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In this paper, we introduce a notion of strongly quasi-local algebras. Theyare defined for each discrete metric space with bounded geometry, and sitbetween the Roe algebra and the quasi-local algebra. We show that stronglyquasi-local algebras are coarse invariants, hence encoding coarse geometricinformation of the underlying spaces. We prove that for a discrete metric spacewith bounded geometry which admits a coarse embedding into a Hilbert space, theinclusion of the Roe algebra into the strongly quasi-local algebra induces anisomorphism in $K$-theory.

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