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

Let $A_i$ be a commutative $C^{\ast }$-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 $$\mathrm{\parallel }T(a+b)\mathrm{\parallel }=\mathrm{\parallel }T(a)+T(b)\mathrm{\parallel }(a,b\in A_1^{+})\mathrm{.}$$ 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 $C^{\ast }$-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.