bounded geometry, and $T_{i}$ be an operator in $\mathscr{B}(\ell^{2}(X_{i}))$
for each $i\in I$.

  1. (1)

Say that $(T_{i})_{i\in I}$ has uniformly finite propagation if there exists
$R>0$ such that for any $f,g\in\ell^{\infty}(X_{i})$ with
$d_{i}(\mathrm{supp}(f),\mathrm{supp}(g))>R$, then $fT_{i}g=0$.

  2. (2)

Given $\varepsilon>0$, $(T_{i})_{i\in I}$ is said to have uniformly finite
$\varepsilon$-propagation if there exists $R>0$ such that for any
$f,g\in\ell^{\infty}(X_{i})$ with
$d_{i}(\mathrm{supp}(f),\mathrm{supp}(g))>R$, then
$\|fT_{i}g\|<\varepsilon\|f\|_{\infty}\|g\|_{\infty}$.

  3. (3)

Say that $(T_{i})_{i\in I}$ is uniformly quasi-local if for any
$\varepsilon>0$, the family $(T_{i})_{i\in I}$ has uniformly finite
$\varepsilon$-propagation.

Recall that we defined the notion of compactly uniform quasi-locality for
operators on source fibres of a given groupoid in Definition 4.5. In fact,
this has close relation with Definition 5.3(3) since we can endow each source
fibre with a metric as follows333We remark that these metrics have also been
considered in [27] to study amenability for certain groupoids, using the
language of length functions on groupoids.. As explained above, this is the
key ingredient to attack Theorem 5.1 where the coarse geometry of the
underlying groupoid plays an important role.

Throughout the rest of this section, let $\mathcal{G}$ be a locally compact,
$\sigma$-compact and étale groupoid with unit space $\mathcal{G}^{(0)}$. We
also fix a sequence of subsets $\\{K_{n}\\}_{n\in\mathbb{N}}$ in $\mathcal{G}$
satisfying the following:

  1. (M.1)

$K_{0}=\mathcal{G}^{(0)}$, and $K_{n}$ is compact and symmetric for $n\geq 1$.

  2. (M.2)

$K_{n}\subset K^{\circ}_{n+1}$ for $n\geq 1$, where $K^{\circ}_{n+1}$ denotes
the interior of $K_{n+1}$.

  3. (M.3)

$K_{n}K_{m}\subset K_{n+m}$ for $n,m\geq 1$ and $\mathcal{G}=\bigcup_{n\geq
1}K_{n}$.

For convenience, when $\mathcal{G}^{(0)}$ is compact we further require that
$K_{0}\subseteq K^{\circ}_{1}$ as well. Using $\\{K_{n}\\}_{n}$, we can endow
a function $d_{x}$ on $\mathcal{G}_{x}\times\mathcal{G}_{x}$ for each
$x\in\mathcal{G}^{(0)}$ as follows:

(5.1)
$d_{x}(\gamma_{1},\gamma_{2})\coloneqq\inf\\{n\in\mathbb{N}:\gamma_{1}\gamma^{-1}_{2}\in
K_{n}\\}\quad\text{ for }\quad\gamma_{1},\gamma_{2}\in\mathcal{G}_{x}.$

It follows from (1)-(3) that $d_{x}$ is indeed a metric on $\mathcal{G}_{x}$.

######  Lemma 5.4.

With the same notation as above, the family
$\\{(\mathcal{G}_{x},d_{x})\\}_{x\in\mathcal{G}^{(0)}}$ has uniformly bounded
geometry.

###### Proof.

Since $\mathcal{G}$ is étale, for each $n\in\mathbb{N}$ there exists a
constant $C_{n}>0$ such that $K_{n}$ can be covered by at most $C_{n}$-many
open bisections, which implies that $\sharp(\mathcal{G}^{y}\cap K_{n})\leq
C_{n}$ for each $y\in\mathcal{G}^{(0)}$. Note that for each $n\in\mathbb{N}$,
$x\in\mathcal{G}^{(0)}$ and $\alpha\in\mathcal{G}_{x}$, an element
$\beta\in\mathcal{G}_{x}$ belongs to the ball $B_{\mathcal{G}_{x}}(\alpha,n)$
if and only if $\alpha\beta^{-1}\in
K_{n}\cap\mathcal{G}^{\mathrm{r}(\alpha)}$. Hence the ball
$B_{\mathcal{G}_{x}}(\alpha,n)$ contains at most $C_{n}$ points, which
concludes the proof. ∎

The following result indicates that the amenability of $\mathcal{G}$ affects
the coarse geometry of the metric family
$\\{(\mathcal{G}_{x},d_{x})\\}_{x\in\mathcal{G}^{(0)}}$. This will be crucial
in the proof of Theorem 5.1.

