$K$-Theoretic Comparison of Roe and Quasi-Local Algebras via Projections

The Roe algebra and the quasi-local algebra are $C^{\ast }$-algebras associated to metric spaces, which play important roles in higher index theory and operator algebras. The key question is whether these two algebras and their $K$-theories are the same, which has drawn considerable attention over the past few years and led to various applications in geometry, topology and analysis. In this paper, we focus on a sparse metric space $X$ (\emph{i.e.}, $X={⨆}_{n\in \mathbb{N}}{X}_n$ for finite subspaces $X_n$ with $d(X_n,X_m)\to +\mathrm{\infty }$ for $n\mathrm{\ne }m$) and block diagonal projections $P$ (\emph{i.e.}, $P:=(\mathrm{S}\mathrm{O}\mathrm{T})$-${\sum }_n{P}_n$ for $P_n\in \mathfrak{B}({\mathrm{\ell }}^2(X_n))$). We prove three main results: (1) If the ranks of the $P_n$ are uniformly bounded, then $P$ is quasi-local if and only if it belongs to the uniform Roe algebra. (2) In general, there exists a ghost projection which is quasi-local but not in the uniform Roe algebra. (3) If $\{X_n{\}}_n$ are expander graphs with an extra girth condition, then the embedding of the uniform Roe algebra into the uniform quasi-local algebra does not induce an isomorphism on their $K_0$-groups. This is the first known negative result on their $K$-theories.