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 cb A(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, Γ, ϕ, ψ). More precisely, we introduce the algebra M r cb (L 1 (G)) of completely bounded right multipliers on L 1 (G) and we show that M r cb (L 1 (G)) can be identified with the algebra of normal completely boundedM -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 r cb (L 1 (G)). We also show that our representation framework allows us to express quantum group "Pontryagin" duality purely as a commutation relation.

Abstract. Let G be a locally compact group. It is shown that there exists a natural completely isometric representation of the completely bounded Fourier multiplier algebra M cb A(G), which is dual to the representation of the measure algebra) are intrinsically characterized, and some commutant theorems are proved. It is also shown that for any amenable group G, there is a natural completely isometric representation of UCB(Ĝ) * on B(L 2 (G)), which can be regarded as a duality result of Neufang's completely isometric representation theorem for LU C(G) * .

In this paper, we consider several interesting multiplier algebras associated with a locally compact quantum group G. Firstly, we study the completely bounded right multiplier algebra M r cb (L1(G)). We show that M r cb (L1(G)) is a dual Banach algebra with a natural operator predual Q r cb (L1(G)), and the completely isometric representation of M r cb (L1(G)) on B(L2(G)), studied recently by Junge, Neufang and Ruan, is actually weak*-weak* continuous if the quantum group G has the right co-approximation property. Secondly, we study the space LUC(G) of left uniformly continuous functionals on L1(G) and its Banach algebra dual LUC(G) * . We prove that LUC(G) is a unital C*-subalgebra of L∞(G) if the quantum group G is semi-regular. We show the connection between LUC(G) * and the quantum measure algebra M (G), as well as their representations on L∞ (G) and B(L2(G)). Finally, we study the right uniformly continuous complete quotient space UCQ r (G) and its Banach algebra dual UCQ r (G) * . For quantum groups G with the right coapproximation property, we establish a completely contractive injection Q r cb (L1(G)) → UCQ r (G) which is compatible with the relation C0(G) ⊆ LUC(G). For co-amenable quantum groups G, we obtain the weak*-weak* homeomorphic and completely isometric algebra isomorphism M r cb (L1(G)) ∼ = M (G) and the completely isometric isomorphism UCQ r (G) ∼ = LUC(G).