From multiplicative unitaries to quantum groups II ✩

It is shown that all important features of a C∗-algebraic quantum group (A,Δ) defined by a modular multiplicative W depend only on the pair (A,Δ) rather than the multiplicative unitary operator W. The proof is based on thorough study of representations of quantum groups. As an application we present a construction and study properties of the universal dual of a quantum group defined by a modular multiplicative unitary—without assuming existence of Haar weights. © 2007 Elsevier Inc. All rights reserved.