Non-commutative stochastic independence and cumulants

作者：Manzel Sarah^*; Schuermann Michael

来源：Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2017, 20(2): 1750010.

DOI：10.1142/S0219025717500102

• Quantum probability
• non-commutative stochastic independence
• cumulants
• monoidal categories

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 Bo. zejko 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.

• 出版日期2017-6