Fixed product preserving mappings on Banach algebras

In this paper, we describe linear maps between complex Banach algebras that preserve products equal to fixed elements. This generalizes some important special cases where the fixed elements are the zero or identity element. First we show that if such map preserves products equal to a finite-rank ope...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2023-01, Vol.517 (1), p.126615, Article 126615
Main Author: Julius, Hayden
Format: Article
Language:English
Subjects:
Banach algebras
Invertibility preservers
Linear preserver problems
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Summary:In this paper, we describe linear maps between complex Banach algebras that preserve products equal to fixed elements. This generalizes some important special cases where the fixed elements are the zero or identity element. First we show that if such map preserves products equal to a finite-rank operator, then it must also preserve the zero product. In several instances, this is enough to show that a product preserving map must be a scalar multiple of an algebra homomorphism. Second, we explore a more general problem concerning the existence of product preserving maps and the relationship between the fixed elements. Lastly, motivated by Kaplansky's problem on invertibility preservers, we show that maps preserving products equal to fixed invertible elements are either homomorphisms or antihomomorphisms multiplied on the left by a fixed element.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2022.126615