonical image of $J_X$ in $M(A{⋊}_{\delta }\hat{S})$ 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.

37 pages; introduction is revised; section 4 is shortened

References in corpus (3)

• K-amenability for amalgamated free products of amenable discrete quantum groups
• Coactions on Cuntz-Pimsner algebras
• A new look at crossed product correspondences and associated C*-algebras

Cited by in corpus (1)

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