Non-linear classification of finite-dimensional simple C*-algebras

A Banach space characterization of simple real or complex C*-algebras is given which even characterizes the underlying field. As an application, it is shown that if A1 and A2 are Birkhoff-James is omorphic simple C*-algebras over the fields F1 and F2, respectively and if A1 is finite-dimensional with dimension greater than one, then F1= F2 and A1 and A2 are (isometrically) *-isomorphic C*-algebras.