The Heyde theorem for locally compact Abelian groups

We prove a group analogue of the well-known Heyde theorem where a Gaussian measure is characterized by the symmetry of the conditional distribution of one linear form given another. Let X be a locally compact second countable Abelian group containing no subgroup topologically isomorphic to the circle group T, G be the subgroup of X generated by all elements of order 2, and Aut(X) be the set of all topological automorphisms of X. Let αj,βj∈Aut(X), j=1,2,…,n, n⩾2, such that for all i≠j. Let ξj be independent random variables with values in X and distributions μj with non-vanishing characteristic functions. If the conditional distribution of L2=β1ξ1+⋯+βnξn given L1=α1ξ1+⋯+αnξn is symmetric, then each μj=γj∗ρj, where γj are Gaussian measures, and ρj are distributions supported in G.