Journal of Operator Theory

Volume 36, Issue 1, Summer 1996 pp. 157-177.

Remarks on braided C*-categories and endomorphisms of C*-algebras

Authors: Anna Paolucci
Author institution:School of Mathematics, University of Leeds, Leeds LS2 9JT, ENGLAND, UNITED KINGDOM

Summary: The duality theory for compact groups of Doplicher and Roberts deals with the category of finite dimensional continuous representations as an abstract $C*$-category. We study braided $C*$-categories for a compact matrix quantum group to model the non commutativity of the tensor product. Let $F_d$ be the category of corepresentations of the quantum group $U_q(d)$. We associate to the Yang-Baxter category $YB(Fd)$ a $C*$-algebra, $(O_d)^{U_q(d)}$. We give conditions for an endomorphism of a unital $C^*$ algebra $\mathcal A$ to determine an action of the strict braided tensor $C^*$ category of corepresentation $U_q(d)$ on $\mathcal A$. Such actions correspond to a *-monomorphism of the fixed point subalgebra $(O_d)^{SU_q(d)}$ into $\mathcal A$ with natural intertwining properties.

Keywords: Braided $C^*$-categories, compact quantum group, endomorphisms, $C^*$-algebras.

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