Constructive Approaches to Lins Theorem for Almost Commuting Matrices

Lins theorem states that for all $epsilon > 0$, there is a $delta > 0$ such that for all $n geq 1$ if self-adjoint contractions $A,B in M_n(mathbb{C})$ satisfy $|[A,B]|leq delta$ then there are self-adjoint contractions $A,B in M_n(mathbb{C})$ with $[A,B]=0$ and $|A-A|,|B-B|<epsilon$. We present full details of the approach in arXiv:0808.2474, which seemingly is the closest result to a general constructive proof of Lins theorem. Constructive results for some special cases are presented along with applications to the problem of almost commuting matrices where $B$ is assumed to be normal and also to macroscopic observables.