On almost commuting unitary matrices
Operator Algebras
2025-12-22 v2 Functional Analysis
Spectral Theory
Abstract
A question going back to Halmos asks when two approximately commuting matrices of a certain kind are close to exactly commuting matrices of the same kind. It has long been known that there is a winding number obstruction for approximately commuting unitary matrices to be close, in a dimension-independent way, to genuinely commuting unitary matrices. In this paper, under the vanishing of the said obstruction, we obtain effective bounds for the distance to commuting unitary matrices in terms of the commutator of the original matrices.
Keywords
Cite
@article{arxiv.2510.03674,
title = {On almost commuting unitary matrices},
author = {Adam Dor-On and Lucas Hall and Ilya Kachkovskiy},
journal= {arXiv preprint arXiv:2510.03674},
year = {2025}
}
Comments
Submitted version. Minor updates in presentation. Added Section 3.4