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 \(\alpha \) be a topological automorphism of the group X such that \(\textrm{Ker}(I+\alpha )=\{0\}\). 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.