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Spectrally bounded -derivations on Banach algebras

Authors:	Tsiu-Kwen Lee Cheng-Kai Liu

Institution:	Department of Mathematics, National Taiwan University, Taipei 106, Taiwan ; Department of Mathematics, National Taiwan University, Taipei 106, Taiwan

Abstract:	Applying the density theorem on algebras with $\phi$ -derivations, we show that if a $\phi$ -derivation $\delta$ of a unital Banach algebra 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 denotes the spectral radius of $a\in A$ .

Keywords:	Radical $\phi $--derivation Banach algebra spectrally bounded mapping

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