Characterization of positive definite, radial functions on free groups

This article studies the properties of positive definite, radial functions on free groups following the work of Haagerup and Knudby [Proc. Amer. Math. Soc. 143 (2015), pp. 1477–1489]. We obtain characterizations of radial functions with respect to the ℓ 2 \ell ^{2} length on the free groups with infinite generators and the characterization of the positive definite, radial functions with respect to the ℓ p \ell ^{p} length on the free real line with infinite generators for 0 > p ≤ 2 0 > p \leq 2 . We obtain a Lévy-Khintchine formula for length-radial conditionally negative functions as well.