Norm additive mappings between commutative $C^{*}$-algebras in the range

Abstract

Let \( A_i \) be a commutative \( C^{*} \)-algebra for \( i = 1, 2 \), and denote by \( A_i^{+} \) its positive cone, consisting of all positive elements of \( A_i \). In this paper, we investigate surjective, not necessarily continuous mappings \( T: A_1^{+} \to A_2^{+} \) that satisfy the norm equality \[ \| T(a + b) \| = \| T(a) + T(b) \| \quad (a, b \in A_1^{+}). \] We prove that such a mapping \( T \) is necessarily additive and positive homogeneous. Furthermore, we show that if the mapping $T:A_{1}^{+}\to A_{2}^{+}$ between the positive cones of two unital commutative $C^{*}$-algebras $A_{i}$ with the unit element \( 1_{A_i} \) for \( i = 1, 2 \), and if \( T \) is also injective, then $T(1_{A_1})^{-1}T$ is a composition operator. This is the submitted version of a paper currently under minor revision for the Journal of Mathematical Analysis and Applications.