2014/07/23 by Dong-woon Kim, Kim, Dong-woon · 1 citation
Mathematics · #46L08 #46L55 #47L65 #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Operator Algebras (math.OA)
paper · pdf · doi:10.48550/arxiv.1407.6106
openalex publication_date 2014/07/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Unifying two notions of an action and coaction of a locally compact group on a C^*-cor\-re\-spond\-ence we introduce a coaction (σ,δ) of a Hopf C^*-algebra S on a C^*-cor\-re\-spond\-ence (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^*-cor\-re\-spond\-ence (X\rtimesσ\widehatS,A\rtimesδ\widehatS) from (σ,δ) and show that it has a representation on the reduced crossed product OX\rtimesζ\widehatS by the induced coaction ζ. This representation is used to prove an isomorphism between the C^*-algebra OX\rtimesζ\widehatS and the Cuntz-Pimsner algebra O_X\rtimesσ\widehatS under the covariance assumption which is guaranteed in particular if the ideal J_X\rtimesσ\widehatS of A\rtimesδ\widehatS is generated by the canonical image of JX in M(A\rtimesδ\widehatS) 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.