Conjugation in representation categories of multiplicative unitaries and their actions on \(C^*\)-algebras
Let \(H\) be a Hilbert space. A ``unitary multiplicative'' is an operator \(V = (V_{ij}) \in {\mathcal L} (H \otimes H)\) which satisfies a ``pentagon equation'' involving \((V_{ij} )\). In many papers, concerned with operator algebras duality, multiplicative unitaries and their representations play an important role. In this paper a conjugation functor on a full category of \({\mathcal R} (v)\), the representation category of a multiplicative unitary \(V\), is defined and studied. The first aim is to give a notion of conjugate object of a corepresentation \(W\) of a multiplicative unitary \(V\) and to show that any such \(W\) has a conjugate provided the regular corepresentation \(V\) has a conjugate. Then examples of selfconjugate regular representations are given. A result is established concerning finite dimensional subrepresentations of tensor product representations. Actions of Hopf algebras arising from certain multiplicative unitaries on Cuntz algebras are defined.
scientific article; zbMATH DE number 847485