REDUCTION OF FREE INDEPENDENCE TO TENSOR INDEPENDENCE

Abstract

In the hierarchy of freeness construction, free independence was reduced to tensor independence in the weak sense of convergence of moments. In this paper we show how to reduce free independence to tensor independence in the strong sense. We construct a suitable unital *-algebra of closed operators "affiliated" with a given unital *-algebra and call the associated closure "monotone". Then we prove that monotone closed operators of the form [Formula: see text] are free with respect to a tensor product state, where X(k) are tensor independent copies of a random variable X and (pk) is a sequence of orthogonal projections. For unital free *-algebras, we construct a monotone closed analog of a unital *-bialgebra called a "monotone closed quantum semigroup" which implements the additive free convolution, without using the concept of dual groups.