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THE BOCHNER–SCHOENBERG-EBERLEIN PROPERTY OF EXTENSIONS OF BANACH ALGEBRAS AND BANACH MODULES

Published online by Cambridge University Press:  09 July 2021

NASRIN ALIZADEH
Affiliation:
Department of Mathematics, Faculty of Science, Urmia University, Urmia, Iran e-mail: [email protected]
ALI EBADIAN*
Affiliation:
Department of Mathematics, Faculty of Science, Urmia University, Urmia, Iran
SAEID OSTADBASHI
Affiliation:
Department of Mathematics, Faculty of Science, Urmia University, Urmia, Iran e-mail: [email protected]
ALI JABBARI
Affiliation:
Department of Mathematics, Faculty of Science, Urmia University, Urmia, Iran e-mail: [email protected]

Abstract

Let A be a Banach algebra and let X be a Banach A-bimodule. We consider the Banach algebra ${A\oplus _1 X}$ , where A is a commutative Banach algebra. We investigate the Bochner–Schoenberg–Eberlein (BSE) property and the BSE module property on $A\oplus _1 X$ . We show that the module extension Banach algebra $A\oplus _1 X$ is a BSE Banach algebra if and only if A is a BSE Banach algebra and $X=\{0\}$ . Furthermore, we consider $A\oplus _1 X$ as a Banach $A\oplus _1 X$ -module and characterise the BSE module property on $A\oplus _1 X$ . We show that $A\oplus _1 X$ is a BSE Banach $A\oplus _1 X$ -module if and only if A and X are BSE Banach A-modules.

Type
Research Article
Copyright
© 2021 Australian Mathematical Publishing Association Inc.

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