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ON COMMUTATIVITY OF BANACH ALGEBRAS WITH DERIVATIONS

Part of: Rings and algebras with additional structure Commutative Banach algebras and commutative topological algebras

Published online by Cambridge University Press:  06 March 2015

SHAKIR ALI  and
ABDUL NADIM KHAN
SHAKIR ALI*
Affiliation:
Department of Mathematics, Faculty of Science, Rabigh, King Abdulaziz University, Jeddah 21589, Saudi Arabia email shakir50@rediffmail.com
ABDUL NADIM KHAN
Affiliation:
Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India email abdulnadimkhan@gmail.com
*
✉shakir50@rediffmail.com
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Abstract

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The aim of this paper is to discuss the commutativity of a Banach algebra $A$ via its derivations. In particular, we prove that if $A$ is a unital prime Banach algebra and $A$ has a nonzero continuous linear derivation $d:A\rightarrow A$ such that either $d((xy)^{m})-x^{m}y^{m}$ or $d((xy)^{m})-y^{m}x^{m}$ is in the centre of $A$ for an integer $m=m(x,y)$ and sufficiently many $x,y$, then $A$ is commutative. We give examples to illustrate the scope of the main results and show that the hypotheses are not superfluous.

Keywords

Banach algebraderivation

MSC classification

Primary: 16W25: Derivations, actions of Lie algebras
Secondary: 46J10: Banach algebras of continuous functions, function algebras
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 91 , Issue 3 , June 2015 , pp. 419 - 425
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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