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

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.