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A new proof for the multiplicative property of the boolean cumulants with applications to the operator-valued case

Mihai Popa — 2009

Colloquium Mathematicae

The paper presents several combinatorial properties of the boolean cumulants. A consequence is a new proof of the multiplicative property of the boolean cumulant series that can be easily adapted to the case of boolean independence with amalgamation over an algebra.

Freeness with amalgamation, limit theorems and S-transform in non-commutative probability spaces of type B

Mihai Popa — 2010

Colloquium Mathematicae

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.

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