Invariant means on weakly almost periodic functionals with application to quantum groups

Let ${\mathcal A}$ be a Banach algebra, and let $\varphi $ be a nonzero character on ${\mathcal A}$ . For a closed ideal I of ${\mathcal A}$ with $I\not \subseteq \ker \varphi $ such that I has a bounded approximate identity, we show that $\operatorname {WAP}(\mathcal {A})$ , the space of weakly almost periodic functionals on ${\mathcal A}$ , admits a right (left) invariant $\varphi $ -mean if and only if $\operatorname {WAP}(I)$ admits a right (left) invariant $\varphi |_I$ -mean. This generalizes a result due to Neufang for the group algebra $L^1(G)$ as an ideal in the measure algebra $M(G)$ , for a locally compact group G. Then we apply this result to the quantum group algebra $L^1({\mathbb G})$ of a locally compact quantum group ${\mathbb G}$ . Finally, we study the existence of left and right invariant $1$ -means on $ \operatorname {WAP}(\mathcal {T}_{\triangleright }({\mathbb G}))$ .