# 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 ($LCQG_{n}$) when the object space of the $LCQG_{n}$ consists of just one object whose elements are, for example, those of a 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 $L^{\infty}(G)$ 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.

With the above definition of $LCG$, and with $G$ being a group, and also the essential data specified in the previous section, $LUC(G)$ is an operator system containing the C*-algebra $C_{o}(G)$.

## 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