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 \(\alpha \) $\alpha$ be a topological automorphism of the group X such that \(\textrm{Ker}(I+\alpha )=\{0\}\) $\text{Ker}(I+\alpha )=\{0\}$ . Let \(\xi _1\) ${\xi }_1$ and \(\xi _2\) ${\xi }_2$ be independent random variables with values in X and distributions \(\mu _1\) ${\mu }_1$ and \(\mu _2\) ${\mu }_2$ with nonvanishing characteristic functions. If the conditional distribution of the linear form \(L_2 = \xi _1 + \alpha \xi _2\) $L_2={\xi }_1+\alpha {\xi }_2$ given \(L_1 = \xi _1 + \xi _2\) $L_1={\xi }_1+{\xi }_2$ is symmetric, then \(\mu _j\) ${\mu }_j$ 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.