Locally compact quantum groups in the universal setting

In this paper we associate to every reduced C*-algebraic quantum group (A, Delta) las defined in [11]) a universal C*-algebraic quantum group (A(u), D elta (u)). We fine l;une a proof of Kirchberg to show that every *-represen tation of a modified L-1-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) carrie s a quantum group structure which is a rich as its reduced companion. We al so establish a bijective correspondence between quantum group morphisms and certain co-actions.