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

Ken Dykema, John Griffin

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 $B$.

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 .
Paper Structure (7 sections, 14 theorems, 99 equations)

This paper contains 7 sections, 14 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Background and Notation
  3. Classes of $B$-valued Haar unitaries
  4. Bipolar decompositions
  5. ${\mathbf C}^2$-valued circular elements
  6. Open Questions
  7. Declaration

Key Result

Theorem 2.9

Let $a\in A$. The following are equivalent:

Theorems & Definitions (41)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Theorem 2.9: BD-paper, Theorem 3.1
  • Definition 2.10
  • ...and 31 more