uniform continuity over locally compact quantum groupoids

0.1 Uniform Continuity over Locally Compact Quantum Groupoids

Let us consider Locally Compact Quantum Groupoids ( $L C Q G n$ ) 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 $L^{\infty}({\mathbb{G}})$ 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 $L C Q G n$ 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 $L C G$ 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. 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