######  Lemma 5.5.

With the same notation as above, the family
$\\{(\mathcal{G}_{x},d_{x})\\}_{x\in\mathcal{G}^{(0)}}$ has uniform Property A
provided $\mathcal{G}$ is amenable.

###### Proof.

Since $\mathcal{G}$ is amenable, Lemma 2.6 implies that for any
$\varepsilon>0$ and positive $n\in\mathbb{N}$, there exists a non-negative
function $h\in C_{c}(\mathcal{G})$ of positive type such that

  * •

$h(x)\leq 1$ for any $x\in\mathcal{G}^{(0)}$, and $h(x)=1$ for any
$x\in\mathrm{r}(K_{n})$;

  * •

$|1-h(\gamma)|<\varepsilon$ for any $\gamma\in K_{n}$.

For each $x\in\mathcal{G}^{(0)}$, we define a function
$k_{x}\colon\mathcal{G}_{x}\times\mathcal{G}_{x}\rightarrow[0,1]$ by

$k_{x}(\gamma,\beta)=h(\gamma\beta^{-1})\quad\text{for}\quad\gamma,\beta\in\mathcal{G}_{x}.$

It follows that $k_{x}$ is a symmetric kernel on $\mathcal{G}_{x}$ of positive
type such that:

  * •

the family $(k_{x})_{x\in\mathcal{G}^{(0)}}$ has uniformly finite propagation;

  * •

for $\gamma\in\mathcal{G}_{x}$ with $\mathrm{r}(\gamma)\in\mathrm{r}(K_{n})$,
then $k_{x}(\gamma,\gamma)=1$;

  * •

for $\gamma,\beta\in\mathcal{G}_{x}$ with $0<d_{x}(\gamma,\beta)\leq n$, then
$|k_{x}(\gamma,\beta)-1|<\varepsilon$.

Note that the kernel $k_{x}$ might _not_ have $(n,\varepsilon)$-variation,
hence we consider another kernel $\widetilde{k}_{x}$ on $\mathcal{G}_{x}$
defined by

$\widetilde{k}_{x}(\gamma,\beta)=\left\\{\begin{array}[]{ll}k_{x}(\gamma,\beta),&\text{if
}\mathrm{r}(\gamma),\mathrm{r}(\beta)\in\mathrm{r}(K_{n});\\\
\delta_{\gamma,\beta},&\text{otherwise}.\end{array}\right.$

Clearly $\widetilde{k}_{x}$ is a normalised symmetric kernel on
$\mathcal{G}_{x}$ of positive type such that

  * •

the family $(\widetilde{k}_{x})_{x\in\mathcal{G}^{(0)}}$ has uniformly finite
propagation;

  * •

$\widetilde{k}_{x}$ has $(n,\varepsilon)$-variation.

The last item comes from the fact that for any $\gamma\in\mathcal{G}_{x}$ with
$\mathrm{r}(\gamma)\notin\mathrm{r}(K_{n})$ and $\beta\in\mathcal{G}_{x}$ with
$\beta\neq\gamma$, we have $d_{x}(\gamma,\beta)>n$. Hence we conclude the
proof. ∎

The following result is basic and the proof is straightforward, hence omitted.

######  Lemma 5.6.

Given $\alpha,\beta\in\mathcal{G}_{x}$ for $x\in\mathcal{G}^{(0)}$ and an
integer $n>0$, we have that $\alpha\beta^{-1}\notin K_{n}$ _if and only if_
either $d_{x}(\alpha,\beta)>n$, or $\alpha=\beta$ with
$\mathrm{r}(\alpha)\notin K_{n}$. If additionally $\mathcal{G}^{(0)}$ is
compact, then $\alpha\beta^{-1}\notin K_{n}$ _if and only if_
$d_{x}(\alpha,\beta)>n$.

