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**Free random variables. A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups.**
*(English)*
Zbl 0795.46049

CRM Monograph Series. 1. Providence, RI: American Mathematical Society (AMS). v, 70 p. (1992).

The purpose of this book is to give an introduction to the free probability theory, which appeared as new branch of non-commutative probability theory about ten years ago. The principal difference of the present approach is using the notion of “freeness” instead of classical “independence”. It turns out that in these frameworks there is the possibility to define an analogue of Gaussian processes for free products of algebras, the analogues of convolution operator and Fourier transform and, moreover, to construct the free harmonic analysis. These topics cover the first three chapters of the book.

There are two nice applications of these results in the book (sections 4 and 5). The first one is the investigation of random matrices (precisely, the selfadjoint matrices having independent entries) based on the free Gaussian processes. The second one, which, perhaps, could be of interest not only for probabilists but also for specialists in the theory of von Neumann algebras, is the new construction of type \(II_ 1\) von Neumann factors (in fact, as some free product of algebras) which, together with hyperfinite ones, are the “best” among other type \(II_ 1\)-factors.

There are two nice applications of these results in the book (sections 4 and 5). The first one is the investigation of random matrices (precisely, the selfadjoint matrices having independent entries) based on the free Gaussian processes. The second one, which, perhaps, could be of interest not only for probabilists but also for specialists in the theory of von Neumann algebras, is the new construction of type \(II_ 1\) von Neumann factors (in fact, as some free product of algebras) which, together with hyperfinite ones, are the “best” among other type \(II_ 1\)-factors.

Reviewer: F.Sukochev (Tashkent)

### MSC:

46L51 | Noncommutative measure and integration |

46L53 | Noncommutative probability and statistics |

46L54 | Free probability and free operator algebras |

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

46L35 | Classifications of \(C^*\)-algebras |

46L10 | General theory of von Neumann algebras |

### Keywords:

freeness; free probability theory; non-commutative probability theory; Gaussian processes for free products of algebras; convolution operator; Fourier transform; free harmonic analysis; random matrices; free Gaussian processes; construction of type \(II_ 1\) von Neumann factors
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\textit{D. V. Voiculescu} et al., Free random variables. A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups. Providence, RI: American Mathematical Society (1992; Zbl 0795.46049)

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