Fixed product preserving mappings on Banach algebras
Abstract
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.
Keywords
Cite
@article{arxiv.2201.12912,
title = {Fixed product preserving mappings on Banach algebras},
author = {Hayden Julius},
journal= {arXiv preprint arXiv:2201.12912},
year = {2022}
}
Comments
9 pages - this is a second version. There is a major flaw in Thm 3.3 from previous version when I approached the boundary of the open set of invertible elements. I removed that argument from the second version