Infinitesimal non-crossing cumulants and free probability of type B

被引:19
作者
Fevrier, Maxime [2 ]
Nica, Alexandru [1 ]
机构
[1] Univ Waterloo, Dept Pure Math, Waterloo, ON N2L 3G1, Canada
[2] Univ Toulouse 3, Inst Math Toulouse, Equipe Stat & Probabil, F-31062 Toulouse 09, France
关键词
Free probability (of type B); Infinitesimal freeness; Infinitesimal non-crossing cumulants; Dual derivation system; PARTITIONS;
D O I
10.1016/j.jfa.2009.10.010
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
Free probabilistic considerations of type B first appeared in the paper of Biane, Good man and Nica [P. Biane, F. Goodman, A. Nica, Non-crossing cumulants of type B, Trans. Amer. Math. Soc. 355 (2003) 2263-2303]. Recently, connections between type B and infinitesimal free probability were put into evidence by Belinschi and Shylakhtenko [S.T. Belinschi, D. Shlyakhtenko, Free probability of type B: Analytic aspects and applications, preprint, 2009, available online at www.arxiv.org under reference arXiv:0903.2721]. The interplay between "type B" and "infinitesimal" is also the object of the present paper. We study infinitesimal freeness for a family of unital subalgebras A(1), ... , A(k) in an infinitesimal noncommutative probability space (A, phi, phi') and we introduce a concept of infinitesimal non-crossing cumulant functionals for (A, phi, phi'), obtained by taking a formal derivative in the formula for usual non-crossing cumulants. We prove that the infinitesimal freeness of A(1), ... , A(k) is equivalent to a vanishing condition for mixed cumulants; this gives the infinitesimal coounterpart for a theorem of Speicher from "usual" free probability. We show that the lattices NC(B)(n) of non-crossing partitions of type B appear in the combinatorial study of (A, phi, phi'), in the formulas for infinitesimal cumulants and when describing alternating products of infinitesimal free random variables. As an application of alternating free products, we observe the infinitesimal analogue for the well-known fact that freeness is preserved under compression with a free projection. As another application, we observe the infinitesimal analogue for a well-known procedure used to construct free families of free Poisson elements. Finally, we discuss situations when the freeness of A(1), ... , A(k) in (A, phi) can be naturally upgraded to infinitesimal freeness in (A, phi, phi'), for a suitable choice of a "companion functional" phi': A -> C. (C) 2009 Elsevier Inc. All rights reserved.
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页码:2983 / 3023
页数:41
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