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

TL;DR

This paper characterizes simple finite-dimensional $C^*$-algebras using Banach space properties and shows that Birkhoff-James isomorphisms imply *-isomorphisms over the same field.

Contribution

It provides a Banach space characterization of simple $C^*$-algebras that determines the underlying field and establishes isometric *-isomorphism conditions.

Findings

Banach space characterization of simple $C^*$-algebras

Field determination from algebraic structure

Isometric *-isomorphism under Birkhoff-James isomorphism

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.