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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Spectrally bounded $\phi $-derivations on Banach algebras


Authors: Tsiu-Kwen Lee and Cheng-Kai Liu
Journal: Proc. Amer. Math. Soc. 133 (2005), 1427-1435
MSC (2000): Primary 47B48, 46H15
Published electronically: November 1, 2004
MathSciNet review: 2111969
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Abstract: Applying the density theorem on algebras with $\phi $-derivations, we show that if a $\phi $-derivation $\delta $ of a unital Banach algebra $A$ is spectrally bounded, then $[\delta (A), A]\subseteq \text{rad}(A)$. Also, $\delta (A)\subseteq \text{rad}(A)$ if and only if $\text{sup}\{r(z^{-1}\delta (z))\mid z\in A\ \text{is invertible}\}<\infty $, where $r(a)$ denotes the spectral radius of $a\in A$.


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Additional Information

Tsiu-Kwen Lee
Affiliation: Department of Mathematics, National Taiwan University, Taipei 106, Taiwan
Email: tklee@math.ntu.edu.tw

Cheng-Kai Liu
Affiliation: Department of Mathematics, National Taiwan University, Taipei 106, Taiwan
Email: ckliu@math.ntu.edu.tw

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07655-5
PII: S 0002-9939(04)07655-5
Keywords: Radical, $\phi $--derivation, Banach algebra, spectrally bounded mapping
Received by editor(s): September 10, 2003
Received by editor(s) in revised form: January 14, 2004
Published electronically: November 1, 2004
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.