Journal of Operator Theory

Volume 47, Issue 1, Winter 2002 pp. 197-212.

Powers of

$R$ -diagonal elements

Authors: Flemming Larsen
Author institution: University of South Denmark, Odense University Campus, Campusvej 55, DK--5230 Odense M, Denmark

Summary: We prove that if

$(a,b)$ is an

$R$ -diagonal pair in some non-commu\-tative probability space

$(A,\phi)$ 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

$[0,\infty[$ , 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.

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