Almost commuting unitaries with spectral gap are near commuting unitaries

Details Almost commuting unitaries with spectral gap are near commuting unitaries Almost commuting unitaries with spectral gap are near commuting unitaries

2008-09-03

arxiv.org/abs/0809.0602v1

Mathematics

Operator Algebras

5 pages

Scientific paper

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.

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