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

Bojan Kuzma, Sushil Singla
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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.
有限维简单 $C^*$ 算法的非线性分类
给出了简单实数或复数$C^*$-代数的巴拿赫空间特征,它甚至描述了底层域的特征。作为一个应用,它证明了如果 $\mathfrak A_1$ 和 $\mathfrak A_2$ 是在 $\mathbb F_1$ 和 $\mathbb F_2$ 域上的伯克霍夫-詹姆斯同构简单 $C^*$ 对象,并且如果 $\mathfrak A_1$ 是有限维的、如果 $\mathfrak A_1$ 是维数大于一的有限维,那么 $\mathbb F_1=\mathbb F_2$ 和 $\mathfrak A_1$ 和 $\mathfrak A_2$ 是(同构的)$\ast$-同构的 $C^*$-代数。
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