Locally compact quantum groups in the universal setting

In this paper we associate to every reduced C∗-algebraic quantum group (A,∆) (as defined in [18]) a universal C∗-algebraic quantum group (Au,∆u). We fine tune a proof of Kirchberg to show that every ∗-representation of a modified L-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 Au. We show that this C∗-algebra Au carries a quantum group structure which is as rich as its reduced companion. Introduction In 1977, S.L. Woronowicz proposed the use of the C∗-language to axiomatize quantizations of locally compact quantum groups. This approach was very successful in the compact case ([43],[40],[38]) and the discrete case ([26],[37],[11]). In both cases the existence of the Haar weights could be proven from a simple set of axioms. The situation for the general non-compact however is less satisfactory. At present, there is still no general definition for a locally compact quantum group in which the existence of the Haar weights is not one of the axioms of the proposed definition. The first attempt to axiomatize locally quantum groups aimed at enlarging the category of locally compact quantum groups in such a way that it contains locally compact groups and the reduced group C∗-algebras. A complete solution for this problem was found independently by M. Enock & J.-M. Schwarz and by Kac & Vainermann (see [12] for a detailed account). The resulting objects are called Kac algebras and their definition was formulated in the von Neumann algebra framework. For quite a time, the main disadvantage of this theory lay in the fact that there was a lack of interesting examples aside from the groups and group duals. S.L. Woronowicz constructed in [44] quantum SU(2), an object which has all the right properties to be called a compact quantum group but does not fit into the framework of Kac algebras. In subsequent papers ([43],[40]), S.L. Woronowicz developed the axiom scheme for compact quantum groups. In contrast to the Kac algebra theory, quantum SU(2) fitted into this category of compact quantum groups.