Left Derivable Maps at Non-Trivial Idempotents on Nest Algebras

• Main Authors: Ghahramani Hoger, Sattari Saman
• Format: Article
• Language: English
• Published: Sciendo 2019-09-01
• Series: Annales Mathematicae Silesianae
• Subjects: nest algebra; left derivable; left derivation; 47B47; 47L35
• Online Access: http://www.degruyter.com/view/j/amsil.2019.33.issue-1/amsil-2019-0001/amsil-2019-0001.xml?format=INT

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.