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 Mcb 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 Qcb r(L 1(double-struck G sign)), and the completely isometric representation of Mcb 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 UCQr (double-struck G sign) and its Banach algebra dual UCQr (double-struck G sign)*. For quantum groups double-struck G sign with the right co-approximation property, we establish a completely contractive injection Qcb r(L1(double-struck G sign)) → UCQr (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 Mcb r(L1(double-struck G sign))≅ M(double-struck G sign) and the completely isometric isomorphism UCQr (double-struck G sign) ≅ LUC(double-struck G sign).

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Journal Proceedings of the London Mathematical Society
Hu, Z. (Zhiguo), Neufang, M, & Ruan, Z.-J. (Zhong-Jin). (2011). Completely bounded multipliers over locally compact quantum groups. Proceedings of the London Mathematical Society, 103(1), 1–39. doi:10.1112/plms/pdq041