Voiculescu, Dan Addition of certain non-commuting random variables. (English) Zbl 0651.46063 J. Funct. Anal. 66, 323-346 (1986). 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 Cited in 14 ReviewsCited in 207 Documents 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 Keywords:free family of elements of a non-commutative unital algebra; logarithm of the Fourier transform; distribution of the sum of a free non-commuting random variable; infinitely divisible probability measures; semigroups of measures in the non-commutative sense Citations:Zbl 0618.46048 PDFBibTeX XMLCite \textit{D. Voiculescu}, J. Funct. Anal. 66, 323--346 (1986; Zbl 0651.46063) Full Text: DOI References: [1] Akhiezer, N. J., The Classical Moment Problem (1961), [Russian] [2] Avitzour, D., Free products of \(C^∗\)-algebras, Trans. Amer. Math. Soc., 271, 423-435 (1982) · Zbl 0513.46037 [3] Clancey, K., Seminormal Operators, (Lecture Notes in Mathematics, Vol. 742 (1979), Springer-Verlag: Springer-Verlag Berlin/New York) · Zbl 0204.16001 [4] Courant, R.; Hilbert, D., Partial Differential Equations, (Methods of Mathematical Physics, Vol. 2 (1962), Interscience: Interscience New York) · Zbl 0729.35001 [5] Cuntz, J., Simple \(C^∗\)-algebras generated by isometries, Comm. Math. Phys., 57, 173-185 (1977) · Zbl 0399.46045 [6] Douglas, R. G., Banach algebra techniques in the theory of Toeplitz operators (1973) · Zbl 0252.47025 [7] Evans, D. E., On \(O_n\), Publ. Res. Inst. Math. Sci., 16, 915-927 (1980) · Zbl 0461.46042 [8] Helton, J. W.; Howe, R., Integral operators: Commutators, traces, index and homology, (Proceedings, Conference on Operator Theory. Proceedings, Conference on Operator Theory, Lecture Notes in Mathematics, Vol. 345 (1973), Springer-Verlag: Springer-Verlag Berlin/New York), 141-209 [9] Paschke, W.; Salinas, N., Matrix algebras over \(O_n\), Michigan Math. J., 26, 3-12 (1979) · Zbl 0412.46049 [10] Pimsner, M.; Popa, S., The Ext-groups of some \(C^∗\)-algebras considered by J. Cuntz, Rev. Roumaine Math. Pures Appl., 23, 1069-1076 (1978) · Zbl 0397.46056 [11] Voiculescu, D., Symmetries of some reduced free product \(C^∗\)-algebras, (Operator Algebras and their Connections with Topology and Ergodic Theory. Operator Algebras and their Connections with Topology and Ergodic Theory, Lecture Notes in Mathematics, Vol. 1132 (1985), Springer-Verlag: Springer-Verlag Berlin/New York), 556-588 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.