Linear sofic approximations of amenable algebras
Benjamin Bachner
TL;DR
The paper defines and studies linear sofic approximations for finitely generated amenable $K$-algebras with no zero divisors, proving a rigidity result: all such approximations are conjugate. It introduces a linear monotiling technique based on locally linearly dependent operators to establish a linear tiling framework, and uses it to prove the main conjugacy theorem, with a parallel to Elek–Szabó's amenable-group result. The results yield a characterization of weak stability for amenable algebras, showing that for a group algebra $K\, ext{Γ}$ of a torsion-free amenable group, weak stability is equivalent to $ ext{Γ}$ being residually finite, and extend to general amenable algebras via the rank metric. An explicit example demonstrates that weak stability does not imply full stability. Overall, the work provides an algebraic analogue of rigidity phenomena for amenable objects and connects stability notions to residual finiteness through a robust linear tiling toolkit.
Abstract
We introduce the notion of linear sofic approximations for algebras, analogous to the concept of sofic approximations for groups. We prove that for a finitely generated amenable $K$-algebra with no zero divisors, all linear sofic approximations are conjugate. This provides an algebraic analogue to Elek and Szabó's theorem for amenable groups. The proof relies on a "linear monotiling" technique, constructed using a theorem by Brešar, Meshulam and Šemrl on locally linearly dependent operators. Finally, we apply this uniqueness result to the problem of weak stability in the rank metric, showing that the group algebra of an amenable group is weakly stable if and only if the group is residually finite.