  We define, for a locally compact quantum group $G$ in the sense of
Kustermans--Vaes, the space of $LUC(G)$ of left uniformly continuous elements
in $L^\infty(G)$. This definition covers both the usual left uniformly
continuous functions on a locally compact group and Granirer's uniformly
continuous functionals on the Fourier algebra. We show that $LUC(G)$ is an
operator system containing the $C^*$-algebra $C_0(G)$ and contained in its
multiplier algebra $M(C_0(G))$. We use this to partially answer an open problem
by Bedos--Tuset: if $G$ is co-amenable, then the existence of a left invariant
mean on $M(C_0(G))$ is sufficient for $G$ to be amenable. Furthermore, we study
the space $WAP(G)$ of weakly almost periodic elements of $L^\infty(G)$: it is a
closed operator system in $L^\infty(G)$ containing $C_0(G)$ and--for
co-amenable $G$--contained in $LUC(G)$. Finally, we show that--under certain
conditions, which are always satisfied if $G$ is a group--the operator system
$LUC(G)$ is a $C^*$-algebra.
