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

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 $\mathrm{A}\mathrm{u}\mathrm{t}({\mathbb{F}}_q)$ 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.