Schoenberg Correspondence on Dual Groups

TL;DR

This paper extends Schoenberg's correspondence to dual groups, showing that conditionally positive functionals generate convolution semigroups of states, thus enabling the construction of Lévy processes in this non-commutative setting.

Contribution

It proves that Schoenberg's correspondence holds for dual groups, facilitating the construction of Lévy processes from conditionally positive functionals in free probability.

Findings

Schoenberg correspondence is valid for dual groups.

Conditionally positive functionals generate convolution semigroups.

Enables Lévy process construction on dual groups.

Abstract

As in the classical case of L\'evy processes on a group, L\'evy processes on a Voiculescu dual group are constructed from conditionally positive functionals. It is essential for this construction that Schoenberg correspondence holds for dual groups: The exponential of a conditionally positive functional is a convolution semigroup of states.