The Cauchy equation and norm additive mappings between positive cones of commutative C⁎-algebras

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 + → A 2 + that satisfy the norm equality ‖ T ( a + b ) ‖ = ‖ T ( a ) + T ( b ) ‖ ( a , b ∈ 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 + → 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.