Invariant means on subspaces of quantum weakly almost periodic functionals

Let M be a Hopf–von Neumann algebra with the predual M_* and WAP(M) the subspace in M composed of weakly almost periodic functionals on M_*. The main example of such an algebra is \(M=L^{\infty}({\mathbb G})\) $$M=L^{\mathrm{\infty }}(\mathbb{G})$$ for a locally compact quantum group \(\mathbb G\) $$\mathbb{G}$$ . A space WAP_iso(M) inside WAP(M) is defined, and it is proved that it carries an invariant mean. It is currently the widest space known to admit invariant means in the quantum setting. In the case when M = L^∞(G) and G is a locally compact group, this space is equal to WAP(G).