LOCALLY COMPACT QUANTUM GROUPS IN THE UNIVERSAL SETTING

In this paper we associate to every reduced C^*-algebraic quantum group (A, Δ) (as defined in [11]) a universal C^*-algebraic quantum group (A_u, Δ_u). We fine tune a proof of Kirchberg to show that every ^*-representation of a modified L¹-space is generated by a unitary corepresentation. By taking the universal enveloping C^*-algebra of a dense sub ^*-algebra of A we arrive at the C^*-algebra A_u. We show that this C^*-algebra A_u carries a quantum group structure which is a rich as its reduced companion. We also establish a bijective correspondence between quantum group morphisms and certain co-actions.