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arXiv · 1611.01262

An alternating moment condition for bi-freeness

Ian Charlesworth
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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 χ-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.

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Ian Charlesworth. 2019-02-08. An alternating moment condition for bi-freeness. https://doi.org/10.1016/j.aim.2019.01.044

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