Almost commuting unitaries with spectral gap are near commuting unitaries

Abstract

Let M_n be the collection of n x n complex matrices equipped with operator norm. Suppose U, V \in M_n are two unitary matrices, each possessing a gap larger than \Delta in their spectrum, which satisfy ||UV-VU|| \le \epsilon. Then it is shown that there are two unitary operators X and Y satisfying XY-YX = 0 and ||U-X|| + ||V-Y|| \le E(\Delta^2/\epsilon) (\epsilon/\Delta^2)^(1/6), where E(x) is a function growing slower than x^(1/k) for any positive integer k.