Freeness with amalgamation, limit theorems and $S$-transform in non-commutative probability spaces of type B
Volume 120 / 2010
Colloquium Mathematicum 120 (2010), 319-329 MSC: 46L54, 46L53. DOI: 10.4064/cm120-2-11
The paper addresses several problems left open by P. Biane, F. Goodman and A. Nica [Trans. Amer. Math. Soc. 355 (2003)]. The main result is that a type B non-commutative probability space can be studied in the framework of freeness with amalgamation. This view allows easy ways of constructing a version of the $S$-transform as well as proving analogues to the Central Limit Theorem and Poisson Limit Theorem.