Norm additive mappings between commutative $C^{*}$-algebras in the range
Daisuke Hirota
TL;DR
The paper investigates norm-based additivity on the positive cones of commutative $C^*$-algebras: given a surjective map $T:C_0^+(L_1)\to C_0^+(L_2)$ with $\|T(f+g)\|=\|T(f)+T(g)\|$, it proves $T$ is additive and positively homogeneous, and in the unital bijective case derives a composition-operator form on spectra via a homeomorphism $\tau$ of maximal ideal spaces: $\widehat{T(a)}(\xi)=\widehat{T(1_{A_1})}(\xi)\widehat{a}(\tau(\xi))$. The results connect norm-preservation to algebraic structure, enabling a full representation of $T$ in terms of the Gelfand transform and spectral maps. An open problem is posed for the noncommutative setting: does the same norm-additivity imply additivity without commutativity? The work thus advances the understanding of how norm identities constrain algebraic structure in operator algebra contexts.
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.