An alternating moment condition for bi-freeness

TL;DR

This paper introduces an equivalent condition for bi-freeness based on alternating moments, extends it to amalgamated and conditional settings, and constructs a bi-free unitary Brownian motion that induces bi-freeness through conjugation.

Contribution

It provides a new characterization of bi-freeness via alternating moment conditions and constructs a bi-free unitary Brownian motion demonstrating asymptotic bi-freeness.

Findings

Equivalent condition for bi-freeness using alternating moments

Extension of characterization to amalgamated and conditional cases

Construction of a bi-free unitary Brownian motion that induces bi-freeness

Abstract

In this note we demonstrate an equivalent condition for bi-freeness, inspired by the well-known "vanishing of alternating centred moments" condition from free probability. We show that all products satisfying a centred condition on maximal monochromatic \chi-intervals have vanishing moments if and only if the family of pairs of faces they come from is bi-free, and show that similar characterisations hold for the amalgamated and conditional settings. In addition, we construct a bi-free unitary Brownian motion and show that conjugation by this process asymptotically creates bi-freeness; these considerations lead to another characterisation of bi-free independence.