Are almost commuting hermitian matrices close to commuting matrices (in the 2-norm)?

There is a recent paper by Glebsky titled "Almost commuting matrices with respect to normalized Hilbert-Schmidt norm" which shows that this is indeed true for any $k$ for Hermitian matrices (and in fact also unitary and normal matrices).

The answer is yes, and much more is true. Any hyperfinite von Neumann algebra (with separable predual) has a unique embedding (up to conjugacy) into the ultra-product of the hyperfinite $II_1$-factor.

This implies in particular, that almost commuting matrices in Hilbert-Schmidt are close to commuting matrices. The proofs goes by contradiction; assume that there is a sequence of counterexamples and construct non-conjugate embeddings. Since any abelian von Neumann algebra is hyperfinite, this yields a contradiction.

Kenley Jung showed that uniqueness of the embedding also implies that the algebra is hyperfinite.

I just found the discussion. In this paper by Filonov and Kachkovskiy there are better estimates than mines and it contains citations of proofs using von Neumann algebras.

(It was a surprise for me too why my paper is in Algebraic Geometry. Probably it is my error. I have not found an easy way to fix it.)