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The Quarterly Journal of Mathematics Advance Access originally published online on February 10, 2006
The Quarterly Journal of Mathematics 2006 57(4):469-478; doi:10.1093/qmath/hal003
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© The author 2006. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oxfordjournals.org

ON A BRESAR–SEMRL CONJECTURE AND DERIVATIONS OF BANACH ALGEBRAS

Mikhail A. Chebotar1,2, Wen-Fong Ke3 and Pjek-Hwee Lee4,{dagger}

1 Department of Management and Information Technology, Southern Taiwan University of Technology, Yung-Kang, Tainan 710, Taiwan
2 Department of Mechanics and Mathematics, Tula State University, Tula, Russia
3 Department of Mathematics, National Cheng Kung University, Tainan 701, Taiwan
4 Department of Mathematics, National Taiwan University, Taipei 106, Taiwan

{dagger} Corresponding author. E-mail: phlee{at}math.ntu.edu.tw


  Abstract

In this paper, we answer a question on derivations of dense algebras of linear operators posed by Bresar and semrl. Our theorem implies the following result: let B be a complex Banach algebra, and let d and g be continuous derivations of B. If dg(x) is quasi-nilpotent for every x isin B, then dg(x)3 lies in the radical of B for every x isin B. This result was proved by Bresar and Semrl with the additional assumption gd = dg.


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