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

I consider on $$M_n(\mathbb C)$$ the normalized $$2$$-norm, i.e. the norm given by $$\|A\|_2 = \sqrt{\mathrm{Tr}(A^* A)/n}$$.

My question is whether a $$k$$-uple of hermitian matrices that are almost commuting (with respect to the $$2$$-norm) is close to a $$k$$-uple of commuting matrices (again with respect to the $$2$$-norm). More precisely, for an integer $$k$$, is the following statement true?

For every $$\varepsilon>0$$, there exists $$\delta>0$$ such that for any $$n$$ and any matrices $$A_1,\dots, A_k\in M_n(\mathbb C)$$ satisfying $$0\leq A_i\leq 1$$ and $$\|A_iA_j - A_j A_i\|_2 \leq \delta$$, there are commuting matrices $$\tilde A_1,\dots,\tilde A_k$$ satisfying $$0\leq \tilde A_i\leq 1$$ and such that $$\|A_i - \tilde A_i\|_2 \leq \varepsilon$$.

The important point is that $$\delta$$ does not depend on $$n$$.

I could not find a reference to this problem in the litterature. However, this question with the $$2$$-norm replaced by the operator norm is well-studied. And the answer is known to be true if $$k=2$$ (a result due to Lin) and false for $$k=3$$, and hence $$k\geq 3$$ (a result of Voiculescu).

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).

• Thanks a lot for your answer. I did not yet have time to read the whole proof, but this seems to completely answer my question. I just wonder why this paper was posted in the algebraic geometry section of arXiv. – Mikael de la Salle Jun 21 '11 at 14:31

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.

• Andreas, is your argument presupposing that the matrices almost commute in HS-norm and are uniformly bounded in operator norm? – Yemon Choi Jun 21 '11 at 21:16
• Thanks Andreas. I already accepted Ashley's answer, but your answer is more what I was hoping for (Glebsky's proof has however the advantage of giving an explicit bound on $\delta$ depending on $k$ and $\varepsilon$). – Mikael de la Salle Jun 22 '11 at 6:42
• I guess that the precise statement is :"Given a non principal ultrafilter $\mathcal U$, any two embeddiing of a hyperfinite von Neumann algebra into $\prod_{\mathcal U} M_n$ (or $R^{\mathcal U}$) are conjugate". (unless I missed something, the ultrapower might depend on $\mathcal U$). By the way, do you have a reference for this statement? – Mikael de la Salle Jun 22 '11 at 6:57
• Yemon: yes, just as in the question. – Andreas Thom Jun 22 '11 at 7:32
• Mikael: you are right, this might depend on the ultrafilter. I would try looking at the paper by Kenley Jung (Math. Ann. 2007 vol. 338 (1) pp. 241-248) and see whether he gives a reference. This is the easy part anyway, it is based on the fact that a full matrix-algebras embeds (unitally) in a unique way (up to conjugacy) into each other. The same holds if you take block-sums of matrix-algebras and remember their relative multiplicities. – Andreas Thom Jun 22 '11 at 7:36

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.)

As part of my dissertation, "Almost Commuting Operators on von Neumann Algebras," I have extended Glebsky's result to the normalized Schatten class for $1 \leq p < \infty$. Moreover, for $p=2$ we recover Filonov and Kachkovskiy's theorem with the same estimate. In our work, however, we use different techniques as the normalized Schatten p-norm does not arise from an inner product for $p \neq 2$.