From multiplicative unitaries to quantum groups II

It is shown that all important features of a $\mathrm{C}^*$-algebraic quantum group $(A,\Delta)$ defined by a modular multiplicative $W$ depend only on the pair $(A,\Delta)$ 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.