Spectrally bounded 𝜙–derivations on Banach algebras

Lee, Tsiu-Kwen; Liu, Cheng-Kai

doi:10.1090/S0002-9939-04-07655-5

Spectrally bounded $\phi$–derivations on Banach algebras
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Proc. Amer. Math. Soc. 133 (2005), 1427-1435 Request permission

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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