2023/01/04 by Malte Gerhold, Gerhold, Malte, Philipp Varšo +1
Computer Science · Mathematics · #05A18 #18M05 #46L53 (Primary) #46L54 (Secondary) #60A05 #Advanced Algebra and Geometry #Advanced Algebra and Logic #FOS: Mathematics #Functional Analysis (math.FA) #Quantum Algebra (math.QA) #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.2301.01816
openalex publication_date 2023/01/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.