Convolution semigroups on locally compact quantum groups and noncommutative Dirichlet forms

TL;DR

This paper explores convolution semigroups on locally compact quantum groups, establishing a correspondence with noncommutative Dirichlet forms, and applies this to characterize properties like Haagerup Property and Property (T).

Contribution

It provides a one-to-one correspondence between symmetric convolution semigroups and noncommutative Dirichlet forms, extending previous results and enabling new characterizations of quantum group properties.

Findings

Established a correspondence between symmetric convolution semigroups and Dirichlet forms.

Provided new characterizations of Haagerup Property and Property (T) for quantum groups.

Developed results on Haagerup's L^p-spaces relevant to the proofs.

Abstract

The subject of this paper is the study of convolution semigroups of states on a locally compact quantum group, generalising classical families of distributions of a L\'{e}vy process on a locally compact group. In particular a definitive one-to-one correspondence between symmetric convolution semigroups of states and noncommutative Dirichlet forms satisfying the natural translation invariance property is established, extending earlier partial results and providing a powerful tool to analyse such semigroups. This is then applied to provide new characterisations of the Haagerup Property and Property (T) for locally compact quantum groups, and some examples are presented. The proofs of the main theorems require developing certain general results concerning Haagerup's $L^{p}$-spaces.