  In this paper we associate to every reduced C*-algebraic quantum group A a
universal C*-algebraic quantum group. 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 uinversal C*-algebra. We show that this
universal C*-algebra carries a quantum group structure which is as rich as its
reduced companion.
