Conjugation in Representation Categories of Multiplicative Unitaries and Their Actions on C*-Algebras

A conjugation functorFon a full subcategory ofR(V), the representation category of a multiplicative unitaryV, is defined. IfVhas a conjugate, it is also regular and the domain ofFis allR(V). Examples of selfconjugate multiplicative unitaries are discussed. A coaction of the HopfC*-algebra associated withVon the Cuntz algebraOdis canonically defined by a unitary objectWofR(V) acting on ad-dimensional Hilbert space. As in the group action case ifd=∞ andWbelongs to the domain ofF, ergodic coactions are often characterized by the absence of finite dimensional subrepresentations ofW. Furthermore model actions of compact quantum group duals onC*-algebras are defined.