A continuous semigroup of notions of independence between the classical and the free one

TL;DR

This paper introduces a continuous family of independence notions bridging classical and free independence in non-commutative probability, exploring their properties, convolutions, and limitations of cumulant analogues.

Contribution

It defines a new continuum of independence concepts, analyzes their associated convolutions, and demonstrates the absence of classical cumulant analogues for these notions.

Findings

Established formulae for new convolutions

Computed simple examples of these convolutions

Proved no reasonable cumulant analogue exists for these notions

Abstract

In this paper, we investigate a continuous family of notions of independence which interpolates between the classical and free ones for non-commutative random variables. These notions are related to the liberation process introduced by D. Voiculescu. To each notion of independence correspond new convolutions of probability measures, for which we establish formulae and of which we compute simple examples. We prove that there exists no reasonable analogue of classical and free cumulants associated to these notions of independence.