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. 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.

Expanded references. 33 pages

References in corpus (14)

• Topological insulators and superconductors
• Floquet Topological Insulator in Semiconductor Quantum Wells
• Topological Criticality in the Chiral-Symmetric AIII Class at Strong Disorder
• Bulk and Boundary Invariants for Complex Topological Insulators: From K-Theory to Physics
• Twisted equivariant matter
• Disordered Topological Insulators: A Non-Commutative Geometry Perspective
• Topological Insulators and C^*-Algebras: Theory and Numerical Practice
• Matrix embeddings on flat $R^3$ and the geometry of membranes
• The non-commutative n-th Chern number
• Smooth gauge and Wannier functions for topological band structures in arbitrary dimensions
• Topology and Phases in Fermionic Systems
• Distance to normal elements in $C^{\ast }$-algebras of real rank zero
• Almost commuting unitary matrices related to time reversal
• Pictures of KK-theory for real C*-algebras and almost commuting matrices

Cited by in corpus (2)

• Local markers for crystalline topology
• Even spheres as joint spectra of matrix models