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

TL;DR

This paper introduces strongly quasi-local algebras for discrete metric spaces, demonstrating their role as coarse invariants and establishing $K$-theory isomorphisms under certain geometric conditions.

Contribution

It defines strongly quasi-local algebras, shows they are coarse invariants, and proves $K$-theory isomorphisms for spaces with coarse embeddings into Hilbert spaces.

Findings

Strongly quasi-local algebras sit between Roe and quasi-local algebras.

They are coarse invariants, capturing geometric information.

Inclusion induces $K$-theory isomorphism for spaces with coarse Hilbert space embeddings.

Abstract

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 and the quasi-local algebra. We show that strongly quasi-local algebras are coarse invariants, hence encoding coarse geometric information of the underlying spaces. We prove that for a discrete metric space with bounded geometry which admits a coarse embedding into a Hilbert space, the inclusion of the Roe algebra into the strongly quasi-local algebra induces an isomorphism in $K$-theory.