Free convolution of measures with unbounded support.(English)Zbl 0806.46070

Some concepts of the noncommutative probability theory of free products are extended to the case of unbounded random variables by which are meant the operators affiliated with a finite von Neumann algebra with a normal faithful finite trace.
First of all, the notions of freeness, additive convolution, and multiplicative convolution are introduced for such variables, and it is shown that the last two have nice continuity properties.
Furthermore, all infinitely divisible distributions with respect to additive as well as to multiplicative free convolution are determined, and all stable distributions with respect to additive free convolution are found. In the course of analysis, the authors extend also analytical methods for the calculation of additive free convolution to the case of unbounded variables.

MSC:

 46L51 Noncommutative measure and integration 46L53 Noncommutative probability and statistics 46L54 Free probability and free operator algebras 60E07 Infinitely divisible distributions; stable distributions
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