Fejér representations for discrete quantum groups and applications

TL;DR

The paper proves that for a discrete quantum group $\mathbb{G}$, the approximation property (AP) is equivalent to the existence of Fejér-type representations for elements in both $C^*$- and von Neumann algebraic crossed products. The authors extend Fejér representation techniques from locally compact groups to discrete quantum groups, using nets in $L^1(\widehat{\mathbb{G}})$ and operator-valued weights to obtain convergent decompositions. They derive new characterizations of invariant $L^{\infty}(\widehat{\mathbb{G}})$-bimodules in $\mathcal{B}(\ell^2(\mathbb{G}))$ and invariant $C(\widehat{\mathbb{G}})$-bimodules in $\mathcal{K}(\ell^2(\mathbb{G}))$, linking them to left ideals $J$ via $\mathrm{Bim}(J^{\perp})$ and $\mathrm{Ran}(J)$, and identifying jointly invariant subspaces with notions of $\Sigma$-harmonic spaces. The Fejér framework is then used to study Fubini crossed products and to establish a slice map property for actions of discrete quantum groups with AP, in both the von Neumann and C*-algebraic settings. Collectively, these results extend and unify Fourier-analytic and amenability-type phenomena in quantum group crossed products, with potential implications for Galois correspondences and harmonic analysis in the quantum group context.

Abstract

We prove that a discrete quantum group $\mathbb{G}$ has the approximation property if and only if a Fejér-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.

Theorems & Definitions (85)

• Theorem 2.1
• Remark 2.2
• Remark 2.3
• Definition 2.4
• Proposition 2.5
• Remark 2.6
• Definition 2.7
• Theorem 2.8
• Remark 2.9
• Definition 2.10