2.1. Basic notation

Here, we collect the notation used throughout the paper.

For a set $X$, denote by $|X|$ the cardinality of $X$. For a subset $A \subseteq X$, denote by $\chi_{A}$ the characteristic function of $A$, and set $\delta_x:=\chi_{\{x\}}$ for $x\in X$.

When $X$ is a locally compact Hausdorff space, denote $C(X)$ the set of complex-valued continuous functions on $X$, and $C_b(X)$ the subset of bounded continuous functions on $X$. The support of $f\in C(X)$ is the closure of $\{x\in X: f(x)\neq 0\}$, written as $\mathrm{supp}(f)$, and denote $C_c(X)$ the set of continuous functions with compact support. We also denote by $C_0(X)$ the set of continuous functions vanishing at infinity with the supremum norm $\|f\|_\infty:=\sup\{|f(x)|: x\in X\}$.

When $X$ is discrete, denote $\ell^\infty(X):=C_b(X)$ and $\ell^2(X)$ the Hilbert space of complex-valued square-summable functions on $X$. Denote $\mathfrak{B}(\ell^2(X))$ the $C^*$-algebra of all bounded linear operators on $\ell^2(X)$, and $\mathfrak{K}(\ell^2(X))$ the $C^*$-subalgebra of all compact operators on $\ell^2(X)$.

For a discrete space $X$, denote $\beta X$ its Stone–Čech compactification and $\partial_\beta X:=\beta X \setminus X$ the Stone–Čech boundary.

For a discrete metric space $(X,d)$, denote $B_{X}(x,r):=\{y\in X: d(x,y)\leq r\}$ for $x\in X$ and $r\geq 0$. For a subset $A\subseteq X$ and $r \gt 0$, denote $\mathcal{N}_r(A):=\{x\in X: d_X(x,A) \leq r\}$. For $R \gt 0$, denote the $R$-entourage by $E_R :=\{(x,y) \in X \times X: d(x,y) \leq R\}$. We say that $(X,d)$ has bounded geometry if for any $r \gt 0$, $\sup_{x\in X} |B_{X}(x,r)| \lt \infty$. Also, say that $(X,d)$ is strongly discrete if the set $\{d(x,y): x,y \in X\}$ is a discrete subset of $\mathbb{R}$.

Convention. We say that ‘ $X$ is a space’ as shorthand for ‘ $X$ is a strongly discrete metric space of bounded geometry’ (as in [Reference Špakula and Willett29]) throughout the rest of this paper.

We remark that although our results hold without the assumption of strong discreteness, we choose to add it so as to simplify the proofs. As discussed in [Reference Špakula and Willett29, Section 2], this will not lose any generality.

Definition 2.2. Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces and $f: X \to Y$ be a map. We say that $f$ is a coarse embedding if there exist functions $\rho_{\pm}: [0,\infty) \to [0,\infty)$ with $\lim_{t\to +\infty}\rho_{\pm}(t) = +\infty$ such that for any $x,y\in X$ we have

\begin{equation*} \rho_-(d_X(x,y)) \leq d_Y(f(x),f(y)) \leq \rho_+(d_X(x,y)). \end{equation*}

If additionally there exists $C \gt 0$ such that $Y=\mathcal{N}_C(f(X))$, then we say that $f$ is a coarse equivalence and $(X,d_X), (Y,d_Y)$ are coarsely equivalent.

2.2. Groupoids and $C^*$-algebras

Here, we collect some basic notions and terminology on groupoids. Details can be found in [Reference Renault38], or [Reference Sims48] in the étale case.

A groupoid is a small category in which every morphism is invertible. More precisely, a groupoid consists of a set $\mathcal{G}$, a subset $\mathcal{G}^{(0)}$ called the unit space, two maps $\mathrm{s},\mathrm{r}: \mathcal{G} \to \mathcal{G}^{(0)}$ called the source and range maps, a composition law:

\begin{equation*} \mathcal{G}^{(2)}:=\{(\gamma_1,\gamma_2) \in \mathcal{G} \times \mathcal{G}: \mathrm{s}(\gamma_1)=\mathrm{r}(\gamma_2)\}\ni(\gamma_1,\gamma_2) \mapsto \gamma_1\gamma_2 \in \mathcal{G}, \end{equation*}

and an inverse map $\gamma \mapsto \gamma^{-1}$, which satisfy a couple of axioms.

For $x \in \mathcal{G}^{(0)}$, denote $\mathcal{G}^x:=\mathrm{r}^{-1}(x)$ and $\mathcal{G}_x:=\mathrm{s}^{-1}(x)$. A subset $Y\subseteq \mathcal{G}^{(0)}$ is called invariant if $\mathrm{r}^{-1}(Y)=\mathrm{s}^{-1}(Y)$, and denote $\mathcal{G}_Y:=\mathrm{r}^{-1}(Y)$.

