## Differential independence via an associative product of infinitely many linear functionals

### Volume 124 / 2011

Colloquium Mathematicum 124 (2011), 79-94
MSC: Primary 46L53; Secondary 46L54.
DOI: 10.4064/cm124-1-6

#### Abstract

We generalize the infinitesimal independence appearing in free probability of type B in two directions: to higher order derivatives and other natural independences: tensor, monotone and Boolean. Such generalized infinitesimal independences can be defined by using associative products of infinitely many linear functionals, and therefore the associated cumulants can be defined. These products can be seen as the usual natural products of linear maps with values in formal power series.