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Addition of certain non-commuting random variables. (English) Zbl 0651.46063
The notion of free family of elements of a non-commutative unital algebra A with a specified state $$\phi$$ was considered by the author in Lect. Notes Math. 1132, 556-588 (1985; Zbl 0618.46048). It was also shown that the distribution of the sum of a free pair of elements depends only on distributions of the elements of the pair. In this paper an analogue of the logarithm of the Fourier transform and the rule of addition in the non-commutative situation is established, which allows to compute the distribution of the sum of a free non-commuting random variable. On analogue of infinitely divisible probability measures and semigroups of measures in the non-commutative sense are considered and descriptions are obtained.
Reviewer: V.I.Ovchinnikov

##### MSC:
 46L51 Noncommutative measure and integration 46L53 Noncommutative probability and statistics 46L54 Free probability and free operator algebras 60E10 Characteristic functions; other transforms 60E07 Infinitely divisible distributions; stable distributions
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