On operator valued Haar unitaries and bipolar decompositions of R-diagonal elements
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866913199923658752 |
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| author | Dykema, Ken Griffin, John |
| author_facet | Dykema, Ken Griffin, John |
| contents | 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 $B$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_17133 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On operator valued Haar unitaries and bipolar decompositions of R-diagonal elements Dykema, Ken Griffin, John Operator Algebras 46L54 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 $B$. |
| title | On operator valued Haar unitaries and bipolar decompositions of R-diagonal elements |
| topic | Operator Algebras 46L54 |
| url | https://arxiv.org/abs/2306.17133 |