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The present material addresses several problems left open in the Trans. AMSpaper " Non-crossing cumulants of type B" of P. Biane, F. Goodman and A. Nica.The main result is that a type B non-commutative probability space can bestudied in the framework of freeness with amalgamation. This view allows easyways of constructing a version of the S-transform as well as proving analogueresults to Central Limit Theorem and Poisson Limit Theorem.

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