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Bounding the matrix norm of a commutator $[A,B]$ in terms of the norms of $A$ and $B$

The setup is as in this question: Given a norm $N$ over ${\bf M}_n(\mathbb C)$, it is a natural question to find the best constant $C_N$ such that $$N([A,B])\le C_N N(A)N(B),\qquad\forall A,B\in{\bf M …