Towards a classification of multi-faced independence: A representation-theoretic approach
Peer reviewed, Journal article
Accepted version
Permanent lenke
https://hdl.handle.net/11250/3139661Utgivelsesdato
2023Metadata
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Sammendrag
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 two-faced independences. Most noteworthy, for many known two-faced independences, we find that they admit continuous deformations within the class of two-faced independences, showing in particular that, in contrast with the single faced case, this class is infinite (and even uncountable).