Additivity of $$\alpha $$ α -elementary maps on triangular algebras

Abstract

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