# Positive definite functions on Coxeter groups with applications to operator spaces and noncommutative probability

## TL;DR

This paper introduces a new class of positive definite functions on Coxeter groups, extending Riesz products, with applications to harmonic analysis, operator spaces, and noncommutative probability.

## Contribution

It develops the Riesz-Coxeter product, extending positive definite functions from Abelian to arbitrary Coxeter groups, and characterizes radial functions on specific groups.

## Key findings

- Defined the Riesz-Coxeter product for Coxeter groups.
- Characterized radial functions on dihedral and permutation groups.
- Applied the theory to harmonic analysis and noncommutative probability.

## Abstract

A new class of positive definite functions related to colour-length function on arbitrary Coxeter group is introduced. Extensions of positive definite functions, called the Riesz-Coxeter product, from the Riesz product on the Rademacher (Abelian Coxeter) group to arbitrary Coxeter group is obtained. Applications to harmonic analysis, operator spaces and noncommutative probability is presented. Characterization of radial and colour-radial functions on dihedral groups and infinite permutation group are shown.