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February 2016 Free probability for pairs of faces II: $2$-variables bi-free partial $R$-transform and systems with rank $\le1$ commutation
Dan-Virgil Voiculescu
Ann. Inst. H. Poincaré Probab. Statist. 52(1): 1-15 (February 2016). DOI: 10.1214/14-AIHP623

Abstract

We compute the generating series for the simplest class of bi-free cumulants, beyond the free cumulants, the two-bands bi-free cumulants of a pair of a left and a right variable. We also consider two-faced systems with a commutation condition implying that two-bands moments, that is expectation values of the product of a monomial of left and a monomial of right variables determine the other moments. Examples include hyponormal operators, dual systems in free entropy theory and bi-partite systems.

Pour une paire de variables, une gauche et une droite, nous calculons la série génératrice des plus simples cumulants bi-libres au-delà des cumulants libres. On considère aussi des systèmes à deux faces, satisfaisant une condition de commutation qui assure que les moments à deux bandes, c’est-à-dire les espérances des produits d’un monôme de variables gauches et d’un monôme de variables droites, déterminent tous les autres moments non-commutatifs. Les opérateurs hyponormaux, les systèmes duaux de la théorie de l’entropie libre, ainsi que les systèmes bipartites, fournissent des exemples.

Citation

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Dan-Virgil Voiculescu. "Free probability for pairs of faces II: $2$-variables bi-free partial $R$-transform and systems with rank $\le1$ commutation." Ann. Inst. H. Poincaré Probab. Statist. 52 (1) 1 - 15, February 2016. https://doi.org/10.1214/14-AIHP623

Information

Received: 3 December 2013; Revised: 9 April 2014; Accepted: 27 April 2014; Published: February 2016
First available in Project Euclid: 6 January 2016

zbMATH: 1187.60010
MathSciNet: MR3449291
Digital Object Identifier: 10.1214/14-AIHP623

Subjects:
Primary: 46L54
Secondary: 47B20

Rights: Copyright © 2016 Institut Henri Poincaré

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Vol.52 • No. 1 • February 2016
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