By virtue of the metrics from (5.1) together with Proposition 4.4 and Lemma
5.6, we reach the following:

######  Lemma 5.7.

With the same notation as above and given
$T\in\mathscr{L}(L^{2}(\mathcal{G}))$, we have:

  1. (1)

if $T\in\mathbb{C}_{u}[\mathcal{G}]$, then
$(\Phi_{x}(T))_{x\in\mathcal{G}^{(0)}}$ has uniformly finite propagation.

  2. (2)

if $T\in C^{*}_{uq}(\mathcal{G})$, then
$(\Phi_{x}(T))_{x\in\mathcal{G}^{(0)}}$ is uniformly quasi-local.

If additionally $\mathcal{G}^{(0)}$ is compact, then both of the converse hold
as well.

To prove Theorem 5.1, we need another description for uniform quasi-locality
in terms of relative commutants introduced in [42] (see also Proposition
4.15). Let us start with some extra notation. Let $(X,d)$ be a discrete metric
space. A function $f\in\ell^{\infty}(X)$ is called $C$-Lipschitz for some
constant $C>0$ if $|f(x)-f(y)|\leq Cd(x,y)$ for any $x,y\in X$. Given $L>0$
and $\varepsilon>0$, we write $T\in Commut_{X}(L,\varepsilon)$ if
$\|[T,f]\|<\varepsilon$ for any $L$-Lipschitz $f\in\ell^{\infty}(X)$ with norm
$1$.

Tracking the parameters in the proof of [43, Lemma 5.2], we obtain the
following version for a metric family:

######  Lemma 5.8.

Let $\\{(X_{i},d_{i})\\}_{i\in I}$ be a family of discrete metric spaces of
uniformly bounded geometry and uniform Property A. For $\varepsilon>0$, $L>0$
and $M>0$, there exists $s>0$ such that for any $i\in I$ and
$T_{i}\in\mathscr{B}(\ell^{2}(X_{i}))$ with $T_{i}\in
Commut_{X_{i}}(L,\frac{\varepsilon}{12})$ and $\|T_{i}\|\leq 2M$, there is a
unit vector $v_{i}\in\ell^{2}(X_{i})$ with
$\mathrm{diam}(\mathrm{supp}(v_{i}))\leq s$ such that

$\|T_{i}v_{i}\|\geq\|T_{i}\|-\frac{\varepsilon}{2}.$

Now we are in the position to prove Theorem 5.1. Before we dive into the
details, let us explain the outline of the proof. The main goal is to
approximate a given equivariant quasi-local operator $T$ using elements in
$\Lambda(C_{c}(\mathcal{G}))$. As indicated in the proof of Lemma 5.5, the
amenability of $\mathcal{G}$ produces a family of kernels
$(k_{x})_{x\in\mathcal{G}^{(0)}}$ which naturally produce a function $g\in
C_{c}(\mathcal{G})$. Moreover, these kernels can be modified to provide
another family $(\widetilde{k}_{x})_{x\in\mathcal{G}^{(0)}}$ with nice
behaviour. Conducting a “uniform” proof of [43, Theorem 3.3], the family of
operators determined by $(\widetilde{k}_{x})_{x\in\mathcal{G}^{(0)}}$ indeed
approximate the given operator $T$. Finally thanks to the quasi-locality of
$T$, this approximating family of operators is simultaneously uniformly close
to the slices of $\Lambda(g)$, which concludes the proof.

###### Proof of Theorem 5.1.

It is clear that $C^{*}_{r}(\mathcal{G})\subseteq
C^{*}_{u}(\mathcal{G})^{\mathcal{G}}\subseteq
C^{*}_{uq}(\mathcal{G})^{\mathcal{G}}$, hence it suffices to prove
$C^{*}_{uq}(\mathcal{G})^{\mathcal{G}}\subseteq C^{*}_{r}(\mathcal{G})$. Fix
an operator $T\in C^{*}_{uq}(\mathcal{G})^{\mathcal{G}}$ and $\varepsilon>0$,
we aim to construct an operator $S\in\Lambda(C_{c}(\mathcal{G}))$ such that
$\|T-S\|\leq\varepsilon$. Throughout the proof, we also fix a sequence of
subsets $\\{K_{n}\\}_{n\in\mathbb{N}}$ of $\mathcal{G}$ satisfying (1)-(3).
Endow a metric $d_{x}$ on $\mathcal{G}_{x}$ defined by (5.1) for each
$x\in\mathcal{G}^{(0)}$.

