The Annals of Probability

A continuous semigroup of notions of independence between the classical and the free one

Florent Benaych-Georges and Thierry Lévy

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Abstract

In this paper, we investigate a continuous family of notions of independence which interpolates between the classical and free ones for noncommutative random variables. These notions are related to the liberation process introduced by Voiculescu. To each notion of independence correspond new convolutions of probability measures, for which we establish formulae and of which we compute simple examples. We prove that there exists no reasonable analogue of classical and free cumulants associated to these notions of independence.

Article information

Source
Ann. Probab., Volume 39, Number 3 (2011), 904-938.

Dates
First available in Project Euclid: 16 March 2011

Permanent link to this document
https://projecteuclid.org/euclid.aop/1300281728

Digital Object Identifier
doi:10.1214/10-AOP573

Mathematical Reviews number (MathSciNet)
MR2789579

Zentralblatt MATH identifier
1222.46049

Subjects
Primary: 46L54: Free probability and free operator algebras 15A52

Keywords
Free probability independence random matrices unitary Brownian motion convolution cumulants

Citation

Benaych-Georges, Florent; Lévy, Thierry. A continuous semigroup of notions of independence between the classical and the free one. Ann. Probab. 39 (2011), no. 3, 904--938. doi:10.1214/10-AOP573. https://projecteuclid.org/euclid.aop/1300281728


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References

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