Invariant means on subspaces of quantum weakly almost periodic functionals

TL;DR

This paper introduces new subspaces within weakly almost periodic functionals on Hopf-von Neumann algebras, demonstrating they support invariant means, thus extending the understanding of invariant means in quantum group settings.

Contribution

It defines and analyzes the spaces $WAP_{iso,l}(M)$ and $WAP_{iso,r}(M)$, proving they admit invariant means and are the broadest known such spaces in the quantum context.

Findings

Spaces $WAP_{iso,l}(M)$ and $WAP_{iso,r}(M)$ carry invariant means.

These spaces are the widest known to admit invariant means in quantum groups.

In the classical case, these spaces coincide with $WAP(G)$.

Abstract

Let $M$ be a Hopf--von Neuman 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 $M=L^\infty(\mathbb G)$ for a locally compact quantum group $\mathbb G$. We define a pair of left/right spaces $WAP_{iso,l}(M)$ and $WAP_{iso,r}(M)$ 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 $WAP(G)$.