Convolution semigroups of states

TL;DR

This paper develops a theory of convolution semigroups of states on quantum groups, establishing their properties and connections to C_0-semigroups, and applies these results to characterize certain functions on compact groups.

Contribution

It introduces a framework for weakly continuous convolution semigroups on C*-bialgebras, linking them to C_0-semigroups and analyzing their continuity properties.

Findings

Bijective correspondence between convolution semigroups and C_0-semigroups on quantum groups.

All weakly continuous convolution semigroups of states are norm-continuous on discrete C*-bialgebras.

Characterization of positive-definite functions on compact groups.

Abstract

Convolution semigroups of states on a quantum group form the natural noncommutative analogue of convolution semigroups of probability measures on a locally compact group. Here we initiate a theory of weakly continuous convolution semigroups of functionals on a C*-bialgebra, the noncommutative counterpart of locally compact semigroup. On locally compact quantum groups we obtain a bijective correspondence between such convolution semigroups and a class of C_0-semigroups of maps which we characterise. On C*-bialgebras of discrete type we show that all weakly continuous convolution semigroups of states are automatically norm-continuous. As an application we deduce a known characterisation of continuous conditionally positive-definite Hermitian functions on a compact group.