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?



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> 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 $\|A\_i - \tilde A\_i\|\_2 \leq \varepsilon$.
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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](http://ams.u-strasbg.fr/mathscinet/search/publdoc.html?pg1=MR&s1=1424963&loc=fromrevtext)) and false for $k=3$, and hence $k\geq 3$ (a result of [Voiculescu](https://mathscinet.ams.org/mathscinet-getitem?mr=617972)). 



Are almost commuting hermitian matrices close to commuting matrices (in the 2-norm)?
There is a [recent paper](http://arxiv.org/abs/1002.3082) 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).


