Characterization of positive definite, radial functions on free groups

Abstract

This article studies the properties of positive definite, radial functions on free groups following the work of Haagerup and Knudby . We obtain characterizations of radial functions with respect to the ${\ell }^2$ length on the free groups with infinite generators and the characterization of the positive definite, radial functions with respect to the ${\ell }^p$ length on the free real line with infinite generators for $0<p\le 2$. We obtain the Levi-Khintchine formulas for length-radial conditionally negative functions as well.

14 pages