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

Abstract

A central question in higher index theory and operator algebras is whether the Roe algebra and the quasi-local algebra associated with a metric space of bounded geometry coincide, or at least have the same $K$-theory. In this paper, we focus on a \emph{sparse} metric space $X$. We prove the following three main results: (1) For a block-diagonal operator $T$ with uniformly bounded block-rank, $T$ is quasi-local if and only if it is in the Roe algebra. (2) In general, we discover a ghost block-diagonal projection which is quasi-local but not in the Roe algebra. (3) For a sequence of expander graphs with sufficiently large girth, the inclusion of the uniform Roe algebra into the uniform quasi-local algebra induces a \emph{non-surjective} map on their $K_0$-groups. This yields the first known $K$-theoretic distinction between the uniform Roe algebra and the uniform quasi-local algebra.