Almost commuting matrices need not be nearly commuting

Abstract:

Let ${\mathcal {M}_n}$ be the collection of $n \times n$ complex matrices with the Hilbert-space-operator norm. There exist two concrete matrices $A,B \in {\mathcal {M}_n}$ with $||A|| = 1 - 1/n,||B|| \leq 1,||AB - BA|| \leq 2/n$, but $||A - R|| + ||B - S|| \geq 1 - 1/n$ for all commuting pairs $R,S \in {\mathcal {M}_n}$. It is shown explicitly that there is a natural obstruction which prevents almost commuting matrices to get close to any commuting pairs.