Almost commuting self-adjoint matrices: The real and self-dual cases

Abstract

We show that a pair of almost commuting self-adjoint, symmetric matrices is close to a pair of commuting self-adjoint, symmetric matrices (in a uniform way). Moreover, we prove that the same holds with self-dual in place of symmetric and also for paths of self-adjoint matrices. Since a symmetric, self-adjoint matrix is real, we get a real version of Huaxin Lin’s famous theorem on almost commuting matrices. Similarly, the self-dual case gives a version for matrices over the quaternions.To prove these results, we develop a theory of semiprojectivity for real [Formula: see text]-algebras and also examine various definitions of low-rank for real [Formula: see text]-algebras.