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

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arxiv 2407.21582 v1 pith:DKYLCEGJ submitted 2024-07-31 math.OA math.FA

classification math.OAmath.FA
keywords mathfrakalgebrasmathbbsimplefinite-dimensionalisomorphicapplicationbanach
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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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  1. Birkhoff-James classification of finite-dimensional $C^*$-algebras

    math.OA 2024-11 conditional novelty 7.0 of 10

    Two finite-dimensional C*-algebras are isomorphic exactly when their Birkhoff-James orthogonality relations coincide, and the underlying field is also determined.

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