On Operator Valued Haar Unitaries and Bipolar Decompositions of R-diagonal Elements

In the context of operator valued W \(^*\) ${}^{\ast }$ -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 \(*\) $\ast$ -free from each other. In particular, we prove, when \(B={\textbf{C}}^2\) $B={\mathbf{C}}^2$ , that if a B-valued circular element has a free bipolar decomposition with v unitary, then it has one where v normalizes B.