Almost commuting matrices, cohomology, and dimension

Journal article, 2023

It is an old problem to investigate which relations for families of commuting matrices are stable under small perturbations, or in other words, which commutative C*-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) <= 2 and H-2(X; Q) = 0. We give several applications to lifting problems for commutative C*-algebras, in particular to liftings from the Calkin algebra and to l-closed C*-algebras in the sense of Blackadar.