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A continuous semigroup of notions of independence between the classical and the free one. (English) Zbl 1222.46049

Summary: 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 D. Voiculescu [Adv. Math. 146, No. 2, 101–166 (1999; Zbl 0956.46045)]. 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.

MSC:

46L54 Free probability and free operator algebras
15B52 Random matrices (algebraic aspects)
46L53 Noncommutative probability and statistics

Citations:

Zbl 0956.46045
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References:

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