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

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).