A REPRESENTATION THEOREM FOR LOCALLY COMPACT QUANTUM GROUPS

Recently, Neufang, Ruan and Spronk proved a completely isometric representation theorem for the measure algebra M(G) and for the completely bounded (Herz-Schur) multiplier algebra McbA(G) on B(L-2(G)), where G is a locally compact group. We unify and generalize both results by extending the representation to arbitrary locally compact quantum groups G = (M, Gamma, phi, psi). More precisely, we introduce the algebra M-cb(r)(L-1(G)) of completely bounded right multipliers on L-1(G) and we show that M-cb(r)(L-1(G)) can be identified with the algebra of normal completely bounded (M) over cap -bimodule maps on B(L-2(G)) which leave the subalgebra M invariant. From this representation theorem, we deduce that every completely bounded right centralizer of L-1(G) is in fact implemented by an element of M-cb(r)(L-1(G)). We also show that our representation framework allows us to express quantum group "Pontryagin" duality purely as a commutation relation.