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 (non-commutative) weak Hopf algebra (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 ${L}^{\mathrm{\infty}}(\mathbb{G})$ 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 ${L}^{\mathrm{\infty}}(G)$ 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 $L\mathit{}U\mathit{}C\mathit{}\mathrm{(}G\mathrm{)}$ which is then an operator system containing the C*-algebra^{} ${C}_{o}\mathrm{}\mathrm{(}G\mathrm{)}$. One may compare the groupoid^{} C*-convolution algebra, ${G}_{C\mathit{}A}$ – obtained in the general case– with the C*-algebra (http://planetmath.org/CAlgebra3) ${C}_{o}\mathit{}\mathrm{(}G\mathrm{)}$ obtained from $L\mathit{}U\mathit{}C\mathit{}\mathrm{(}G\mathrm{)}$ in the particular case of uniform continuity over a locally compact (http://planetmath.org/LocallyCompact) group.
References
- 1 M. Buneci. 2003. Groupoid Representations^{}, 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 |