ALMOST COMMUTING MATRICES, COHOMOLOGY, AND DIMENSION

It is an old problem to investigate which relations for families of commuting matrices are stable under small perturbations, or in otherwords, 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.