## Multiplication of certain non-commuting random variables.(English)Zbl 0662.46069

Let (A,$$\phi)$$ be a unital algebra over $${\mathbb{C}}$$ with a state $$\phi$$ :A$$\to {\mathbb{C}}$$, $$\phi (1)=1$$. A pair of subalgebras $$A_ 1$$ and $$A_ 2$$ is called free if for any $$a_ 1\in A_ 1$$, $$a_ 2\in A_ 2$$, $$\phi (a_ 1)=0$$ $$\phi (\alpha_ 2)=0$$ implies $$\phi (a_ 1a_ 2)=0$$. A pair of elements a, b is called free if subalgebras generated by a and b are free. An element a is considered as “random variable”, which generates the sequence of moments $$\mu (x^ n)=\phi (a^ n)$$. An analogue of the characteristic function $$S_ a(z)$$ is defined as follows: $$S_ a(z)$$ is a formal series such that $$S_ a(z)=\chi (z)z^{-1}(1+z)$$, where $$\chi (\psi (z))=z$$ and $$\psi (z)=\sum_{n\geq 1}\phi (a^ n)z^ n$$. It is shown that $$S_{ab}(z)=S_ a(z)\cdot S_ b(z)$$ if $$\phi$$ (a)$$\neq 0$$ and $$\phi$$ (b)$$\neq 0$$. The computation of the moments of ab with the help of moments of a and b in general situation is also considered.
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