Invariant means on subspaces of quantum weakly almost periodic functionals

Abstract

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 $M=L^{\infty }(\mathbb{G})$ 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 $M=L^{\infty }(G)$ and $G$ is a locally compact group, these spaces are equal to $WAP(G)$.

References in corpus (3)

• Admissibility Conjecture and Kazhdan's Property (T) for quantum groups
• Non-commutative separate continuity and weakly almost periodicity for Hopf von Neumann algebras
• Duals of quantum semigroups with involution