Convolution powers in the operator-valued framework Academic Article uri icon


  • We consider the framework of an operator-valued noncommutative probability space over a unital C*-algebra B. We show how for a B-valued 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 B-valued free Brownian motion. We also obtain two results on the operator-valued analytic function theory related to convolution powers μ{squared plus}η. One of the results concerns the analytic subordination of the Cauchy-Stieltjes transform of μ{squared plus}η with respect to the Cauchy-Stieltjes transform of μ. The other one gives a B-valued version of the inviscid Burgers equation, which is satisfied by the Cauchy-Stieltjes transform of a B-valued free Brownian motion. © 2012 American Mathematical Society.

author list (cited authors)

  • Anshelevich, M., Belinschi, S. T., Fevrier, M., & Nica, A.

citation count

  • 7

publication date

  • October 2012