Coarse directed limits of metric spaces admitting coarse embeddings into Hilbert spaces

Abstract

For an inductive system of metric spaces that are coarsely embeddable into Hilbert spaces, its directed limit in the category of metric spaces may not admit a coarse embedding into a Hilbert space. In this article, we will show that the corresponding directed limit in the category of coarse spaces (with morphisms being bornologous maps) is always coarsely embeddable into a Hilbert space.