A locally compact Hausdorff groupoid is a groupoid with a locally compact Hausdorff topology such that the structure maps are continuous. Such a groupoid is called étale if the range map is a local homeomorphism.

Now we recall the groupoid $C^*$-algebras. Let $\mathcal{G}$ be a locally compact, Hausdorff and étale groupoid with unit space $\mathcal{G}^{(0)}$. The space $C_c(\mathcal{G})$ is a $\ast$-involutive algebra with respect to the following operations: for $f,g \in C_c(\mathcal{G})$,

\begin{eqnarray*} (f \ast g)(\gamma) &=& \sum_{\alpha \in \mathcal{G}_{\mathrm{s}(\gamma)}} f(\gamma \alpha^{-1}) g(\alpha), \\ f^*(\gamma) &=& \overline{f(\gamma^{-1})}. \end{eqnarray*}

The maximal (full) groupoid $C^*$-algebra $C^*_{\mathrm{max}}(\mathcal{G})$ is the completion of $C_c(\mathcal{G})$ with respect to the norm $\|f\|_{\mathrm{max}}:=\sup \|\pi(f)\|$, where the supremum is taken over all $\ast$-representations $\pi$ of $C_c(\mathcal{G})$. The regular representation at $x \in \mathcal{G}^{(0)}$, denoted by $\lambda_x: C_c(\mathcal{G}) \to \mathfrak{B}(\ell^2(\mathcal{G}_x))$, is defined by:

(2.1)\begin{equation} \big(\lambda_x(f)\xi\big)(\gamma):=\sum_{\alpha \in \mathcal{G}_x} f(\gamma \alpha^{-1})\xi(\alpha), \quad \text{where}~ f \in C_c(\mathcal{G})\text{~and~}\xi \in \ell^2(\mathcal{G}_x). \end{equation}

The reduced norm on $C_c(\mathcal{G})$ is $\|f\|_r:=\sup_{x \in \mathcal{G}^{(0)}} \|\lambda_x(f)\|$, and the reduced groupoid $C^*$-algebra $C^*_r(\mathcal{G})$ is the completion of the $\ast$-algebra $C_c(\mathcal{G})$ with respect to this norm. Then each $\lambda_x$ can be extended to a homomorphism $\lambda_x: C^*_r(\mathcal{G}) \to \mathfrak{B}(\ell^2(\mathcal{G}_x))$.

2.3. Uniform Roe algebras and the coarse groupoid

Let $(X,d)$ be a discrete metric space. Each $T \in \mathfrak{B}(\ell^2(X))$ can be written in the matrix form $T=(T(x,y))_{x,y\in X}$, where $T(x,y)=\langle T \delta_y, \delta_x \rangle \in \mathbb{C}$. Hence, $T \in \mathfrak{B}(\ell^2(X))$ can be regarded as an element in $\ell^\infty(X \times X)$. Denote by $\|T\|$ the operator norm of $T$ in $\mathfrak{B}(\ell^2(X))$, and $\|T\|_\infty$ the supremum norm when regarding $T$ as a function in $\ell^\infty(X \times X)$.

Given an operator $T \in \mathfrak{B}(\ell^2(X))$, we define the support of $T$ to be

\begin{equation*} \mathrm{supp}(T):=\{(x,y) \in X \times X: T(x,y) \neq 0\}, \end{equation*}

and the propagation of $T$ to be

\begin{equation*} \mathrm{prop}(T):= \sup\{d(x,y): (x,y) \in \mathrm{supp}(T)\}. \end{equation*}

Now we recall the notion of coarse groupoid from [Reference Skandalis, Tu and Yu49]. For a space $(X,d)$, the coarse groupoid $G(X)$ is defined (as a topological space) by:

\begin{equation*} G(X):=\bigcup_{r \gt 0}{\overline{E_r}}^{\beta (X \times X)} \subseteq \beta (X \times X). \end{equation*}

The projection maps $\mathrm{r}, \mathrm{s}: X \times X \to X$ onto the first and second coordinate can be extended to maps $G(X) \to \beta X$, still denoted by $\mathrm{r}$ and $\mathrm{s}$. It was shown in [Reference Skandalis, Tu and Yu49, Lemma 2.7] that the map $(\mathrm{r},\mathrm{s}): G(X) \to \beta X \times \beta X$ is injective, and hence $G(X)$ can be regarded as a subset of $\beta X \times \beta X$ and then endowed with the groupoid structure from Example 2.3, called the coarse groupoid of $X$. It was shown in [Reference Skandalis, Tu and Yu49, Proposition 3.2] that the coarse groupoid $G(X)$ is locally compact, Hausdorff, étale and principal. Clearly, the unit space of $G(X)$ can be identified with $\beta X$.

