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Free probabilistic considerations of type B first appeared in a paper byBiane, Goodman and Nica in 2003. Recently, connections between type B andinfinitesimal free probability were put into evidence by Belinschi andShlyakhtenko (arXiv:0903.2721). The interplay between "type B" and"infinitesimal" is also the object of the present paper. We study infinitesimalfreeness for a family of unital subalgebras A_1, ..., A_k in an infinitesimalnoncommutative probability space (A, phi, phi'), and we introduce a concept ofinfinitesimal non-crossing cumulant functionals for (A, phi, phi'), obtained bytaking a formal derivative in the formula for usual non-crossing cumulants. Weprove that the infinitesimal freeness of A_1, ... A_k is equivalent to avanishing condition for mixed cumulants; this gives the infinitesimalcounterpart for a theorem of Speicher from "usual" free probability. We showthat the lattices of non-crossing partitions of type B appear in thecombinatorial study of (A, phi, phi'), in the formulas for infinitesimalcumulants and when describing alternating products of infinitesimally freerandom variables. As an application of alternating free products, we observethe infinitesimal analogue for the well-known fact that freeness is preservedunder compression with a free projection. As another application, we observethe infinitesimal analogue for a well-known procedure used to construct freefamilies of free Poisson elements. Finally, we discuss situations when thefreeness of A_1, ..., A_k in (A, phi) can be naturally upgraded toinfinitesimal freeness in (A, phi, phi'), for a suitable choice of a "companionfunctional" phi'.

Infinitesimal non-crossing cumulants and free probability of type B