Let $H$ be a Hilbert space. In this article, under appropriate “regularity” conditions, we associate to every multiplicative unitary $V\in {\cal L}(H\otimes H)$, 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 Pontrjagyn dual and we give their modular theory.