$K$-Theoretic Comparison of Roe and Quasi-Local Algebras via Projections
The Roe algebra and the quasi-local algebra are $C^*$-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.}, $X = \bigsqcup_{n\in \mathbb{N}} X_n$ for finite subspaces $X_n$ with $d(X_n, X_m) \to +\infty$ for $n\neq m$) and block diagonal projections $P$ (\emph{i.e.}, $P:=\mathrm{(SOT)}$-$\sum_{n} P_n$ for $P_n \in \mathfrak{B}(\ell^2(X_n))$). 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.