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 asa particular case of locally compact quantum groupoids ($LCQ{G}_n$) when theobject space of the $LCQ{G}_n$ 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 $L^{\mathrm{\infty }}(G)$ 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 $L\mathit{}C\mathit{}G$, and with $G$ being a group, and also the essential data specified in the previous section, $L\mathit{}U\mathit{}C\mathit{}\mathrm{(}G\mathrm{)}$ is an operator system containing the C*-algebra $C_o\mathrm{}\mathrm{(}G\mathrm{)}$.