Invariant means on subspaces of quantum weakly almost periodic functionals

Let $M$ be a Hopf–von Neuman algebra with the predual $M_{\ast }$ and $WAP(M)$ the subspace in $M$ composed of weakly almost periodic functionals on $M_{\ast }$. The main example of such an algebra is $M=L^{\mathrm{\infty }}(\mathbb{G})$ for a locally compact quantum group $\mathbb{G}$. We define a pair of left/right spaces $WA{P}_{iso,l}(M)$ and $WA{P}_{iso,r}(M)$ inside $WAP(M)$ and prove that they carry invariant means. These spaces are currently the widest known to admit invariant means in the quantum setting. In the case when $M=L^{\mathrm{\infty }}(G)$ and $G$ is a locally compact group, these spaces are equal to $WAP(G)$.