Towards a classification of multi-faced independences: a combinatorial approach
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2023
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| _version_ | 1866916286319034368 |
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| author | Gerhold, Malte Varšo, Philipp |
| author_facet | Gerhold, Malte Varšo, Philipp |
| contents | We determine a set of necessary conditions on a partition-indexed family of complex numbers to be the "highest coefficients" of a positive and symmetric multi-faced universal product; i.e. the product associated with a multi-faced version of noncommutative stochastic independence, such as bifreeness. The highest coefficients of a universal product are the weights of the moment-cumulant relation for its associated independence. We show that these conditions are almost sufficient, in the sense that whenever the conditions are satisfied, one can associate a (automatically unique) symmetric universal product with the prescribed highest coefficients. Furthermore, we give a quite explicit description of such families of coefficients, thereby producing a list of candidates that must contain all positive symmetric universal products. We discover in this way four (three up to trivial face-swapping) previously unknown moment-cumulant relations that give rise to symmetric universal products; to decide whether they are positive, and thus give rise to independences which can be used in an operator algebraic framework, remains an open problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_01816 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Towards a classification of multi-faced independences: a combinatorial approach Gerhold, Malte Varšo, Philipp Functional Analysis Quantum Algebra 46L53 (Primary), 05A18, 60A05, 18M05, 46L54 (Secondary) We determine a set of necessary conditions on a partition-indexed family of complex numbers to be the "highest coefficients" of a positive and symmetric multi-faced universal product; i.e. the product associated with a multi-faced version of noncommutative stochastic independence, such as bifreeness. The highest coefficients of a universal product are the weights of the moment-cumulant relation for its associated independence. We show that these conditions are almost sufficient, in the sense that whenever the conditions are satisfied, one can associate a (automatically unique) symmetric universal product with the prescribed highest coefficients. Furthermore, we give a quite explicit description of such families of coefficients, thereby producing a list of candidates that must contain all positive symmetric universal products. We discover in this way four (three up to trivial face-swapping) previously unknown moment-cumulant relations that give rise to symmetric universal products; to decide whether they are positive, and thus give rise to independences which can be used in an operator algebraic framework, remains an open problem. |
| title | Towards a classification of multi-faced independences: a combinatorial approach |
| topic | Functional Analysis Quantum Algebra 46L53 (Primary), 05A18, 60A05, 18M05, 46L54 (Secondary) |
| url | https://arxiv.org/abs/2301.01816 |