Summary: We give a self-contained and elementary proof of Huaxin Lin’s theorem that pairs of almost commuting selfadjoint matrices are near commuting pairs of selfadjoint matrices (in a uniform way). As in Lin’s proof, the result is obtained by showing that a certain corona \(C^*\)-algebra has property (FN), i.e. any normal element can be approximated by a normal element with finite spectrum.
We prove a generalization of Lin’s theorem, that almost commuting selfadjoint elements in any \(C^*\)-algebra with property (IR) (a property weaker than stable rank one) are close to commuting selfadjoint elements (again in a uniform way).
Using similar methods, we give a necessary and sufficient condition for a \(C^*\)-algebra to have property (FN), and from this it follows in particular that every \(C^*\)-algebra of real rank zero, stable rank one and with trivial \(K_1\)-group has property (FN).