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

TL;DR

This paper explores the structure of operator valued R-diagonal and circular elements in free probability, focusing on bipolar decompositions and conditions under which these elements can be expressed with certain properties.

Contribution

It introduces new results on bipolar decompositions of operator valued R-diagonal and circular elements, especially when the unitary component normalizes the algebra.

Findings

Differentiates classes of Haar unitaries in operator valued free probability.

Establishes conditions for bipolar decompositions with unitary components that normalize the algebra.

Proves that for B=C^2, circular elements with free bipolar decompositions can have unitary parts that normalize B.

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 $B$.