On operator valued Haar unitaries and bipolar decompositions of R-diagonal elements
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 where is self-adjoint and is a partial isometry, and we study such decompositions of operator valued R-diagonal and circular elements that are free, meaning that and are *-free from each other. In particular, we prove, when B=C^2, that if a -valued circular element has a free bipolar decomposition with unitary, then it has one where normalizes .
Keywords
Cite
@article{arxiv.2306.17133,
title = {On operator valued Haar unitaries and bipolar decompositions of R-diagonal elements},
author = {Ken Dykema and John Griffin},
journal= {arXiv preprint arXiv:2306.17133},
year = {2024}
}
Comments
The revision includes a few minor changes in the first four sections and a rewriting of Section 5. The rewriting highlights the results that are valid for general B by presenting them as lemmas, and then proceeding to consider the case of two-dimensional B for the remaining calculations. This paper will appear in the journal Integral Equations and Operator Theory