Strongly quasi-local algebras and their $K$-theories

In this paper, we introduce a notion of strongly quasi-local algebras. They are defined for each discrete metric space with bounded geometry, and sit between the Roe algebra quasilocal algebra. We show that algebras coarse invariants, hence encoding geometric information underlying spaces. prove geometry which admits embedding into Hilbert space, inclusion induces an isomorphism in $K$-theory.