Fixed product preserving mappings on Banach algebras

TL;DR

This paper characterizes linear maps between Banach algebras that preserve products equal to fixed elements, extending known cases and connecting to invertibility preservation, with implications for algebra homomorphisms.

Contribution

It generalizes product-preserving maps to fixed elements beyond zero or identity, linking such maps to scalar multiples of homomorphisms and exploring their structure.

Findings

Maps preserving products with finite-rank operators also preserve zero products.

Such maps are often scalar multiples of algebra homomorphisms.

Maps preserving products with invertible elements are homomorphisms or antihomomorphisms up to a fixed element.

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.