Characterizing Lie derivations on triangular algebras by local actions

Let U = Tri(A,M,B) be a triangular algebra, where A, B are unital algebras over a field F and M is a faithful (A,B)-bimodule. Assume that ξ ∈ F and L : U → U is a map. It is shown that, under some mild conditions, L is linear and satisfies L([X, Y ]) = [L(X), Y ] + [X,L(Y )] for any X,Y ∈ U with [X, Y ] = XY − Y X = 0 if and only if L(X) = φ(X) + ZX + f(X) for all A, where φ is a linear derivation, Z is a central element and f is a central valued linear map. For the case 1 6= ξ ∈ F , L is additive and satisfies L([X,Y ]ξ) = [L(X), Y ]ξ + [X,L(Y )]ξ for any X, Y ∈ U with [X, Y ]ξ = XY − ξY X = 0 if and only if L(I) is in the center of U and L(A) = φ(A) + L(I)A for all A, where φ is an additive derivation satisfying φ(ξA) = ξφ(A) for each A. In addition, all additive maps L satisfying L([X, Y ]ξ) = [L(X), Y ]ξ + [X,L(Y )]ξ for any X,Y ∈ U with XY = 0 are also characterized.