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arXiv · 2306.17133

On operator valued Haar unitaries and bipolar decompositions of R-diagonal elements

Ken Dykema·John Griffin
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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=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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Ken Dykema, John Griffin. 2023-06-29. On operator valued Haar unitaries and bipolar decompositions of R-diagonal elements. https://arxiv.org/abs/2306.17133

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