Towards a Classification of Multi-Faced Independence: A Representation-Theoretic Approach

TL;DR

This paper explores the classification of multi-faced independence in non-commutative probability, introducing universal lifts and revealing new examples and continuous deformations of 2-faced independences.

Contribution

It formalizes the concept of universal lifts for constructing multi-faced independences and classifies these lifts for tensor and free products, uncovering new examples.

Findings

Universal lifts can construct well-behaved multi-faced independences.

Complete classification of lifts for tensor and free products.

Existence of uncountably many continuous deformations of 2-faced independences.

Abstract

We attack the classification problem of multi-faced independences, the first non-trivial example being Voiculescu's bi-freeness. While the present paper does not achieve a complete classification, it formalizes the idea of lifting an operator on a pre-Hilbert space in a "universal" way to a larger product space, which is key for the construction of (old and new) examples. It will be shown how universal lifts can be used to construct very well-behaved (multi-faced) independences in general. Furthermore, we entirely classify universal lifts to the tensor product and to the free product of pre-Hilbert spaces. Our work brings to light surprising new examples of 2-faced independences. Most noteworthy, for many known 2-faced independences, we find that they admit continuous deformations within the class of 2-faced independences, showing in particular that, in contrast with the single faced case, this class is infinite (and even uncountable).