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

TL;DR

This paper characterizes certain norm-preserving mappings between positive cones of commutative $C^{*}$-algebras, proving they are additive, positively homogeneous, and, under injectivity, related to composition operators.

Contribution

It establishes that norm additive mappings between positive cones are necessarily additive and positive homogeneous, and characterizes injective cases as composition operators.

Findings

Mappings are additive and positively homogeneous.

Injective mappings correspond to composition operators.

Results apply to positive cones of unital commutative $C^{*}$-algebras.

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.