# locally compact quantum groups: uniform continuity

## 0.1 Uniform continuity over locally compact quantum groups (LCG)

One can consider locally compact quantum groups^{} ($LCG$) to be defined as
a particular case of locally compact quantum groupoids^{} ($LCQ{G}_{n}$) when the
object space of the $LCQ{G}_{n}$ consists of just one object whose elements are, for example,
those of a (non-commutative) Hopf algebra^{}. This is also consistent with the definition introduced by Kustermans and Vaes for a locally compact quantum group by including a Haar measure system associated with the quantum group^{}.

## 0.2 Operator system containg the C*-algebra

Let us consider $LCG$ to be a locally compact quantum group. Then consider the space $LUC(G)$ of left uniformly continuous elements in ${L}^{\mathrm{\infty}}(G)$ introduced in ref. [1]. The definition according to V. Runde (loc. cit.) covers both the space of left uniformly continuous functions on a locally compact group and (Granirer’s) uniformly continuous functionals on the Fourier algebra.

## References

- 1 V. Runde. 2008. Uniform continuity over locally compact quantum groups. http://arxiv.org/PS_cache/arxiv/pdf/0802/0802.2053v4.pdf(math.OA -arxiv/0802.2053v4).

Title | locally compact quantum groups: uniform continuity |

Canonical name | LocallyCompactQuantumGroupsUniformContinuity |

Date of creation | 2013-03-22 18:21:14 |

Last modified on | 2013-03-22 18:21:14 |

Owner | bci1 (20947) |

Last modified by | bci1 (20947) |

Numerical id | 14 |

Author | bci1 (20947) |

Entry type | Example |

Classification | msc 81T05 |

Classification | msc 57T05 |

Classification | msc 81R15 |

Classification | msc 22A22 |

Classification | msc 81R50 |

Classification | msc 54E15 |

Synonym | uniform continuity over topological groups associated with Hopf algebras |

Related topic | QuantumGroups |

Related topic | C_cG |

Related topic | LocallyCompact |

Related topic | LocallyCompactGroupoids |

Related topic | WeakHopfCAlgebra2 |

Related topic | LocallyCompactQuantumGroup |

Defines | left uniformly continuous functions on a locally compact group |

Defines | uniformly continuous functionals on the Fourier algebra |