Non-linear classification of finite-dimensional simple $C^{\ast }$-algebras

A Banach space characterization of simple real or complex $C^{\ast }$-algebras is given which even characterizes the underlying field. As an application, it is shown that if ${\mathfrak{A}}_1$ and ${\mathfrak{A}}_2$ are Birkhoff-James isomorphic simple $C^{\ast }$-algebras over the fields ${\mathbb{F}}_1$ and ${\mathbb{F}}_2$, respectively and if ${\mathfrak{A}}_1$ is finite-dimensional with dimension greater than one, then ${\mathbb{F}}_1={\mathbb{F}}_2$ and ${\mathfrak{A}}_1$ and ${\mathfrak{A}}_2$ are (isometrically) $\ast$-isomorphic $C^{\ast }$-algebras.