B-valued Free Convolution for Unbounded Operators
نویسنده
چکیده
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 Ã. For a random variable X ∈ Ã we study the Cauchy transform GX and show that the von Neumann algebra (B ∪ {X}) ′′ can be recovered from this function. In the case where B is finite dimensional, we show that, when X,Y ∈ Ã 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 functions that arise as the Cauchy transform of affiliated operators are the limit points of the Cauchy transforms of bounded operators in a suitable topology.
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