POWERS OF R-DIAGONAL ELEMENTS

Abstract

We prove that if (a; b) is an R-diagonal pair in some non-commutative probability space (A; ') then (a p ; b p ) is R-diagonal too and we compute the determining series f (a p ;b p ) in terms of the distribution of ab. We give estimates of the upper and lower bounds of the support of free multiplicative convolution of probability measures compactly supported on [0; 1[, and use the results to give norm estimates of powers of R-diagonal elements in finite von Neumann algebras. Finally we compute norms, distributions and R-transforms related to powers of the circular element. 1 Introduction and Preliminaries In the setup of Free Probability Theory we study certain random variables. By a noncommutative probability space (A; ') we mean a unital algebra A (over the complex numbers) equipped with a unital functional '. If A is a von Neumann algebra and ' is normal we call (A; ') a non-commutative W -probability space. We write (M; ) for a non-commutative W -probability space ...