2019/02/27 by Enders, Dominic, Shulman, Tatiana
#FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.1902.10451
We 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 H2(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.