Linear sofic approximations of amenable algebras

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.

💡 Research Summary

The paper introduces “linear sofic approximations” for associative algebras, mirroring the well‑known notion of sofic approximations for groups but using the normalized rank metric on matrices instead of the Hamming metric on permutations. An asymptotic homomorphism ϕ_k : A → M_{n_k}(K) is called a linear sofic approximation when it is asymptotically multiplicative and, for every non‑zero element a∈A, the normalized rank of ϕ_k(a) tends to 1 along a non‑principal ultrafilter. This “maximally separating’’ condition is shown to be equivalent to the existence of d‑approximations for all d (Proposition 3.3).

The main structural result (Theorem 1.1) states that if A is a finitely generated amenable K‑algebra without zero divisors, then any two linear sofic approximations of A are conjugate in the metric ultraproduct Q^ω GL_{n_k}(K). In other words, there is a unique conjugacy class of linear sofic approximations for such algebras. The proof proceeds by developing a “linear monotiling’’ technique, the algebraic analogue of Ornstein–Weiss quasi‑tilings for amenable groups. The key ingredients are:

1. Følner subspaces: Amenability of A provides a sequence of finite‑dimensional subspaces W_n that are almost invariant under multiplication by any fixed finite set.
2. Locally linearly dependent operators: The Brešar‑Meshulam‑Šemrl theorem guarantees, for any finite set of operators that are locally linearly dependent, a non‑trivial linear combination of rank at most d‑1. This yields many “root vectors’’ for a given finite‑dimensional subspace W.
3. Root vectors and tiles: Lemma 4.1 shows that for a d‑approximation ϕ_k and a subspace W, one can find a vector u such that the map w↦ϕ_k(w)u is injective and multiplicative on W, while avoiding a prescribed small exceptional subspace. These vectors serve as the building blocks of tiles.
4. Quasi‑tiling of the algebra: By selecting a finite collection of subspaces (tiles) that together cover almost all of A (up to a small error) and using the root vectors, one constructs matrices M_k that almost intertwine two given approximations on each tile. The error on each tile can be made arbitrarily small by choosing d large enough.

Putting these pieces together yields matrices M_k with rk(ϕ_k(a)−M_kψ_k(a)M_k^{-1})→0 for every a∈A, establishing conjugacy.

The paper then applies this rigidity theorem to the problem of weak stability in the rank metric. A linear sofic approximation is weakly stable if it can be approximated arbitrarily well (in rank) by an actual representation. Theorem 1.2 proves that for a torsion‑free amenable group Γ whose group algebra KΓ has no zero divisors, KΓ is weakly stable if and only if Γ is residually finite. The forward direction uses Theorem 1.1 to replace any linear sofic approximation by a conjugate of a genuine permutation approximation; the converse follows from classical results that residually finite amenable groups admit Følner monotiles that can be turned into exact permutations. The result extends to any amenable algebra that admits a representation separating its elements maximally in rank.

An explicit example is given: the group algebra of Abels’ group A_p is weakly stable but not stable, mirroring known results for permutation stability.

The paper concludes by noting that while the converse of Theorem 1.1 (uniqueness ⇒ amenability) is open for algebras, the techniques introduced—particularly the linear monotiling built on the Brešar‑Meshulam‑Šemrl theorem—are substantially more efficient than classical group tilings and may be adaptable to non‑commutative or operator‑algebraic contexts. Future work is suggested on extending these rigidity phenomena beyond amenable algebras, improving amplification tricks for linear sofic algebras, and exploring connections with C*‑algebraic soficity.