2015/07/09 by John D. Williams, Williams, John D.
Mathematics · #46E40 #46L54 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #Probability (math.PR) #math.FA #math.OA #math.PR #msc:46E40 #msc:46L54
paper · pdf · doi:10.48550/arxiv.1507.02580
Several small errors corrected. Streamlined proofs. To be published in Indiana University Journal or Mathematics
arxiv created 2015/12/17 · arxiv updated 2015/12/18
Consider the B-valued probability space (A, E, B), where A is a tracial von Neumann algebra. We extend the theory of operator valued free probability to the algebra of affiliated operators A. For a random variable X ∈ Asa we study the Cauchy transform GX and show that the operator algebra (B ∪ \X\)" can be recovered from this function. In the case where B is finite dimensional, we show that, when X, Y ∈ Asa are assumed to be B-free, the R-transforms are defined on universal subsets of the resolvent and satisfy RX + RY = RX + Y. Examples indicating a failure of the theory for infinite dimensional B are provided. Lastly, we show that the class of functions that arise as the Cauchy transform of affiliated operators is, in a natural way, the closure of the set of Cauchy transforms of bounded operators.