  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 $\mathfrak A_1$ and $\mathfrak A_2$ are Birkhoff-James isomorphic
simple $C^*$-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 $C^*$-algebras.
