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- Title
Non-commutative stochastic independence and cumulants.
- Authors
Manzel, Sarah; Schürmann, Michael
- Abstract
In a fundamental lemma we characterize 'generating functions' of certain functors on the category of algebraic non-commutative probability spaces. Special families of such generating functions correspond to 'unital, associative universal products' on this category, which again define a notion of non-commutative stochastic independence. Using the fundamental lemma, we prove the existence of cumulants and of 'cumulant Lie algebras' for all independences coming from a unital, associative universal product. These include the five independences (tensor, free, Boolean, monotone, anti-monotone) appearing in Muraki's classification, c-free independence of Bożejko and Speicher, the indented product of Hasebe and the bi-free independence of Voiculescu. We show how the non-commutative independence can be reconstructed from its cumulants and cumulant Lie algebras.
- Subjects
NONCOMMUTATIVE differential geometry; CUMULANTS; STOCHASTIC processes; ALGEBRAIC spaces; LIE algebras
- Publication
Infinite Dimensional Analysis, Quantum Probability & Related Topics, 2017, Vol 20, Issue 2, p-1
- ISSN
0219-0257
- Publication type
Article
- DOI
10.1142/S0219025717500102