locally compact quantum groups: uniform continuity

0.1 Uniform continuity over locally compact quantum groups (LCG)

One can consider locally compact quantum groups ( $L C G$ ) to be defined as a particular case of locally compact quantum groupoids ( $LCQG_{n}$ ) when the object space of the $LCQG_{n}$ consists of just one object whose elements are, for example, those of a (non-commutative) Hopf algebra. 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 group.

0.2 Operator system containg the C*-algebra

Let us 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. [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 $L C G$ , and with $G$ being a group, and also the essential data specified in the previous section, $LUC(G)$ is an operator system containing the C*-algebra $C_{o}(G)$ .

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