A LIOUVILLE THEOREM FOR MATRIX-VALUED HARMONIC FUNCTIONS ON NILPOTENT GROUPS

Abstract

Let $\sigma$ be a non-degenerate positive $M_n$-valued measure on a locally compact group $G$ with $\|\sigma\|\,{=}\,1$. An $M_n$-valued Borel function $f$ on $G$ is called $\sigma$-harmonic if $f(x) = \int_G f(xy^{-1})\,d\sigma(y)$ for all $x\,{\in}\,G$. Given such a function $f$ which is bounded and left uniformly continuous on $G$, it is shown that every central element in $G$ is a period of $f$. Further, it is shown that $f$ is constant if $G$ is nilpotent or central.