Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2015; 52(4): 1123-1132

Printed July 31, 2015

https://doi.org/10.4134/BKMS.2015.52.4.1123

Copyright © The Korean Mathematical Society.

Derivations on convolution algebras

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