By Proposition 4.4, there exists $n_{0}\in\mathbb{N}$ such that for any
$x\in\mathcal{G}^{(0)}$ and $A_{x},B_{x}\subseteq\mathcal{G}_{x}$ with
$A_{x}\cap(K_{n_{0}}\cdot B_{x})=\emptyset$, then
$\|\chi_{A_{x}}\Phi_{x}(T)\chi_{B_{x}}\|\leq\frac{\varepsilon}{12}$. Moreover,
it follows from Lemma 5.7(2) that the family
$(\Phi_{x}(T))_{x\in\mathcal{G}^{(0)}}$ is uniformly quasi-local. Hence
following the proof of [42, Theorem 2.8, “(ii) $\Rightarrow$ (i)”], there
exists $L>0$ (which only depends on $T$ and $\varepsilon$) such that
$\Phi_{x}(T)\in Commut_{\mathcal{G}_{x}}(L,\frac{\varepsilon}{48})$ for any
$x\in\mathcal{G}^{(0)}$.

Thanks to Lemma 5.4 and Lemma 5.5, we can apply Lemma 5.8 for
$\frac{\varepsilon}{2}$, $L$ and $M\coloneqq\|T\|$ and obtain a constant $s>0$
satisfying the condition therein. We set
$N\coloneqq\sup_{x\in\mathcal{G}^{(0)}}\sup_{\gamma\in\mathcal{G}_{x}}\sharp
B_{\mathcal{G}_{x}}(\gamma,s+\frac{1}{L})$ and
$\widetilde{\varepsilon}=\min\\{\frac{\varepsilon}{8MN},1\\}$, and choose an
integer $\widetilde{n}>s+\frac{1}{L}+n_{0}$.

Since $\mathcal{G}$ is amenable, it follows from Lemma 2.6 that there exists a
non-negative function $h\in C_{c}(\mathcal{G})$ of positive type such that

  * •

$h(x)\leq 1$ for any $x\in\mathcal{G}^{(0)}$, and $h(x)=1$ for any
$x\in\mathrm{r}(K_{\widetilde{n}})$;

  * •

$|1-h(\gamma)|<\widetilde{\varepsilon}$ for any $\gamma\in K_{\widetilde{n}}$.

For each $x\in\mathcal{G}^{(0)}$ and $\gamma,\beta\in\mathcal{G}_{x}$, we
define $k_{x}(\gamma,\beta)=h(\gamma\beta^{-1})$ and

$\widetilde{k}_{x}(\gamma,\beta)=\left\\{\begin{array}[]{ll}h(\gamma\beta^{-1}),&\text{if
}\mathrm{r}(\gamma),\mathrm{r}(\beta)\in\mathrm{r}(K_{\widetilde{n}});\\\
\delta_{\gamma,\beta},&\text{otherwise}.\end{array}\right.$

The proof of Lemma 5.5 shows that $\widetilde{k}_{x}$ is a normalised
symmetric kernel on $\mathcal{G}_{x}$ of positive type and
$(\widetilde{n},\widetilde{\varepsilon})$-variation such that the family
$(\widetilde{k}_{x})_{x\in\mathcal{G}^{(0)}}$ has uniformly finite
propagation.

For each $x\in\mathcal{G}^{(0)}$, we consider the operator
$T_{x}^{\prime}=m_{\widetilde{k}_{x}}(\Phi_{x}(T))-\Phi_{x}(T)$ where
$m_{\widetilde{k}_{x}}$ is the Schur multiplier. It is clear that

$\sup_{x\in\mathcal{G}^{(0)}}\|T_{x}^{\prime}\|\leq 2M\quad\text{and}\quad
T_{x}^{\prime}\in Commut_{\mathcal{G}_{x}}(L,\frac{\varepsilon}{24}).$

For the family $\\{T^{\prime}_{x}\\}_{x}$ and the parameters above, Lemma 5.8
implies that for each $x\in\mathcal{G}^{(0)}$ there exists a unit vector
$v_{x}\in\ell^{2}(\mathcal{G}_{x})$ with
$\mathrm{diam}(\mathrm{supp}(v_{x}))\leq s$ such that

