A Banach space theoretical characterization of abelian 𝐶*-algebras

Abstract

A Banach space theoretical characterization of abelian C ∗ C^* - algebras among all C ∗ C^* -algebras is given. As an application, it is shown that if A A and B B are C ∗ C^* -algebras (nonlinearly) isomorphic to each other with respect to the structure of Birkhoff-James orthogonality, and if either A A or B B is abelian, then they are ∗ * -isomorphic. Moreover, it is pointed out that the same kind of characterization is not valid for preduals of abelian von Neumann algebras.