Composition in Modulus Maps on Semigroups of Continuous Functions

TL;DR

This paper investigates surjective maps between subsemigroups of continuous functions that preserve a specific norm-based metric, demonstrating these maps are essentially composition in modulus maps.

Contribution

It extends the understanding of norm-preserving maps on subsemigroups of continuous functions, showing they are characterized as composition in modulus maps.

Findings

Surjections preserving a specific metric are composition in modulus maps.

The study generalizes previous results on norm multiplicative and additive conditions.

Provides a characterization of structure-preserving maps on function semigroups.

Abstract

For locally compact Hausdorff spaces $X$ and $Y$, and function algebras $A$ and $B$ on $X$ and $Y$, respectively, surjections $T:A \longrightarrow B$ satisfying norm multiplicative condition $\|Tf\, Tg\|_Y =\|fg\|_X$, $f,g\in A$, with respect to the supremum norms, and those satisfying $\||Tf|+|Tg|\|_Y=\||f|+|g|\|_X$ have been extensively studied. Motivated by this, we consider certain (multiplicative or additive) subsemigroups $A$ and $B$ of $C_0(X)$ and $C_0(Y)$, respectively, and study surjections $T: A \longrightarrow B$ satisfying the norm condition $\rho(Tf, Tg)=\rho(f,g)$, $f,g \in A$, for some class of two variable positive functions $\rho$. It is shown that $T$ is also a composition in modulus map.