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 βiα−1 i ± βj α−1 j ∈ Aut(X) for all i = j. Let ξj be independent random variables with values in X and distributions μj with nonvanishing 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. © 2010 Elsevier Inc. All rights reserved