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Let \(X\) be a uniformly discrete metric space. We say that \(X\) has
property A if for every \(\epsilon > 0\) and \(R>0\) there exists a
collection of finite subest \(\{ A_{x} \}_{x \in X}, A_{x} \subseteq X
\times \mathbb{N}\) for every \(x \in X\), and a constant \(S > 0\) such
that
- \(\frac{\#(A_{x}\Delta A_{y})}{\#(A_{x} \cap A_{y})} \leq \epsilon \text{ when } d(x,y) \leq R, \text{ and}\)
- \(A_{x} \subseteq B(x,S) \times \mathbb{N}\).
Let \(X\) be a uniformly discrete metric space with bounded
geometry. \(X\) has property A if and only if for every \(\epsilon >
0\) and \(R > 0\) there exists a map \(\xi: X \to \ell^1(X)_{1,+}, x \maps
\to \xi_{x}\), and a number \(S > 0\) such that
- \(\left\| \xi_{x} - \xi_{y} \right\| \leq \epsilon \text{ if } d(x,y) \leq R \text{ and} \)
- \(\supp \xi_{x} \subseteq B(x,S)\).
Let \(X\) be a metric space with property A and let \(\{ A_{x} \}\) and \(S
> 0\) satisfy the conditions of property A for a given \(\epsilon > 0\)
and \(R > 0\). If \(d(x,y) \leq R\) then
\[
1 \leq \frac{\#A_{x}}{\#(A_{x} \cap A_{y})} \leq 1 + \epsilon \text{ and }
\frac{1}{1+\epsilon} \leq \frac{\#A_{x}}{\#A_{y}} \leq 1 + \epsilon.
\]
Let \(\{ X_{i} \}_{i \in \mathbb{N}}\) be a sequence of finite metric
spaces. The coarse disjoint union of \(X_{i}\) has property A if and
only if for every \(\epsilon > 0\), \(R > 0\) and for all but finitely
many \(i \in \mathbb{N}\) the space \(X_{i}\) has a Higson-Roe function
\(\xi^i\) for \(\epsilon\) and \(R\) satisfying \(\supp \xi^i_{g} \subseteq
B(x,S)\), where \(S > 0\) independent of \(i\).