Left derivable maps at non-trivial idempotents on nest algebras
Abstract
Let Alg𝓝 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∈Alg𝓝 with range P(𝕏)∈𝓝, and δ:Alg𝓝→Alg𝓝 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∈Alg𝓝 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 Alg𝓝 with the property that δ(P) = 2Pδ(P) or δ(P) = 2Pδ(P) - Pδ(I) for every idempotent P in Alg𝓝.
Keywords
nest algebra; left derivable; left derivation
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Department of Mathematics, University of Kurdistan, Iran Iran, Islamic Republic of
Department of Mathematics, University of Kurdistan, Iran Iran, Islamic Republic of
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