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

Voiculescu, Dan

J. Oper. Theory 18, No. 2, 223-235 (1987).

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

Keywords:

distribution of multiplication of non-commuting random variables; sequence of moments; characteristic function