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 asa particular case of locally compact quantum groupoids ( $LCQG_{n}$ ) when theobject 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).

单词	LocallyCompactQuantumGroupsUniformContinuity
释义	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 asa particular case of locally compact quantum groupoids ( $LCQG_{n}$ ) when theobject 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).
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