Additivity of α-elementary maps on triangular algebras

Let T be a triangular algebra and S an arbitrary ring. We study the additivity of surjective maps M : T → S and M ∗ : S → T preserving a family of sums of triple products on these rings. In this paper, we give sufficient conditions on T such that both M and M ∗ are additives. In particular, if T is a standard subalgebra of a nest algebra, then M and M ∗ are additives.