uniform continuity over locally compact quantum groupoids

0.1 Uniform Continuity over Locally Compact Quantum Groupoids

Let us consider Locally Compact Quantum GroupoidsPlanetmathPlanetmathPlanetmath (LCQGn) defined as locally compact groupoidsPlanetmathPlanetmath endowed with a Haar systemPlanetmathPlanetmath, ν, (𝒢,ν):=([𝔾,G2,μ],ν), or as derived from a (non-commutative) weak Hopf algebra (WHA), with the additional condition of uniform continuity over 𝔾 defined as follows . Let us also consider a space LUC(𝔾) of left uniformly continuous elements in L(𝔾) defined over G2, which is endowed with the induced product topology from the subset G2 of composable pairs in the topological groupoidPlanetmathPlanetmath 𝔾. This step completesPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath 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(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 functionalsMathworldPlanetmathPlanetmathPlanetmath on the Fourier algebra.) Also consider LUC(G) which is then an operator system containing the C*-algebraPlanetmathPlanetmath Co(G). One may compare the groupoidPlanetmathPlanetmathPlanetmath C*-convolution algebra, GCA – obtained in the general case– with the C*-algebra (http://planetmath.org/CAlgebra3) Co(G) obtained from LUC(G) in the particular case of uniform continuity over a locally compact (http://planetmath.org/LocallyCompact) group.


  • 1 M. Buneci. 2003. Groupoid RepresentationsPlanetmathPlanetmathPlanetmathPlanetmath, 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