Given $f \in C_c(G(X))$, we define an operator $\theta(f)$ on $\ell^2(X)$ by setting its matrix coefficients to be $\theta(f)(x,y):=f(x,y)$ for $x,y\in X$. We have:

Regarding $T \in \mathfrak{B}(\ell^2(X))$ as an element in $\ell^\infty(X \times X)$, we denote by $\overline{T}$ the continuous extension of $T$ on $\beta(X \times X)$. Then $\mathrm{supp}(\overline{T}) = \overline{\mathrm{supp}(T)}$ and we have:

Furthermore, Property A and coarse embeddability for $(X,d)$ can also be described using the coarse groupoid $G(X)$. Let us recall the following:

Definition 2.8 ([Reference Anantharaman-Delaroche and Renault1])

A locally compact, Hausdorff and étale groupoid $\mathcal{G}$ is said to be (topologically) amenable if for any $\varepsilon \gt 0$ and compact $K \subseteq \mathcal{G}$, there exists $f \in C_c(\mathcal{G})$ with range in $[0,+\infty)$ such that for any $\gamma \in K$ we have

\begin{equation*} \big| \sum_{\alpha \in \mathcal{G}_{\mathrm{r}(\gamma)}} f(\alpha) - 1 \big| \lt \varepsilon \quad \text{and} \quad \sum_{\alpha \in \mathcal{G}_{\mathrm{r}(\gamma)}} |f(\alpha) - f(\alpha\gamma)| \lt \varepsilon. \end{equation*}

Recall that open or closed subgroupoids of amenable étale groupoids are amenable, and amenability is preserved under taking groupoid extensions. See [Reference Anantharaman-Delaroche and Renault1, Section 5] for details. Also, recall the following:

Tu introduced the notion of a-T-menability for groupoids (see [Reference Tu50] for a precise definition). Here, we only need the following significant result by Tu:

Finally, we record the following result for coarse groupoids:

2.4. Limit spaces and limit operators

Here, we recall the limit operator theory for metric spaces developed in [Reference Špakula and Willett29]. We will freely use the notion of ultrafilters, and related materials can be found in [Reference Brown and Ozawa9, Appendix A] and [Reference Roe44, Chapter 7.4].

Recall that $t: D \to R$ with $D,R \subseteq X$ is a partial translation if $t$ is a bijection from $D$ to $R$, and $\sup_{x\in X}d(x,t(x))$ is finite.

Denote by $X(\omega)$ all ultrafilters on $X$ compatible with $\omega$. A compatible family for $\omega$ is a collection of partial translations $\{t_\alpha\}_{\alpha \in X(\omega)}$ such that each $t_\alpha$ is compatible with $\omega$ and $t_\alpha(\omega)=\alpha$. Define a function $d_{\omega}: X(\omega) \times X(\omega) \to [0,\infty)$ by

\begin{equation*}d_{\omega}(\alpha, \beta):=\lim_{x\to \omega}d(t_\alpha(x),t_\beta(x)).\end{equation*}

It is shown in [Reference Špakula and Willett29, Proposition 3.7] that $d_\omega$ is a uniformly discrete metric of bounded geometry on $X(\omega)$, independent of the choice of $\{t_{\alpha}\}$. This leads to the following:

Limit spaces can be described in terms of the coarse groupoid $G(X)$:

Lemma 2.14 ([Reference Špakula and Willett29, Lemma C.3])

Given a non-principal ultrafilter $ \omega \in \beta X$, the map

\begin{equation*}F : X(\omega) \rightarrow G(X)_\omega ~, \quad \alpha \mapsto (\alpha, \omega)\end{equation*}

is a bijection. Hence, $X(\omega)$ is the smallest invariant subset of $\beta X$ containing $\omega$.

We record the following observation, whose proof is straightforward and almost identical to that of Lemma 2.14 (originally from [Reference Roe44, discussion in 10.18-10.24]):

Lemma 2.15. Let $t: D \to R$ be a partial translation on $X$. Then we have:

\begin{equation*} \overline{\mathrm{gr}(t)}^{\beta X \times \beta X} = \mathrm{gr}(t) \sqcup \bigcup_{\omega \in \partial_\beta X} \big\{(\alpha, \omega): \omega (D) =1 \text{and } \alpha = t(\omega)\big\}. \end{equation*}

Here, we write $\mathrm{gr}(t)$ for the graph of $t$.

