On Operator Valued Haar Unitaries and Bipolar Decompositions of R-diagonal Elements

Pub Date : 2024-02-26 DOI:10.1007/s00020-024-02760-z

引用次数: 0

Abstract

In the context of operator valued W \(^*\) -free probability theory, we study Haar unitaries, R-diagonal elements and circular elements. Several classes of Haar unitaries are differentiated from each other. The term bipolar decomposition is used for the expression of an element as vx where x is self-adjoint and v is a partial isometry, and we study such decompositions of operator valued R-diagonal and circular elements that are free, meaning that v and x are \(*\) -free from each other. In particular, we prove, when \(B={\textbf{C}}^2\) , that if a B-valued circular element has a free bipolar decomposition with v unitary, then it has one where v normalizes B.

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论R对角元素的算子值哈尔单元和双极分解

摘要在无算子值 W （^*\）概率论的背景下，我们研究了哈尔单元、R 对角元素和圆元素。我们将几类 Haar 单元相互区分开来。双极分解这一术语用于表达一个元素为 vx，其中 x 是自共轭的，v 是部分等距的，我们研究了算子值 R 对角元素和圆元素的这种分解，它们是自由的，这意味着 v 和 x 彼此是 \(*\) -free 的。特别是，我们证明，当 \(B={\textbf{C}}^2\) 时，如果一个 B 值圆周元素有一个自由的双极分解，且 v 是单元的，那么它就有一个 v 使 B 正常化的双极分解。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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