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arXiv · 1712.04431

The Fra\"iss\'e limit of matrix algebras with the rank metric

Aaron Anderson
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Abstract

We realize the $\mathbb{F}_q$-algebra $M(\mathbb{F}_q)$ studied by von Neumann and Halperin as the Fra\"iss\'e limit of the class of finite-dimensional matrix algebras over a finite field $\mathbb{F}_q$ equipped with the rank metric. We then provide a new Fra\"iss\'e-theoretic proof of uniqueness of such an object. Using the results of Carderi and Thom, we show that the automorphism group of $\mathrm{Aut}(\mathbb{F}_q )$ is extremely amenable. We deduce a Ramsey-theoretic property for the class of algebras $M(\mathbb{F}_q)$, and provide an explicit bound for the quantities involved.

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Aaron Anderson. 2017-12-12. The Fra\"iss\'e limit of matrix algebras with the rank metric. https://arxiv.org/abs/1712.04431

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