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 M_cbA(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 double-strock G sign = (M, Γ, φ, ψ). More precisely, we introduce the algebra M_cb^r(L₁(G)) of completely bounded right multipliers on L₁(double-strock G sign) and we show that $M_cb^r(L_L(G)) can be identified with the algebra of normal completely bounded M̂-bimodule maps on B(L₂(G)) which leave the subalgebra M invariant. From this representation theorem, we deduce that every completely bounded right centralizer of L₁(double-strock G sign) is in fact implemented by an element of M_cb^r(L₁(G)). We also show that our representation framework allows us to express quantum group " Pontryagin" duality purely as a commutation relation.