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 alpha(j), beta(j) is an element of Aut(X), j = 1,2, ..., n, n >= 2, such that beta(i)alpha(-1)(j) +/- beta(j)alpha(-1)(j) is an element of Aut(X) for all i not equal j. Let xi(j) be independent random variables with values in X and distributions mu(j) with non-vanishing characteristic functions. If the conditional distribution of L(2) = beta(1)xi(1) + ... + beta(n)xi(n) given L(1) = alpha(1)xi(1) + ... + alpha(n)xi(n) is symmetric, then each mu(j) = gamma(j) * rho(j), where gamma(j) are Gaussian measures, and rho(j) are distributions supported in G. (C) 2010 Elsevier Inc. All rights reserved.