Almost commuting self-adjoint matrices --- the real and self-dual cases

TL;DR

This paper proves that almost commuting self-adjoint matrices, including real and self-dual cases relevant in physics, are close to exactly commuting matrices, extending Huaxin Lin's theorem to these settings and developing related algebraic theory.

Contribution

It extends the theory of almost commuting matrices to real and self-dual cases, including applications to real C*-algebras and paths of matrices, with implications in physics.

Findings

Almost commuting symmetric matrices are close to commuting symmetric matrices.

Self-dual matrices over quaternions exhibit similar approximation properties.

Develops a theory of semiprojectivity for real C*-algebras.

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. The notion of self-dual Hermitian matrices is important in physics when studying fermionic systems that have time reversal symmetry. 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. We prove analogous results for element of real C^*-algebras of "low rank." In particular, these stronger results apply to paths of almost commuting Hermitian matrices that are real or self-dual. Along the way we develop a theory of semiprojectivity for real C^*-algebras.