Completely bounded multipliers over locally compact quantum groups

In this paper, we consider several interesting multiplier algebras associated with a locally compact quantum group double-struck G sign. Firstly, we study the completely bounded right multiplier algebra M_cb ^r(L₁(double-struck G sign)). We show that M _cb^r(L₁(double-struck G sign)) is a dual Banach algebra with a natural operator predual Q_cb^r(L ₁(double-struck G sign)), and the completely isometric representation of M_cb^r(L₁(double-struck G sign)) on ℬ(L₂(double-struck G sign)), studied recently by Junge, Neufang and Ruan, is actually weak*-weak* continuous if the quantum group double-struck G sign has the right co-approximation property. Secondly, we study the space LUC(double-struck G sign) of left uniformly continuous functionals on L₁(double-struck G sign) and its Banach algebra dual LUC(double-struck G sign)*. We prove that LUC(double-struck G sign) is a unital C*-subalgebra of L_∞(double-struck G sign) if the quantum group double-struck G sign is semi-regular. We show the connection between LUC(double-struck G sign)* and the quantum measure algebra M(double-struck G sign), as well as their representations on L _∞(double-struck G sign) and ℬ(L₂(double-struck G sign)). Finally, we study the right uniformly continuous complete quotient space UCQ^r (double-struck G sign) and its Banach algebra dual UCQ^r (double-struck G sign)*. For quantum groups double-struck G sign with the right co-approximation property, we establish a completely contractive injection Q_cb^r(L₁(double-struck G sign)) → UCQ^r (double-struck G sign) which is compatible with the relation C₀(double-struck G sign) ⊆ LUC(double-struck G sign). For co-amenable quantum groups double-struck G sign, we obtain the weak*-weak* homeomorphic and completely isometric algebra isomorphism M_cb^r(L₁(double-struck G sign))≅ M(double-struck G sign) and the completely isometric isomorphism UCQ^r (double-struck G sign) ≅ LUC(double-struck G sign).