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首页> 外文期刊>Infinite dimensional analysis, quantum probability, and related topics >REDUCTION OF FREE INDEPENDENCE TO TENSOR INDEPENDENCE
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REDUCTION OF FREE INDEPENDENCE TO TENSOR INDEPENDENCE

机译：将自由独立性降低为张量独立性

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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 X' = Σ from x = 1 to ∞ of X(k) (direct X)-bar p_k, X" = Σ from x = 1 to ∞ of p_k (direct X)-bar 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.
机译：在自由性构建的层次结构中，在瞬间收敛的弱意义上，自由独立性降低为张量独立性。在本文中，我们将展示如何从强烈的意义上将自由独立性降低为张量独立性。我们构造与给定的单位*-代数“关联”的封闭运算符的合适单位*-代数，并将关联的闭包称为“单调”。然后，我们证明，从x = 1到X（k）（直接X）-bar p_k的X = 1到∞，形式为X'=Σ的单调封闭算符，从x = 1到p_k（直接X）-∞的X“ =Σ X（k）相对于张量积状态是自由的，其中X（k）是随机变量X的与张量无关的副本，而（p_k）是正交投影的序列。对于单位自由*-代数，我们构造*单一双线性代数的单调闭环类似物，称为“单调闭环量子半群”，可实现加法自由卷积，而无需使用双群的概念。

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