Left Derivable Maps at Non-Trivial Idempotents on Nest Algebras

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 Alg 𝒩. We show that is a (generalized) Jordan left derivation. Moreover, in a strongly operator topology we characterize continuous linear maps on some nest algebras Alg 𝒩 with the property that δ (P ) = 2Pδ (P ) or δ (P ) = 2P δ (P ) − Pδ (I) for every idempotent P in Alg 𝒩.

Generalized Jordan left derivations in rings with involution

In the present paper we study generalized left derivations on Lie ideals of rings with involution. Some of our results extend other ones proven previously just for the action of generalized left derivations on the whole ring. Furthermore, we prove that every generalized Jordan left derivation on a 2-torsion free *-prime ring with involution is a generalized left derivation.