Addition of certain non-commuting random variables.

*(English)*Zbl 0651.46063The 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 |

##### 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
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, () · Zbl 0204.16001 |

[4] | Courant, R; Hilbert, D, Partial differential equations, () · 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 On, 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, (), 141-209 |

[9] | Paschke, W; Salinas, N, Matrix algebras over On, 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, (), 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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.