# uniform continuity over locally compact quantum groupoids

## 0.1 Uniform Continuity over Locally Compact Quantum Groupoids

Let us consider Locally Compact Quantum Groupoids ($LCQGn$) defined as locally compact groupoids endowed with a Haar system, $\nu$, $(\mathcal{G},\nu):=([{\mathbb{G}},G_{2},\mu],\nu)$, or as derived from a (WHA), with the additional condition of uniform continuity over ${\mathbb{G}}$ defined as follows . Let us also consider a space $LUC({\mathbb{G}})$ of left uniformly continuous elements in $L^{\infty}({\mathbb{G}})$ defined over $G_{2}$, which is endowed with the induced product topology from the subset $G^{2}$ of composable pairs in the topological groupoid ${\mathbb{G}}$. This step completes the construction of uniform continuity over $LCQGn$ that can be then compared with the results obtained from ‘quantum groupoids’ derived from a weak Hopf algebra.

### 0.1.1 C*-algebra Comparison and Example

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. [2]. (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.) Also consider $LUC(G)$ which is then an operator system containing the C*-algebra $C_{o}(G)$. One may compare the groupoid C*-convolution algebra, $G_{CA}$ – obtained in the general case– with the C*-algebra (http://planetmath.org/CAlgebra3) $C_{o}(G)$ obtained from $LUC(G)$ in the particular case of uniform continuity over a locally compact (http://planetmath.org/LocallyCompact) group.

## References

• 1 M. Buneci. 2003. , Publs: Ed. Mirton, Timishoara.
• 2 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 uniform continuity over locally compact quantum groupoids Canonical name UniformContinuityOverLocallyCompactQuantumGroupoids Date of creation 2013-03-22 18:21:21 Last modified on 2013-03-22 18:21:21 Owner bci1 (20947) Last modified by bci1 (20947) Numerical id 24 Author bci1 (20947) Entry type Topic Classification msc 46L05 Classification msc 81R50 Classification msc 46M20 Classification msc 18B40 Classification msc 22A22 Related topic QuantumGroups Related topic QuantumGroupoids2 Related topic LocallyCompact Related topic LocallyCompactGroupoids Related topic UniformContinuity Related topic RepresentationsOfLocallyCompactGroupoids Related topic HomomorphismsOfCAlgebrasAreContinuous Related topic CAlgebra Defines uniform continuity over topological groupoids