Derivations on convolution algebras
Bull. Korean Math. Soc. 2015 Vol. 52, No. 4, 1123-1132
https://doi.org/10.4134/BKMS.2015.52.4.1123
Published online July 31, 2015
Mohammad Javad Mehdipour and Zahra Saeedi
Shiraz University of Technology, Shiraz University of Technology
Abstract : In this paper, we investigate derivations on the noncommutative Banach algebra $L_0^\infty (\omega )^\ast$ equipped with an Arens product. As a main result, we prove the Singer-Wermer conjecture for the noncommutative Banach algebra $L_0^\infty (\omega )^\ast$. We then show that a derivation on $L_0^\infty (\omega )^\ast$ is continuous if and only if its restriction to $\hbox{rad}(L_0^\infty(\omega )^\ast)$ is continuous. We also prove that there is no nonzero centralizing derivation on $L_0^\infty (\omega )^\ast$. Finally, we prove that the space of all inner derivations of $L_0^\infty (\omega )^\ast$ is continuously homomorphic to the space $L_0^\infty (\omega )^\ast/L^1(\omega)$.
Keywords : derivation, inner derivation, centralizing, automatic continuity
MSC numbers : Primary 47B47, 46H40, 16W25
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