## Independence and Convolutions in Noncommutative Probability

**From** 2013-04-01 **to** 2015-03-31, Grant Agreement terminated

## Project details

### Total cost:

EUR 194 046,6

### EU contribution:

EUR 194 046,6

### Coordinated in:

France
## Objective

"Noncommutative probability, also called quantum probability or algebraic

probability theory, is an extension of classical probability theory where the

algebra of random variables is replaced by a possibly noncommutative

algebra. A surprising feature of noncommutative probability is the existence

of many very different notions of independence. The most prominent among them

is freeness or free probability, which was introduced by Voiculescu to study

questions in operator algebra theory. In the last twenty-five years, free

probability has turned into a very active and very competitive research area,

in which analogues for many important probabilistic notions like limit

theorems, infinite divisibility, and L\'evy processes have been discovered. It

also turned out to be closely related to random matrix theory, which has

important applications in quantum physics and telecommunication.

The current project proposes to study the mathematical theory of independence

in noncommutative probability, and the associated convolution products. We

will concentrate on the following topics:

(1) Applications of monotone independence to free probability. Some

applications have been found already, but recent work indicates that much more

is possible.

(2) Analysis of infinitely divisible distributions in classical and free

probability. Common complex analysis methods will be used for both classes,

and we expect more insight into their mutual relations.

(3) Application and development of Lenczewski's matricial free

independence. This concept introduces very new ideas, whose better

understanding will certainly lead to new interesting results.

The methods we will use in this project come not only from noncommutative

probability, but also from functional analysis, complex analysis, combinatorics, classical probability, random matrices, and graph theory."

## Related information

## Coordinator

UNIVERSITE DE FRANCHE-COMTE

France

**EU contribution: **EUR 194 046,6

1 RUE CLAUDE GOUDIMEL

25000 BESANCON

France

**Record Number**: 108064 / **Last updated on**: 2015-03-11
**Last updated on** 2015-03-11