Almost commuting matrices need not be nearly commuting

Abstract

Let M n {\mathcal {M}_n} be the collection of n × n n \times n complex matrices with the Hilbert-space-operator norm. There exist two concrete matrices A , B ∈ M n A,B \in {\mathcal {M}_n} with | | A | | = 1 − 1 / n , | | B | | ≤ 1 , | | A B − B A | | ≤ 2 / n ||A|| = 1 - 1/n,||B|| \leq 1,||AB - BA|| \leq 2/n , but | | A − R | | + | | B − S | | ≥ 1 − 1 / n ||A - R|| + ||B - S|| \geq 1 - 1/n for all commuting pairs R , S ∈ M n 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.