The Fraïssé limit of matrix algebras with the rank metric

Abstract

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