A continuous field of Roe 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., the \u Spakula's 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.

11 pages, one of two main results of the previous version (about continuous fields) is generalized, and an error on the fiber at $\infty$ is fixed; another main result (about direct limits) does not allow a similar generalizations, so is skipped in the current version