On Norm-Additive Maps Between the Maximal Groups of Positive Continuous Functions

Abstract

Assume that X and Y are compact Hausdorff spaces. We call $C_+(X)=\{f\in C(X):f(x)>0\,\,\,\text {for all}\,\,\,x\in X\}$ the maximal positive continuous functions group of C(X). A map $T:C_+(X)\rightarrow C_+(Y)$ is called norm-additive, if $\Vert Tf+ Tg\Vert =\Vert f+ g\Vert$ for all $f,g\in C_+(X)$. We show that any norm-additive map between $C_+(X)$ and $C_+(Y)$ is a composition operator, and hence the restriction of a norm-additive map between C(X) and C(Y).