Almost commuting matrices, cohomology, and dimension

Abstract

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.

Almost commuting matrices and stability for product groups

“…It was first explicitly posed by Rosenthal [Ro69] for the normalized Hilbert-Schmidt norm and by Halmos [Ha76] for the operator norm. Almost commuting matrices have since been studied extensively and found applications to several areas of mathematics, including operator algebras and group theory, quantum physics and computer science (see, e.g, the introductions of [LS13,ES19]). The most interesting case of this question is when the matrices are contractions, and "almost" and "close" are taken independent of their sizes.…”

Obstructions to countable saturation in corona algebras

“…We are grateful to Logan Hoehn (see the last paragraph of §3) and to Søren Eilers for turning our attention to [27] and [7].…”

“…The reason for restricting our attention to abelian C * -algebras is that, rather inconveniently for our purposes, the zero-set of ϕ n is not definable in the class of all C * -algebras. The reason for this is that the algebra C(S n−1 ) is not weakly semiprojective (see [27] and the introduction to [7]). However, the machinery of this section may be applicable to other weakly stable formulas (see [22]).…”