### Abstract

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}(L_{1}(double-struck G sign)). We show that M _{cb}^{r}(L_{1}(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}(L_{1}(double-struck G sign)) on ℬ(L_{2}(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 L_{1}(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(L_{2}(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}(L_{1}(double-struck G sign)) → UCQ^{r} (double-struck G sign) which is compatible with the relation C_{0}(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}(L_{1}(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).

Original language | English (US) |
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Pages (from-to) | 1-39 |

Number of pages | 39 |

Journal | Proceedings of the London Mathematical Society |

Volume | 103 |

Issue number | 1 |

DOIs | |

State | Published - Jul 1 2011 |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Proceedings of the London Mathematical Society*,

*103*(1), 1-39. https://doi.org/10.1112/plms/pdq041