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

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 $α$ be a topological automorphism of the group $X$ such that ${\rm Ker}(I+α)=\{0\}$. Let $ξ_1$ and $ξ_2$ be independent random variables with values in $X$ and distributions $μ_1$ and $μ_2$ with nonvanishing characteristic functions. If the conditional distribution of the linear form $L_2 = ξ_1 + αξ_2$ given $L_1 = ξ_1 +ξ_2$ is symmetric, then $μ_j$ 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.

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