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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).

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4 Answers 4

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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).

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    $\begingroup$ 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. $\endgroup$
    – Mikael de la Salle
    Commented Jun 21, 2011 at 14:31
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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.

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  • $\begingroup$ Andreas, is your argument presupposing that the matrices almost commute in HS-norm and are uniformly bounded in operator norm? $\endgroup$
    – Yemon Choi
    Commented Jun 21, 2011 at 21:16
  • $\begingroup$ 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 $\varepsilon$). $\endgroup$
    – Mikael de la Salle
    Commented Jun 22, 2011 at 6:42
  • $\begingroup$ 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? $\endgroup$
    – Mikael de la Salle
    Commented Jun 22, 2011 at 6:57
  • $\begingroup$ Yemon: yes, just as in the question. $\endgroup$
    – Andreas Thom
    Commented Jun 22, 2011 at 7:32
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    $\begingroup$ 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. $\endgroup$
    – Andreas Thom
    Commented Jun 22, 2011 at 7:36
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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.)

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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 $p \neq 2$.

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