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On norm-additive maps between the maximal groups of positive continuous functions. (English) Zbl 1439.46019

Summary: 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 \(C(Y)\).

MSC:

46E15 Banach spaces of continuous, differentiable or analytic functions
46E25 Rings and algebras of continuous, differentiable or analytic functions
47B33 Linear composition operators
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