Reduction of free independence to tensor independence

Details Reduction of free independence to tensor independence Reduction of free independence to tensor independence

2002-10-23

arxiv.org/abs/math/0210358v2

Inf. Dim. Anal. Quant. Probab. Rel. Topics Vol. 7 (2004), 337-360

Mathematics

Quantum Algebra

23 pages, latex, no figures, new abstract and introduction

Scientific 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 $$ X'= \sum_{k=1}^{\infty}X(k)\bar{\otimes} p_{k}, X''=\sum_{k=1}^{\infty} p_{k}\bar{\otimes}X(k) $$ are free with respect to a tensor product state, where $X(k)$ are tensor independent copies of a random variable $X$ and $(p_{k})$ 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.

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