On positive definiteness over locally compact quantum groups

TL;DR

This paper extends classical results on positive-definite functions from groups to the setting of locally compact quantum groups, providing new theorems and characterizations in the quantum framework.

Contribution

It generalizes key properties and theorems of positive-definite functions to locally compact quantum groups, advancing the mathematical understanding of quantum harmonic analysis.

Findings

Generalized theorems on square roots of positive-definite functions

Compared topologies for positive-definite functions in quantum groups

Characterized amenability and separation properties in quantum setting

Abstract

The notion of positive-definite functions over locally compact quantum groups was recently introduced and studied by Daws and Salmi. Based on this work, we generalize various well-known results about positive-definite functions over groups to the quantum framework. Among these are theorems on "square roots" of positive-definite functions, comparison of various topologies, positive-definite measures and characterizations of amenability, and the separation property with respect to compact quantum subgroups.