Characterizations of left derivable maps at non-trivial idempotents on nest algebras

Abstract

Let $Alg\mathcal{N}$ be a nest algebra associated with the nest $\mathcal{N}$ on a (real or complex) Banach space $\X$. Suppose that there exists a non-trivial idempotent $P\in Alg\mathcal{N}$ with range $P(\X) \in \mathcal{N}$ and $\delta :Alg\mathcal{N}\to Alg\mathcal{N}$ is a continuous linear mapping (generalized) left derivable at $P$, i.e. $\delta (ab)=a\delta (b)+b\delta (a)$ ($\delta (ab)=a\delta (b)+b\delta (a)-ba\delta (I)$) for any $a,b\in Alg\mathcal{N}$ with $ab=P$. we show that $\delta$ is a (generalized) Jordan left derivation. Moreover, we characterize the strongly operator topology continuous linear maps $\delta$ on some nest algebra $Alg\mathcal{N}$ with property that $\delta (P)=2P\delta (P)$ or $\delta (P)=2P\delta (P)-P\delta (I)$ every idempotent $P$ in $Alg\mathcal{N}$.

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• Left derivable or Jordan left derivable mappings on Banach algebras