On a theorem of K. Schmidt

Let Y be a locally compact group, Aut(Y) be the group of topologicalautomorphisms of Y and (Y) be the set of continuous positivedefinite functions on Y which have unit value at the identity.A function (Y²) is said to be of product type if there aresuch functions _j (Y) that (u, v) = ₁(u)₂(v). Define the mappingT: Y² Y² by the formula T(u, v) = (A₁ uA₂ v, A₃ u A₄ v), whereA_j Aut(Y), and assume that T is a one-to-one transform. K.Schmidt proved: (i) if both (u, v) and (T(u, v)) are of producttype, then the functions _j are infinitely divisible; (ii) ifY is Abelian, both (u, v) and (T(u, v)) are of product type,and (u, v) 0, then the functions _j are Gaussian. We show thatstatement (i), generally, is not valid, but K. Schmidt's proofholds true if (u, v) 0. We also give another proof of statement(ii). Our proof uses neither the Levy–Khinchin formulafor a continuous infinitely divisible positive definite functionnor (i) on which K. Schmidt's proof is based.