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

Part of: Commutative Banach algebras and commutative topological algebras

Published online by Cambridge University Press:  09 July 2021

NASRIN ALIZADEH Open the ORCID record for NASRIN ALIZADEH [Opens in a new window] ,
ALI EBADIAN Open the ORCID record for ALI EBADIAN [Opens in a new window] ,
SAEID OSTADBASHI Open the ORCID record for SAEID OSTADBASHI [Opens in a new window]  and
ALI JABBARI Open the ORCID record for ALI JABBARI [Opens in a new window]
NASRIN ALIZADEH
Affiliation:
Department of Mathematics, Faculty of Science, Urmia University, Urmia, Iran e-mail: n.alizadeh@yahoo.com
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: s.ostadbashi@urmia.ac.ir
ALI JABBARI
Affiliation:
Department of Mathematics, Faculty of Science, Urmia University, Urmia, Iran e-mail: jabbari_al@yahoo.com
*
e-mail: ebadian.ali@gmail.com
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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.

Keywords

BSE Banach algebraBSE Banach modulecharacterlocally compact groupmodule extension Banach algebramultipliervector field

MSC classification

Primary: 46J05: General theory of commutative topological algebras
Secondary: 46J20: Ideals, maximal ideals, boundaries
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 105 , Issue 1 , February 2022 , pp. 134 - 145
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Copyright
© 2021 Australian Mathematical Publishing Association Inc.

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