MULTIPLICATIVE UNITARIES AND DUALITY FOR CROSSED-PRODUCTS OF C-ASTERISK-ALGEBRAS

Let H be a Hilbert space. A unitary operator V is-an-element-of L (H x H) is said to be multiplicative if it satisfies the pentagone equation V12V13V23=V23V12. In many papers concerned on operator algebras with duality ([13], [44], [17], [6], [19], [11]), a multiplicative unitary plays a fundamental role. In this paper we look for additional conditions that a multiplicative unitary should satisfy in order to correspond to a ''locally compact quantum group''. We introduce two conditions: ''regularity'' and ''irreducibility''. To any multiplicative unitary satisfying these conditions we associate two pairwise dual Hopf C*-algebras. Moreover, we establish Takesaki-Takai duality results, using an adaptation of the method of [5]. If the Hilbert space is finite dimensional or if the unitary V satisfies a commutativity condition, regularity and irreducibility are automatic. If the unitary V is of compact or discrete type, its regularity implies its irreducibility.