A continuous field of Roe algebras

TL;DR

This work studies how Roe-type algebras vary when a metric measure space $X$ is approximated by controlled Delone subsets. It constructs a tautological family of algebras $D\mapsto C^*_u(D)$ together with $X\mapsto C^*_k(X)$ and proves that for sequences $D_n\to X$ these algebras form a continuous field over $\mathbb{N}\cup\{\infty\}$, provided $X$ is proper with bounded geometry and no isolated points. The authors develop partitions of unity adapted to Delone sets, build explicit isometries between $\ell^2(D)$ and $L^2(X)$, and show density results ensuring the continuity of the field; this extends previous work by replacing the Roe algebra $C^*(X)$ with $C^*_k(X)$ and generalizing to wider metric spaces. The construction clarifies how coarse geometric invariants encoded in Roe-type algebras behave under discrete approximations and provides a robust framework for coarse index theory in this setting.

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 $C^*_u(D)$ (resp., the \u Spakula's version $C_k^*(X)$ of the Roe algebra of $X$) we get a tautological family of $C^*$-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 $C^*_u(D_n)$, together with $C^*_k(X)$, form a continuous field of $C^*$-algebras over $\mathbb N\cup\{\infty\}$ when $X$ is a proper metric measure space of bounded geometry with no isolated points.

Theorems & Definitions (35)

• Definition 1.1
• Theorem 1.2
• proof
• Lemma 1.3
• proof
• Definition 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof