<abstract xmlns="http://www.w3.org/1999/xhtml"><p>Let <italic>Alg</italic> 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 <italic>P</italic> ∈ <italic>Alg</italic> 𝒩 with range <italic>P</italic> (𝕏) ∈ 𝒩, and <italic>δ</italic> : <italic>Alg</italic> 𝒩 → <italic>Alg</italic> 𝒩 is a continuous linear mapping (generalized) left derivable at <italic>P</italic>, i.e. <italic>δ</italic> (<italic>ab</italic>) = <italic>aδ</italic> (<italic>b</italic>) + <italic>bδ</italic> (<italic>a</italic>) (<italic>δ</italic> (<italic>ab</italic>) = <italic>aδ</italic>(<italic>b</italic>) + <italic>bδ</italic>(<italic>a</italic>) <italic>− baδ</italic>(<italic>I</italic>)) for any <italic>a, b</italic> ∈ <italic>Alg</italic> 𝒩 with <italic>ab</italic> = <italic>P</italic>, where <italic>I</italic> is the identity element of <italic>Alg</italic> 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 <italic>Alg</italic> 𝒩 with the property that <italic>δ</italic> (<italic>P</italic> ) = 2<italic>Pδ</italic> (<italic>P</italic> ) or <italic>δ</italic> (<italic>P</italic> ) = 2<italic>P δ</italic> (<italic>P</italic> ) <italic>− Pδ</italic> (<italic>I</italic>) for every idempotent <italic>P</italic> in <italic>Alg</italic> 𝒩.</p></abstract>

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.

Left Derivable Maps at Non-Trivial Idempotents on Nest Algebras

Annales Mathematicae Silesianae

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"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...

Left Derivable Maps at Non-Trivial Idempotents on Nest Algebras

Published Online: Jul 18, 2019

Page range: 97 - 105

Received: Jan 14, 2018

Accepted: Feb 10, 2019

DOI: https://doi.org/10.2478/amsil-2019-0001

Keywords
nest algebra, left derivable, left derivation

© 2019 Hoger Ghahramani et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Left Derivable Maps at Non-Trivial Idempotents on Nest Algebras

Published Online: Jul 18, 2019

Page range: 97 - 105

Received: Jan 14, 2018

Accepted: Feb 10, 2019

DOI: https://doi.org/10.2478/amsil-2019-0001

Keywordsnest algebra, left derivable, left derivation

© 2019 Hoger Ghahramani et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
nest algebra, left derivable, left derivation