Consequently, we have the following:

Lemma 2.16. For any $S\geq 0$, we have:

\begin{equation*} \overline{E_S}^{\beta X \times \beta X} = E_S \sqcup \bigcup_{\omega \in \partial_\beta X} \big\{(\alpha, \gamma)\in X(\omega) \times X(\omega): d_\omega(\alpha, \gamma) \leq S \big\}. \end{equation*}

Proof. Since $(X,d)$ has bounded geometry, we decompose

\begin{equation*} E_S= \mathrm{gr}(t_1) \sqcup \cdots \sqcup \mathrm{gr}(t_N) \end{equation*}

where each $t_i: D_i \to R_i$ is a partial translation. Hence, we have

\begin{equation*} \overline{E_S} = \overline{\mathrm{gr}(t_1)} \cup \cdots \cup \overline{\mathrm{gr}(t_N)}. \end{equation*}

Applying Lemma 2.15, $\overline{E_S}$ is contained in the right-hand side of the statement.

Conversely, given $\omega \in \partial_\beta X$ and $\alpha, \gamma \in X(\omega)$ with $d_\omega(\alpha, \gamma) \leq S$, we have $(X(\omega), d_\omega) = (X(\gamma),d_\gamma)$. Take a partial translation $t: D \to R$ such that $\gamma(D) =1$ and $\alpha = t(\gamma)$. Note that

\begin{equation*} d_\gamma(\alpha, \gamma) = \lim_{x \to \gamma} d(t(x),x) \leq S. \end{equation*}

Hence, for $D':=\{x\in D: d(t(x),x) \leq S\}$, we have $\gamma(D')=1$. Consider the restriction of $t$ on $D^{\prime}$, denoted by $t^{\prime}$. Then $t^{\prime}$ is also a partial translation and $\alpha = t'(\gamma)$. By Lemma 2.15, we obtain that $(\alpha, \gamma) \in \overline{\mathrm{gr}(t')}$, which is contained in $\overline{E_S}$ as desired.

Now we recall the notion of limit operators for metric spaces:

Definition 2.17 ([Reference Špakula and Willett29, Definition 4.4])

For a non-principal ultrafilter $\omega$ on $X$, fix a compatible family $\{t_\alpha\}_{\alpha \in X(\omega)}$ for $\omega$ and let $T \in C^*_u(X)$. The limit operator of $T$ at $\omega$, denoted by $\Phi_\omega(T)$, is an $X(\omega)$-by- $X(\omega)$ indexed matrix defined by

\begin{equation*} \Phi_\omega(T)_{\alpha\gamma}:=\lim_{x \to \omega}T_{t_{\alpha}(x)t_{\gamma}(x)} \quad \text{for} \quad \alpha, \gamma \in X(\omega). \end{equation*}

It is shown in [Reference Špakula and Willett29, Chapter 4] that the above definition does not depend on the choice of the compatible family $\{t_\alpha\}_{\alpha \in X(\omega)}$. Furthermore, the limit operator $\Phi_\omega(T)$ is indeed a bounded operator on $\ell^2(X(\omega))$ and moreover, $\Phi_\omega(T) \in C^*_u(X(\omega))$.

The following is implicitly mentioned in the proof of [Reference Špakula and Willett29, Lemma C.3]:

Lemma 2.18. For a non-principal ultrafilter $\omega$ on $X$ and $T \in C^*_u(X)$, we have

\begin{equation*} \Phi_\omega(T)_{\alpha\gamma} = \overline{T}(\alpha, \gamma) \quad \text{for} \quad \alpha, \gamma \in X(\omega). \end{equation*}

Since the limit operator $\Phi_\omega(T)$ contains the information of the asymptotic behaviour of $T$ ‘in the $\omega$-direction’, we introduce the following:

6.1. Partial Property A

Recall from Proposition 5.1 that a space $X$ has Property A if and only if the groupoid $G(X)_{\partial_\beta X}$ is amenable, which characterizes $I(X) = \tilde{I}(X)$. Together with Proposition 5.2, this inspires us to introduce the following:

It is clear from the definition that $X$ has Property A if and only if it has partial Property A relative to $X$. On the other hand, it follows from Proposition 5.2 that if $X$ has partial Property A relative to some invariant open $U$, then we have $I(U) = \tilde{I}(U)$. The rest of this section is devoted to studying the converse.

