Free Probability and Operators

Date: March 22, 2024

Time: 4:00PM - 5:00PM

Location: BLOC 304

Speaker: Ken Dykema, TAMU

Title: On operator-valued R-diagonal and Haar unitary elements (Joint work with John Griffin)

Abstract: R-diagonal elements are naturally defined by conditions on the free cumulants of the pair consisting of the element and its adjoint. In the tracial, scalar-valued context, it is known (due to pioneering work of Nica and Speicher) that being R-diagonal is equivalent to having the same *-distribution as an element with a polar decomposition z=u|z|, where u and |z| are *-free and where u is a Haar unitary. In the operator-valued context (namely, B-valued where B is an operator algebra), this is no longer the case. Freeness need not occur, and even notions of Haar unitary are more complicated in the operator-valued setting. We will (1) examine different notions of operator-valued Haar unitary (2) introduce the notion of a free bipolar decomposition and (3) discuss a specific result about free bipolar decompositions of B-valued circular elements (which are a very special case of B-valued R-diagonal elements) when B is two-dimensional.