Free Probability and Operators
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Date Time 
Location  Speaker 
Title – click for abstract 

03/22 4:00pm 
BLOC 304 
Ken Dykema TAMU 
On operatorvalued Rdiagonal and Haar unitary elements (Joint work with John Griffin)
Rdiagonal elements are naturally defined by conditions on
the free cumulants of the pair consisting of the element and its
adjoint. In the tracial, scalarvalued context, it is known (due to
pioneering work of Nica and Speicher) that being Rdiagonal is
equivalent to having the same *distribution as an element with a polar
decomposition z=uz, where u and z are *free and where u is a Haar
unitary. In the operatorvalued context (namely, Bvalued 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
operatorvalued setting. We will (1) examine different notions of
operatorvalued Haar unitary (2) introduce the notion of a free bipolar
decomposition and (3) discuss a specific result about free bipolar
decompositions of Bvalued circular elements (which are a very special
case of Bvalued Rdiagonal elements) when B is twodimensional. 
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