Heyde theorem for locally compact Abelian groups containing no subgroups topologically isomorphic to the 2-dimensional torus

Abstract: We prove the following group analogue of the well-known Heyde theorem on a characterization of the Gaussian distribution on the real line. Let $X$ be a second countable locally compact Abelian group containing no subgroups topologically isomorphic to the 2-dimensional torus. Let $G$ be the subgroup of $X$ generated by all elements of $X$ of order $2$ and let $\alpha$ be a topological automorphism of the group $X$ such that ${\rm Ker}(I+\alpha)={0}$. Let $\xi_1$ and $\xi_2$ be independent random variables with values in $X$ and distributions $\mu_1$ and $\mu_2$ with nonvanishing characteristic functions. If the conditional distribution of the linear form $L_2 = \xi_1 + \alpha\xi_2$ given $L_1 = \xi_1 +\xi_2$ is symmetric, then $\mu_j$ are convolutions of Gaussian distributions on $X$ and distributions supported in $G$. We also prove that this theorem is false if $X$ is the 2-dimensional torus.