Firstly, we aim to unpack the groupoid language and provide a concrete geometric description for partial Property A, which resembles the definition of Property A (see Definition 2.1). The following is the main result:

To prove Proposition 6.2, we start with the following lemma:

Proof. By definition, $X$ has partial Property A relative to $U$ if and only if for any $\varepsilon \gt 0$ and compact $K \subseteq G(X)_{U^c}$, there exists $g \in C_c(G(X)_{U^c})$ with range in $[0,1]$ such that for any $\gamma \in K$ we have

\begin{equation*} \sum_{\alpha \in \mathcal{G}_{\mathrm{r}(\gamma)}} g(\alpha) =1 \quad \text{and} \quad \sum_{\alpha \in \mathcal{G}_{\mathrm{r}(\gamma)}} |g(\alpha) - g(\alpha\gamma)| \lt \varepsilon. \end{equation*}

Since the restriction map $C_c(G(X)) \to C_c(G(X)_{U^c})$ is surjective, $g$ can be regarded as a function in $C_c(G(X))$. Taking $f$ to be the restriction of $g$ on $X \times X$, then $f \in \ell^\infty(X \times X)$ and there exists $S \gt 0$ such that $\mathrm{supp}(f) \subseteq E_S$ for some $S \gt 0$. Moreover, $g = \overline{f}$. Note that $G(X)_{U^c} = \bigcup_{R \gt 0} (\overline{E_R} \cap G(X)_{U^c})$, and hence $K \subseteq \overline{E_R} \cap G(X)_{U^c}$ for some $R \gt 0$. By Lemma 2.14 and Lemma 2.16, we conclude the proof.

Proof of Proposition 6.2

Sufficiency: For any $\varepsilon, R \gt 0$, choose $S \gt 0$, $D \subseteq X$ and $g: X \times X \to [0,1]$ satisfying the conditions (1)-(3) for $\varepsilon$ and $3R$. Take a map $p: \mathcal{N}_R(D) \to D$ such that $p|_D = \mathrm{Id}_D$ and $d(p(x), x) \leq R$. Now we define:

\begin{equation*} f(x,y) = \begin{cases} ~g(x,p(y)), & y\in \mathcal{N}_R(D); \\ ~g(x,y), & \text{otherwise}. \end{cases} \end{equation*}

It is clear that $\mathrm{supp}(f) \subseteq E_{R+S}$. Moreover, for any $y_1, y_2 \in \mathcal{N}_R(D)$ with $d(y_1, y_2) \leq R$, we have $d(p(y_1),p(y_2)) \leq 3R$ and hence

\begin{equation*} \sum_{x\in X} |f(x,y_1) - f(x,y_2)| = \sum_{x\in X} |g(x,p(y_1)) - g(x,p(y_2))| \leq \varepsilon. \end{equation*}

Therefore, (enlarging $S$ to $S+R$), we obtain that for any $\varepsilon, R \gt 0$, there exist $S \gt 0$, a subset $D \subseteq X$ with $\overline{D} \supseteq U^c$ and a function $f: X \times X \to [0,1]$ satisfying:

1. (1) $\mathrm{supp}(f) \subseteq E_S$;

2. (2) for any $x\in D$, we have $\sum_{z\in X} f(z,x) =1$;

3. (3) for any $x\in D$ and $y\in X$ with $d(x,y) \leq R$, we have $\sum_{z\in X} |f(z,x) - f(z,y)| \leq \varepsilon$.

Now we fix $\varepsilon, R \gt 0$ and take such $S, D$ and function $f$.

Given $\omega \in U^c$, we have $\omega(D)=1$. Choose $\{t_\alpha:D_\alpha \to R_\alpha\}$ to be a compatible family for $\omega$. Applying [Reference Špakula and Willett29, Proposition 3.10], there exists $Y_\omega \subseteq X$ with $\omega(Y)=1$ and a local coordinate system $\{h_y: B(\omega, R+S) \to B(y,R+S)\}_{y\in Y_\omega}$ such that the map

\begin{equation*} h_y: B(\omega, R+S) \to B(y,R+S), \quad \alpha \mapsto t_\alpha(y) \end{equation*}

is a surjective isometry for each $y\in Y_\omega$. Replacing $Y$ by $Y_\omega \cap D$, we assume that $Y_\omega \subseteq D$. Note that $\mathrm{supp}(f) \subseteq E_S$, and hence applying Lemma 2.16 we have

(6.1)\begin{align} \sum_{\alpha \in X(\omega)} \overline{f}(\alpha, \omega) & = \sum_{\alpha \in B(\omega, S)} \overline{f}(\alpha, \omega) = \sum_{\alpha \in B(\omega, S)} \lim_{x\to \omega} f(t_\alpha(x),x)\nonumber\\ &= \lim_{x\to \omega, x\in Y_\omega} \sum_{\alpha \in B(\omega, S)} f(t_\alpha(x),x) \nonumber \\ &= \lim_{x\to \omega, x\in Y_\omega} \sum_{z\in B(x,S)} f(z,x) = \lim_{x\to \omega, x\in Y_\omega} \sum_{z\in X} f(z,x), \end{align}

where the last item equals $1$ thanks to the assumption.

