SOME GEOMETRIC PROPERTIES OF THE SUBORDINATION FUNCTION ASSOCIATED TO AN OPERATOR-VALUED FREE CONVOLUTION SEMIGROUP

Serban Belinschi (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219

In his article "On the free convolution with a semicircular distribution," Biane found very useful characterizations of the boundary values of the imaginary part of the Cauchy-Stieltjes transform of the free additive convolution of a probability measure on R with a Wigner (semicircular) distribution. Biane's methods were recently extended by Huang to measures which belong to the partial free convolution semigroups introduced by Nica and Speicher. The main purpose of this short note is to further extend some of Biane's methods and results to free convolution powers of operator-valued distributions and to free convolutions with operator-valued semicirculars.

Mots clés: free probability; noncommutative functions; free convolution semigroups

Domaines: Algèbres d'opérateurs [math.OA]

Serban Belinschi. SOME GEOMETRIC PROPERTIES OF THE SUBORDINATION FUNCTION ASSOCIATED TO AN OPERATOR-VALUED FREE CONVOLUTION SEMIGROUP. 2015. ⟨hal-01134078v1⟩