2023/06/29 by Ken Dykema, Dykema, Ken, John S. Griffin +1
Mathematics · #46L54 #FOS: Mathematics #Mathematical Analysis and Transform Methods #Operator Algebras (math.OA) #Random Matrices and Applications #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2306.17133
openalex publication_date 2023/06/29 · openalex created_date 2023/07/01 · openalex updated_date 2026/07/28
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=C2, that if a B-valued circular element has a free bipolar decomposition with v unitary, then it has one where v normalizes B.