On the other hand, for any $\alpha \in B(\omega, R)$ we have $\lim_{x\to \omega} d(t_\alpha(x),x) \leq R$. Hence, shrinking $Y_\omega$ if necessary, we can assume $d(t_\alpha(x),x) \leq R$ for any $x\in Y_\omega$. Hence

(6.2)\begin{equation} \sum_{\gamma\in X(\omega)} |\overline{f}(\gamma, \alpha) - \overline{f}(\gamma, \omega)| = \sum_{\gamma\in B(\omega, R+S)} |\overline{f}(\gamma, \alpha) - \overline{f}(\gamma, \omega)|\qquad\qquad\qquad\qquad \end{equation}

\begin{align*} &\qquad\qquad\qquad\qquad\qquad\! = \lim_{x\to \omega, x\in Y_\omega}\sum_{\gamma\in B(\omega, R+S)} |f(t_\gamma(x), t_\alpha(x)) - f(t_\gamma(x), x)|\\ &\qquad\qquad\qquad\qquad\qquad\! = \lim_{x\to \omega, x\in Y_\omega}\sum_{z\in B(x, R+S)} |f(z, t_\alpha(x)) - f(z, x)| \\ &\qquad\qquad\qquad\qquad\qquad\! = \lim_{x\to \omega, x\in Y_\omega}\sum_{z\in X} |f(z, t_\alpha(x)) - f(z, x)| \leq \varepsilon. \end{align*}

Therefore, applying Lemma 6.3, we conclude the sufficiency.

Necessity: Given $\varepsilon, R \gt 0$, Lemma 6.3 provides $S \gt 0$ and $f: X \times X \to [0,1]$ satisfying the conditions (1)-(3) therein. For $\omega \in U^c$, choose a compatible family $\{t_\alpha:D_\alpha \to R_\alpha\}$. Applying [Reference Špakula and Willett29, Proposition 3.10], there exists $Y_\omega \subseteq X$ with $\omega(Y)=1$ and a local coordinate system $\{h_y: B(\omega, R+S) \to B(y,R+S)\}_{y\in Y_\omega}$ such that

\begin{equation*} h_y: B(\omega, R+S) \to B(y,R+S), \quad \alpha \mapsto t_\alpha(y) \end{equation*}

is a surjective isometry for each $y\in Y_\omega$. By the calculations in (6.1), we obtain that

\begin{equation*} \lim_{x\to \omega, x\in Y_\omega} \sum_{z\in X} f(z,x) =1. \end{equation*}

Hence, for the given $\varepsilon$, there exists $Y'_\omega \subseteq Y_\omega$ with $\omega(Y'_\omega)=1$ such that

\begin{equation*} \sum_{z\in X} f(z,x) \in (1-\varepsilon, 1+\varepsilon), \quad \forall x\in Y'_\omega. \end{equation*}

On the other hand, for any $\alpha \in B(\omega, R)$ the calculations in (6.2) show that

\begin{equation*} \lim_{x\to \omega, x\in Y'_\omega}\sum_{z\in X} |f(z, t_\alpha(x)) - f(z, x)| = \sum_{\gamma\in X(\omega)} |\overline{f}(\gamma, \alpha) - \overline{f}(\gamma, \omega)| \leq \varepsilon. \end{equation*}

For $x\in Y'_\omega$, [Reference Špakula and Willett29, Proposition 3.10] implies that $\{t_\alpha(x): \alpha \in B(\omega, R)\} = B(x,R)$. Hence, there exists $Y^{\prime\prime}_\omega \subseteq Y'_\omega$ with $\omega(Y^{\prime\prime}_\omega)=1$ such that for any $x \in Y^{\prime\prime}_\omega$ and $y\in B(x,R)$, we have

\begin{equation*} \sum_{z\in X} |f(z, y) - f(z, x)| \lt 2\varepsilon. \end{equation*}

Taking $D:=\bigcup_{\omega \in U^c} Y^{\prime\prime}_\omega$, clearly $\overline{D} \supseteq U^c$. Moreover, the analysis above shows that:

• • for any $x\in D$ we have $\sum_{z\in X} f(z,x) \in (1-\varepsilon, 1+\varepsilon)$;

• • for any $x\in D$ and $y\in B(x,R)$, we have $\sum_{z\in X} |f(z, y) - f(z, x)| \lt 2\varepsilon$.

Finally, using a standard normalization argument, we conclude the proof.

