COMPLETIONS OF QUANTUM GROUP ALGEBRAS IN CERTAIN NORMS AND OPERATORS WHICH COMMUTE WITH MODULE ACTIONS

Let L-cb(1)(G) (respectively L-M(1)(G)) denote the closure of the quantum group algebra L-1(G) of a locally compact quantum group G, in the space of completely bounded (respectively bounded) double centralizers of L-1(G). In this paper, we study quantum group generalizations of various results from Fourier algebras of locally compact groups. In particular, left invariant means on L-cb(1)(G)* and on L-M(1)(G)* are defined and studied. We prove that the set of left invariant means on L-infinity(G) and on L-cb(1)(G)* (L-M(1)(G)*) have the same cardinality. We also study the left uniformly continuous functionals associated with these algebras. Finally, for a Banach A-bimodule (sic) of a Banach algebra A we establish a contractive and injective representation from the dual of a left introverted subspace of A* into B-A(sic*). As an application of this result we show that if the induced representation Phi : LUCcb(G)* -> B-Lcb1(G) L-infinity(G) is surjective, then L-cb(1)(G) has a bounded approximate identity. We also obtain a characterization of co-amenable quantum groups in terms of representations of quantum measure algebras M(G)