Minimal supports, projection localization, and quasi-local ghosts

Let X be a metric space whose balls have uniformly bounded cardinality at each fixed radius. We prove that every uniformly quasi-local sequence of nonzero projections on ℓ²(X) admits rank-one subprojections with uniform finite-propagation approximation, independently of the parent ranks. If every strong subseries of the parent projections lies in the uniform Roe algebra, the rank-one subprojections can be chosen on the full sequence with the same property. This answers the selection question posed by Braga, Farah, and Vignati in 2022. It also gives the geometric characterization of projection localization left open in that work: localization for finite-rank projections is equivalent to localization for rank-one projections and to absence of sparse subspaces consisting of ghostly measured asymptotic expanders. These conditions are further equivalent to compactness of every quasi-local ghost projection on every sparse subspace. In particular, this compactness holds for sparse spaces coarsely embeddable into any L^p-space, for 1 ≤ p < ∞. The selection and compactness results extend to locally compact projections with arbitrary Hilbert-space coefficients. Source repository: https://github.com/eulogica/minimal-support-selection