Setting $\xi_y(x):=f(x,y)$ for the function $f$ in Proposition 6.2, we can rewrite Proposition 6.2 as follows:

Now we provide an alternative picture for partial Property A using the notion of ideals in spaces (see Definition 3.10). Recall that for an ideal $\textbf{L}$ in $X$, we denote $U(\textbf{L}):= \bigcup_{Y \in \textbf{L}} \overline{Y}$. The following lemma is easy, and the proof is left to the readers.

Thanks to Lemma 6.5, we can rewrite Proposition 6.2:

6.2. Partial operator norm localization property

Let $\nu$ be a positive locally finite Borel measure on $X$ and $\mathcal{H}$ be a separable infinite-dimensional Hilbert space. For an operator $T \in \mathfrak{B}(L^2(X,\nu) \otimes \mathcal{H})$, we can also define its propagation as in Section 2.3. We introduce the following:

Using a similar argument as for [Reference Sako46, Proposition 3.1] together with Lemma 6.5, we have the following. We leave the details to the readers.

Now we can mimic the proof of [Reference Sako46, Theorem 4.1] using Proposition 6.2, Remark 6.4, Corollary 6.6 together with an analogue of [Reference Sako46, Proposition 3.1], and reach the following. The proof is almost identical, and hence the details are left to the readers.

Now we study a permanence property of partial ONL. Let $X$ be a space and $\textbf{L}$ an ideal in $X$. Assume that $X= X_1 \cup X_2$. Consider

(6.3)\begin{equation} \textbf{L}_i:=\{Y \cap X_i: Y \in \textbf{L}\} \quad \text{for} \quad i=1,2. \end{equation}

Then it is routine to check that $\textbf{L}_i$ is an ideal in $X_i$ for $i=1,2$, and

\begin{equation*} \textbf{L}=\{Y_1 \cup Y_2: Y_i \in \textbf{L}_i, i=1,2\}. \end{equation*}

One way to prove Proposition 6.10 is to follow the proof of [Reference Chen, Wang and Wang16, Lemma 3.3] with minor changes. Here, we choose another approach using Proposition 6.9. Firstly, we have the following (the proof is straightforward, hence omitted):

Although $U \cap \overline{X_i}$ is invariant in $\beta X_i$, generally it is not invariant in $\beta X$. This coincides with the fact that $\chi_{X_i} C^*_u(X) \chi_{X_i} \cong C^*_u(X_i)$ is just a subalgebra in $C^*_u(X)$ rather than an ideal. Instead, we consider the spatial ideal $I_{X_i}$, and prove the following permanence property for partial Property A:

Proof. For any $R \gt 0$ and $i=1,2$, set $U_i(R):=\overline{\bigcup_{Y \in \textbf{L}(U_i)} \mathcal{N}_R(Y)}$, which is an invariant open subset of $\overline{\mathcal{N}_R(X_i)} = \beta (\mathcal{N}_R(X_i))$. Proposition 6.2 implies that $\mathcal{N}_R(X_i)$ has partial Property A relative to $U_i(R)$, i.e., $G(\mathcal{N}_R(X_i))_{\overline{\mathcal{N}_R(X_i)} \setminus U_i(R)}$ is amenable. Hence, as a subgroupoid, $G(\mathcal{N}_R(X_i))_{\overline{\mathcal{N}_R(X_i)} \setminus U}$ is amenable, which further implies that the groupoid $\bigcup_{R \gt 0}G(\mathcal{N}_R(X_i))_{\overline{\mathcal{N}_R(X_i)} \setminus U}$ is amenable.

By Lemma 3.12, $\bigcup_R \overline{\mathcal{N}_R(X_i)}\setminus U$ is invariant in $\beta X$. Lemma 3.14 implies that

\begin{equation*} G(X)_{\bigcup_R \overline{\mathcal{N}_R(X_i)}\setminus U} = \bigcup_{R \gt 0} G(\mathcal{N}_R(X_i))_{\overline{\mathcal{N}_R(X_i)} \setminus U}, \end{equation*}

which is, hence, amenable. Note that $\bigcup_R \overline{\mathcal{N}_R(X_1)} \cup \bigcup_R \overline{\mathcal{N}_R(X_2)} = \beta X$, and hence

\begin{equation*} G(X)_{\beta X \setminus U} = G(X)_{\bigcup_R \overline{\mathcal{N}_R(X_1)}\setminus U} \cup G(X)_{\bigcup_R \overline{\mathcal{N}_R(X_2)}\setminus U} \end{equation*}

is amenable as required.

Combining Proposition 6.9 and Lemma 6.12, we conclude Proposition 6.10.

