locally compact quantum groups: uniform continuity


0.1 Uniform continuity over locally compact quantum groups (LCG)

One can consider locally compact quantum groupsPlanetmathPlanetmath (LCG) to be defined as a particular case of locally compact quantum groupoidsPlanetmathPlanetmath (LCQGn) when the object space of the LCQGn consists of just one object whose elements are, for example, those of a (non-commutative) Hopf algebraPlanetmathPlanetmath. 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 groupPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath.

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(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 sectionPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, LUC(G) is an operator system containing the C*-algebraPlanetmathPlanetmath Co(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