Rocky Mountain Journal of Mathematics

Full-derivable points of $\mathcal {J}$-subspace lattice algebras

Xiaofei Qi and Jinchuan Hou

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Abstract

Let $\mathcal{L}$ be a $\mathcal{J}$-subspace lattice on a complex Banach space $X$ and $\mbox{\rm Alg\,}{\mathcal L}$ the associated $\mathcal{J}$-subspace lattice algebra. We say that an operator $Z\in {\rm Alg\,}{\mathcal L}$ is a full-derivable point of $\mbox{\rm Alg\,}{\mathcal L}$ if every linear map $\delta$ from $\mbox{\rm Alg\,}{\mathcal L}$ into itself derivable at $Z$ (i.e., $\delta(A)B+A\delta(B)=\delta(Z)$ for any $A,B \in {\rm Alg\,}{\mathcal L}$ with $AB=Z$) is a derivation and is a full-generalized-derivable point of $\mbox{\rm Alg\,}{\mathcal L}$ if every linear map $\delta$ from $\mbox{\rm Alg\,}{\mathcal L}$ into itself generalized derivable at $Z$ (i.e., $\delta(A)B+A\delta(B)-A\delta(I)B=\delta(Z)$ for any $A,B \in {\rm Alg\,}{\mathcal L}$ with $AB=Z$) is a generalized derivation. In this paper, we prove that if $Z\in\mbox{\rm Alg\,}{\mathcal L}$ is an injective operator or an operator with dense range, then $Z$ is a full-derivable point as well as a full-generalized-derivable point of ${\rm Alg\,}{\mathcal L}$.

Article information

Source
Rocky Mountain J. Math., Volume 45, Number 1 (2015), 345-358.

Dates
First available in Project Euclid: 7 April 2015

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1428412784

Digital Object Identifier
doi:10.1216/RMJ-2015-45-1-345

Mathematical Reviews number (MathSciNet)
MR3334215

Zentralblatt MATH identifier
1309.47041

Subjects
Primary: 47B47: Commutators, derivations, elementary operators, etc. 47L35: Nest algebras, CSL algebras

Keywords
$\mathcal J$-subspace lattice algebra derivations generalized derivations full-derivable point

Citation

Qi, Xiaofei; Hou, Jinchuan. Full-derivable points of $\mathcal {J}$-subspace lattice algebras. Rocky Mountain J. Math. 45 (2015), no. 1, 345--358. doi:10.1216/RMJ-2015-45-1-345. https://projecteuclid.org/euclid.rmjm/1428412784


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