## Regular representation of Woronowicz’s quantum displacements groups. (Représentation régulière du groupe quantique des déplacements de Woronowicz.)(French)Zbl 0840.46036

Connes, A. (ed.), Recent advances in operator algebras. Collection of talks given in the conference on operator algebras held in Orléans, France in July 1992. Paris: Société Mathématique de France, Astérisque. 232, 11-48 (1995).
Summary: Let $$H$$ be a Hilbert space. In this article, under appropriate “regularity” conditions, we associate to every multiplicative unitary $$V\in {\mathcal L}(H\otimes H)$$, a pair of Hopf $$C^*$$-algebras in duality. We show that the regular representation of the quantum $$E_\mu(2)$$ group of Woronowicz is a multiplicative unitary satisfying our conditions and we calculate its covariant representations. We also calculate the Haar measures of $$E_\mu(2)$$ and its Pontryagin dual and we give their modular theory.
For the entire collection see [Zbl 0832.00041].

### MSC:

 46L05 General theory of $$C^*$$-algebras 46L60 Applications of selfadjoint operator algebras to physics 46M05 Tensor products in functional analysis 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory