Almost commuting matrices, cohomology, and dimension

Abstract

We investigate which relations for families of commuting matrices are stable under small perturbations, or in other words, which commutative $C^{\ast }$-algebras $C(X)$ are matricially semiprojective. Extending the works of Davidson, Eilers-Loring-Pedersen, Lin and Voiculescu on almost commuting matrices, we identify the precise dimensional and cohomological restrictions for finite-dimensional spaces $X$ and thus obtain a complete characterization: $C(X)$ is matricially semiprojective if and only if $\dim (X)\le 2$ and $H^2(X;\mathbb{Q})=0$. We give several applications to lifting problems for commutative $C^{\ast }$-algebras, in particular to liftings from the Calkin algebra and to $l$-closed $C^{\ast }$-algebras in the sense of Blackadar.

arXiv commentAccepted to Annales scientifiques de l'Ecole normale superieure