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Homelocally compact quantum groups: uniform continuity

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# locally compact quantum groups: uniform continuity

# 0.1 Uniform continuity over locally compact quantum groups (LCG)

One can consider locally compact quantum groups ($LCG$) 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 $LCG$ 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 $LCG$, 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. (math.OA -arxiv/0802.2053v4).

## Mathematics Subject Classification

81T05*no label found*57T05

*no label found*81R15

*no label found*22A22

*no label found*81R50

*no label found*54E15

*no label found*

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