Two finite-dimensional C*-algebras are isomorphic exactly when their Birkhoff-James orthogonality relations coincide, and the underlying field is also determined.
Non-linear classification of finite-dimensional simple $C^*$-algebras
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abstract
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.
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Birkhoff-James classification of finite-dimensional $C^*$-algebras
Two finite-dimensional C*-algebras are isomorphic exactly when their Birkhoff-James orthogonality relations coincide, and the underlying field is also determined.