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.

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 $C_o(G)$ .

Bibliography

1

V. Runde. 2008. Uniform continuity over locally compact quantum groups. (math.OA -arxiv/0802.2053v4).