A continuous field of Roe-type algebras

Abstract

Let $X$ be a metric measure space. A Delone subset $D\subset X$ is a uniformly discrete set coarsely equivalent to $X$. We consider the space ${\mathcal{D}}_F$ of controlled Delone subsets of $X$ with an appropriate metric and show that it, together with $X$ itself, is a compact space. By assigning to each point $D$ of ${\mathcal{D}}_F$ (resp., to $X$) the uniform Roe algebra $C_u^{\ast }(D)$ (resp., a certain version $C_k^{\ast }(X)$ of the Roe algebra of $X$), we get a tautological family of $C^{\ast }$-algebras. For a sequence $\{D_n{\}}_{n\in \mathbb{N}}$ of controlled Delone subsets convergent to $X$, we show that the corresponding uniform Roe algebras $C_u^{\ast }(D_n)$, together with $C_k^{\ast }(X)$, form a continuous field of $C^{\ast }$-algebras over $\mathbb{N}\cup \{\infty \}$ when $X$ is a proper metric measure space of bounded geometry with no isolated points.