## Abstract

We study locally compact quantum groups G through the convolution algebras L_{1} (G) and (T(L2(G)), ⋊). We prove that the reduced quantum group C^{*}-algebra C_{0} (G) can be recovered from the convolution ⋊ by showing that the right T(L_{2}(G))-module (K(L_{2}(G)) ⋊ T(L_{2}(G))) is equal to C_{0}(G). On the other hand, we show that the left T(L _{2}(G))-module (T(L _{2}(G)) ⋊ K(L2(G))) is isomorphic to the reduced crossed product C_{0} (Ĝ)_{r} × C_{0} (G), and hence is a much larger C^{*}-subalgebra of B(L_{2} (G)). We establish a natural isomorphism between the completely bounded right multiplier algebras of L_{1}(G) and (T(L2(G)),⋊), and settle two invariance problems associated with the representation theorem of Junge-Neufang-Ruan (2009). We characterize regularity and discreteness of the quantum group G in terms of continuity properties of the convolution ⋊ on T(L _{2}(G)). We prove that if G is semi-regular, then the space (T(L_{2}(G)) ⋊ B(L_{2}(G))) of right G-continuous operators on L_{2}(G), which was introduced by Bekka (1990) for L_{∞}(G), is a unital C ^{*}-subalgebra of B(L_{2}(G)). In the representation framework formulated by Neufang-Ruan-Spronk (2008) and Junge-Neufang-Ruan, we show that the dual properties of compactness and discreteness can be characterized simultaneously via automatic normality of quantum group bimodule maps on B(L_{2}(G)). We also characterize some commutation relations of completely bounded multipliers of (T(L _{2}(G)), [⋊) over B(L_{2} (G)).

Original language | English (US) |
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Pages (from-to) | 1043-1072 |

Number of pages | 30 |

Journal | Canadian Journal of Mathematics |

Volume | 65 |

Issue number | 5 |

DOIs | |

State | Published - 2013 |

## Keywords

- Banach algebras
- Locally compact quantum groups and associated

## ASJC Scopus subject areas

- Mathematics(all)