Separation properties for positive-definite functions on locally compact quantum groups and for associated von Neumann algebras

TL;DR

This paper establishes strong separation results for quantum positive-definite functions on locally compact quantum groups, linking amenability to approximation properties of associated von Neumann algebras.

Contribution

It introduces new separation theorems using Godement mean, strengthening the connection between quantum group amenability and positive-definite function approximation.

Findings

Strong separation results for quantum positive-definite functions

Von Neumann algebras of unimodular discrete quantum groups have the matrix ε-separation property

Enhanced understanding of non-w*CPAP in quantum group von Neumann algebras

Abstract

Using Godement mean on the Fourier-Stieltjes algebra of a locally compact quantum group we obtain strong separation results for quantum positive-definite functions associated to a subclass of representations, strengthening for example the known relationship between amenability of a discrete quantum group and existence of a net of finitely supported quantum positive-definite functions converging pointwise to $I$. We apply these results to show that von Neumann algebras of unimodular discrete quantum groups enjoy a strong form of non-$w^*$-CPAP, which we call the matrix $\epsilon$-separation property.