Convolution powers in the operatorvalued framework
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We consider the framework of an operatorvalued noncommutative probability space over a unital C*algebra B. We show how for a Bvalued distribution μ one can define convolution powers μ{squared plus}η (with respect to free additive convolution) and μ{multiset union}η (with respect to Boolean convolution), where the exponent η is a suitably chosen linear map from B to B, instead of being a nonnegative real number. More precisely, μ{multiset union}η is always defined when η is completely positive, while μ{squared plus}η is always defined when η is completely positive (with "1" denoting the identity map on B). In connection to these convolution powers we define an evolution semigroup {Bη  η: B → B, completely positive}, related to the Boolean Bercovici Pata bijection. We prove several properties of this semigroup, including its connection to the Bvalued free Brownian motion. We also obtain two results on the operatorvalued analytic function theory related to convolution powers μ{squared plus}η. One of the results concerns the analytic subordination of the CauchyStieltjes transform of μ{squared plus}η with respect to the CauchyStieltjes transform of μ. The other one gives a Bvalued version of the inviscid Burgers equation, which is satisfied by the CauchyStieltjes transform of a Bvalued free Brownian motion. © 2012 American Mathematical Society.
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Anshelevich, M., Belinschi, S. T., Fevrier, M., & Nica, A.
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