Linear sofic approximations of amenable algebras

TL;DR

This paper introduces linear sofic approximations for algebras, proves their conjugacy for certain amenable algebras, and connects this to weak stability and residual finiteness of groups.

Contribution

It develops the concept of linear sofic approximations for algebras and establishes their uniqueness for finitely generated amenable algebras without zero divisors.

Findings

All linear sofic approximations are conjugate for the specified class of algebras.

The group algebra of an amenable group is weakly stable iff the group is residually finite.

Introduces a linear monotiling technique based on locally linearly dependent operators.

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 Szab\'o's theorem for amenable groups. The proof relies on a "linear monotiling" technique, constructed using a theorem by Bre\v{s}ar, Meshulam and \v{S}emrl 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.