Left derivable maps at non-trivial idempotents on nest algebras - Ghahramani, Hoger; Sattari, Saman (Annales Mathematicae Silesianae, 2019)
Let Alg N be a nest algebra associated with the nest N on a (real or complex) Banach space X. Suppose that there exists a non-trivial idempotent P∈Alg N with range P(X)∈?N, and δ:Alg N→Alg N is a continuous linear mapping (generalized) left derivable at P, i.e. δ(ab) = aδ(b) + bδ(a) (δ(ab) = aδ(b) + bδ(a) - baδ(I)) for any a,b∈Alg N with ab = P, where I is the identity element of AlgN. We show that δ is a (generalized) Jordan left derivation. Moreover, in a strongly operator topology we characterize continuous linear maps on some nest algebras Alg???? with the property that δ(P) = 2Pδ(P) or δ(P) = 2Pδ(P) - Pδ(I) for every idempotent P in Alg N.