6.3. Countable generatedness

Here, we introduce the extra assumption we need to characterize $\tilde{I}(U) = I(U)$.

The property of being countably generated leads to the following:

Lemma 6.15. Let $\textbf{L}$ be a countably generated ideal in $X$. Then there exists a countable subset $\{Y_1, Y_2, \cdots, Y_n, \cdots\}$ in $\textbf{L}$ such that

\begin{equation*} \textbf{L}=\{Z\subseteq X: ~\exists~n\in \mathbb{N} \text{such that } Z \subseteq Y_n\}. \end{equation*}

To prove Lemma 6.15, we need an auxiliary result on the structure of $\textbf{L}(\mathcal{S})$. The proof is straightforward, hence omitted.

Lemma 6.16. For a set $\mathcal{S}$ of subsets of $X$, denote $\mathcal{S}^{(1)}:=\{A_1 \cup \cdots \cup A_n: A_i \in \mathcal{S}, n\in \mathbb{N}\}$ and $\mathcal{S}^{(2)}:=\{\mathcal{N}_k(\tilde{A}): \tilde{A} \in \mathcal{S}^{(1)}, k \in \mathbb{N}\}$. Then we have:

\begin{equation*} \textbf{L}(\mathcal{S}) = \{Z: \exists~Y \in \mathcal{S}^{(2)} \text{such that } Z \subseteq Y\}. \end{equation*}

The following example shows that not every ideal is countably generated.

Example 6.17. Let $X = \mathbb{N} \times \mathbb{N}$, equipped with the metric induced from the Euclidean metric $d_E$ on $\mathbb{R}^2$. For each $\theta \in [0, \frac{\pi}{2}]$ and $k\in \mathbb{N}$, we define

\begin{equation*} \ell_\theta:=\{(x,y) \in \mathbb{R} \times \mathbb{R}: y=\tan(\theta)x\} \end{equation*}

and

\begin{equation*} S_{\theta,k}:=\{(x,y) \in X: d_E((x,y), \ell_\theta) \leq k\} = \mathcal{N}_k(\ell_\theta) \cap X. \end{equation*}

Consider $\mathcal{S}:=\{S_{\theta,k}: \theta \in [0, \frac{\pi}{2}], k\in \mathbb{N}\}$ and set $\mathcal{S}^{(1)}$ as in Lemma 6.16. For any $R \gt 0$ and $A= A_1 \cup \cdots \cup A_n\in \mathcal{S}^{(1)}$ where $A_i \in \mathcal{S}$, we have $\mathcal{N}_R(A) = \mathcal{N}_R(A_1) \cup \cdots \cup \mathcal{N}_R(A_n)$, contained in some element in $\mathcal{S}$. Hence, applying Lemma 6.16, the ideal $\textbf{L}(\mathcal{S})$ generated by $\mathcal{S}$ is:

(6.4)\begin{equation} \textbf{L}(\mathcal{S})= \{Z: \exists~Y \in \mathcal{S}^{(1)} \text{such that } Z \subseteq Y\}. \end{equation}

We claim that $\textbf{L}(\mathcal{S})$ is not countably generated. Otherwise, there exists a countable subset $\mathcal{S}^{\prime}$ generating $\textbf{L}(\mathcal{S})$. Moreover, according to (6.4) we can assume that

\begin{equation*} \mathcal{S}' = \{Y_n: n\in \mathbb{N}\} \quad \text{where} \quad Y_n=S_{\theta_{n,1}, k_n} \cup \cdots \cup S_{\theta_{n,p_n}, k_{n}} \in \mathcal{S}^{(1)}. \end{equation*}

Choose $\theta \in [0,\frac{\pi}{2}] \setminus \{\theta_{n,i}: i=1,2,\cdots, p_n; n\in \mathbb{N}\}$, and set $Y=S_{\theta,1}$. Since $\mathcal{S}^{\prime}$ generates $\textbf{L}(\mathcal{S})$, Lemma 6.16 implies that there exist $R \gt 0$ and $Y_{m_1}, \cdots, Y_{m_l} \in \mathcal{S}^{\prime}$ such that

\begin{equation*} S_{\theta,1} =Y \subseteq \mathcal{N}_R(Y_{m_1}\cup \cdots\cup Y_{m_l}). \end{equation*}

The right-hand side is contained in a finite union of some $R^{\prime}$-neighbourhood of lines (in $\mathbb{R}^2$ crossing the origin) with slopes in the set

\begin{equation*} \big\{\tan(\theta_{n,i}): i=1,2,\cdots, p_n; n\in \mathbb{N}\big\}. \end{equation*}

This leads to a contradiction due to the choice of $\theta$.