Conjugation in representation categories of multiplicative unitaries and their actions on \(C^*\)-algebras
type: Item
instance of: scholarly article
title: Conjugation in representation categories of multiplicative unitaries and their actions on \(C^*\)-algebras
zbMATH Open document ID: 0844.22014
author: Claudia Pinzari
published in: Journal of Functional Analysis
publication date: 1996-08-26T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 22D25
Mathematics Subject Classification ID: 46L89
zbMATH DE Number: 847485
zbMATH Keywords: unitary multiplicative
zbMATH Keywords: pentagon equation
zbMATH Keywords: Hilbert space
zbMATH Keywords: operator
zbMATH Keywords: operator algebras
zbMATH Keywords: duality
zbMATH Keywords: representations
zbMATH Keywords: conjugation functor
zbMATH Keywords: tensor product representations
zbMATH Keywords: Hopf algebras
zbMATH Keywords: Cuntz algebras
reviewed by: H. Hogbe Nlend
MaRDI profile type: Publication
full work available at URL: jfan.1996.0014
OpenAlex ID: W2039407037
DOI: 10.1006/JFAN.1996.0014
description: scientific article; zbMATH DE number 847485