Norms of commutators If an $n$ by $n$ complex matrix $A$ has trace zero, then it is a commutator, which means that there are $n$ by $n$ matrices $B$ and $C$ so that $A= BC-CB$. What is the order of the best constant $\lambda=\lambda(n)$ so that you can always choose $B$ and $C$ to satisfy the inequality $\|B\|\cdot \|C\| \le \lambda \|A\|$?
Added June 10:  Gideon Schechtman showed me that for normal $A$ you can take $B$ a permutation matrix and $\|C\|\le \|A\|$ s.t. $A=BC-CB$.
 A: Almost the references cited below discuss upper bounds (i.e., norm(commutator) $\le$ something). One of the most relevant results is in reference #3 that I alluded to in my comment above.

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*A short note on the Frobenius norm of the commutator

*The Frobenius norm and the commutator (PDF)

*How big can the commutator of two matrices be and how big is it typically?

*Commutators, Pinching, and Spectral variation (Bhatia and Kittaneh)

*Norm inequalities for commutators of normal operators
If you chase the citations to these papers in google scholar, you will find several more very interesting and relevant papers—though, I have not been able to (yet) find a paper that discusses lower-bounds like yours.
A: In a recent paper ([1]), Ravichandran and Srivastava (RS) study pavings for collections of matrices. Their main theorem claims to yield an improvement to the bound obtained by Johnson, Ozawa, and Schechtman (JOS). However, as noted by YCor in a comment, RS [1] cite the JOS work as satisfying a bound on $\max(\|B\|,\|C\|)$, instead of a bound on the product $\|B\| \|C\|$ as in Bill Johnson's answer above. 
But as YCor notes, we can scale $B$ by $\|A\|$ (or both $B$ and $C$ by suitably, e.g., $\sqrt{\|A\|}$), to recover the inequality for the case noted in the OP and in the JOS paper.
In particular, Ravichandran and Srivastava's results imply the following:

Corollary (Corollary 3 in [1]). Every zero trace matrix $A \in M_n(\mathbb{C})$ may be written as $A=[B,C]$ such that $\|B\|$, $\|C\| \le K\log^2(n)\|A\|$ for some universal constant $K$.

(By suitable scaling, this translates into $\|B'\|\|C'\| \le K^2\log^4(n)\|A\|$, for $[B',C']=A$). 
[1]. M. Ravichandran and N. Srivastava. Asymptotically Optimal Multi-Paving. arXiv. Jun 2017.
A: Ozawa, Schechtman, and I finally wrote up what we know on this question. The estimate is that for every $\epsilon > 0$ there is a constant $C_\epsilon$ so that for every $n$, $\lambda(n)\le C_\epsilon n^{\epsilon}$. The paper can be downloaded from the arXiv.
