Fej\'er representations for discrete quantum groups and applications

TL;DR

This paper establishes a connection between the approximation property of discrete quantum groups and Fejér-type representations of their associated algebras, extending known results and providing new characterizations in the quantum setting.

Contribution

It proves the equivalence between the approximation property and Fejér representations for discrete quantum groups and extends several classical results to this quantum context.

Findings

Characterization of invariant bimodules in quantum groups

Extension of classical results to quantum groups with approximation property

Analysis of Fubini crossed products in the quantum setting

Abstract

We prove that a discrete quantum group $\mathbb{G}$ has the approximation property if and only if a Fej\'{e}r-type representation holds for its $C^*$-algebraic or von Neumann algebraic crossed products. As applications, we extend several results from the literature to the context of discrete quantum groups with the approximation property. Additionally, we provide new characterizations of invariant $L^\infty(\widehat{\mathbb{G}})$-bimodules of $\mathcal{B}(\ell^2(\mathbb{G}))$ and invariant $C(\widehat{\mathbb{G}})$-bimodules of $\mathcal{K}(\ell^2(\mathbb{G}))$, some of which are new in the group setting. Finally, we study Fubini crossed products of discrete quantum group actions.