QUANTUM GROUPS AND DUALITY

We present an approach to the duality theory for quantum groups which is explicitly modelled on the construction of the group C*-algebra C*(G) from C0 (G), namely: in the dual of C0(G), single out the ideal of all elements absolutely continuous with respect to Haar measure, renorm with a C*-norm determined by the representations of this ideal, and complete. We thus obtain a C*-algebra whose *-representations are in one-to-one correspondence with the representations of the quantum group. This C*-algebra is itself a (non-compact) quantum group, and we verify a duality theorem for it.