(5.2) $\|T^{\prime}_{x}v_{x}\|\geq\|T^{\prime}_{x}\|-\frac{\varepsilon}{4}.$

Setting
$V_{x}=\\{\gamma\in\mathcal{G}_{x}:d_{x}(\mathrm{supp}(v_{x}),\gamma)<\frac{1}{L}\\}$
for each $x\in\mathcal{G}^{(0)}$, we have

(5.3)
$\|T^{\prime}_{x}v_{x}\|\leq\|\chi_{V_{x}}(T^{\prime}_{x}v_{x})\|+\|\chi_{\mathcal{G}_{x}\backslash
V_{x}}(T^{\prime}_{x}v_{x})\|\leq\|\chi_{V_{x}}(T^{\prime}_{x}v_{x})\|+\frac{\varepsilon}{24},$

where the second inequality comes from $T^{\prime}_{x}\in
Commut_{\mathcal{G}_{x}}(L,\frac{\varepsilon}{24})$. For $\gamma\in V_{x}$, we
have

$\displaystyle|(T^{\prime}_{x}v_{x})(\gamma)|$
$\displaystyle=\left|\sum_{\alpha\in\mathrm{supp}(v_{x})}\left((\widetilde{k}_{x}(\gamma,\alpha)-1)\Phi_{x}(T)_{\gamma,\alpha}\right)v_{x}(\alpha)\right|$
$\displaystyle\leq\left(\sum_{\alpha\in\mathrm{supp}(v_{x})}\left|(\widetilde{k}_{x}(\gamma,\alpha)-1)\right|^{2}\right)^{1/2}\|\Phi_{x}(T)v_{x}\|\leq\widetilde{\varepsilon}\sqrt{N}M\leq\frac{\varepsilon}{8\sqrt{N}},$

where the second inequality comes from the Cauchy-Schwarz inequality and the
third one is due to that $\widetilde{k}_{x}$ has
$(\widetilde{n},\widetilde{\varepsilon})$-variation. Here
$\Phi_{x}(T)_{\gamma,\alpha}=\langle\delta_{\gamma},\Phi_{x}(T)(\delta_{\alpha})\rangle$,
as defined in Section 2.5. Hence

(5.4) $\|\chi_{V_{x}}(T^{\prime}_{x}v_{x})\|^{2}=\sum_{\gamma\in
V_{x}}|(T^{\prime}_{x}v_{x})(\gamma)|^{2}\leq\frac{\varepsilon^{2}}{64}.$

Combining (5.2), (5.3) and (5.4) together, we obtain that for any
$x\in\mathcal{G}^{(0)}$,

(5.5)
$\|m_{\widetilde{k}_{x}}(\Phi_{x}(T))-\Phi_{x}(T)\|=\|T^{\prime}_{x}\|\leq\|T^{\prime}_{x}v_{x}\|+\frac{\varepsilon}{4}\leq\|\chi_{V_{x}}(T^{\prime}_{x}v_{x})\|+\frac{\varepsilon}{24}+\frac{\varepsilon}{4}<\frac{\varepsilon}{2}.$

On the other hand, it follows from Proposition 3.30 that $T=\Lambda(f_{T})$
for some $f_{T}\in C_{b}(\mathcal{G})$. Hence for $x\in\mathcal{G}^{(0)}$ and
$\gamma,\alpha\in\mathcal{G}_{x}$ we have

$(m_{k_{x}}(\Phi_{x}(T)))_{\gamma,\alpha}=k_{x}(\gamma,\alpha)f_{T}(\gamma\alpha^{-1})=h(\gamma\alpha^{-1})f_{T}(\gamma\alpha^{-1}),$

which implies that

(5.6) $m_{k_{x}}(\Phi_{x}(T))=\Phi_{x}(\Lambda(h\cdot f_{T})).$

Set
$C_{x}\coloneqq\\{\gamma\in\mathcal{G}_{x}:\mathrm{r}(\gamma)\in\mathrm{r}(K_{\widetilde{n}})\\}$
and $D_{x}\coloneqq\mathcal{G}_{x}\backslash C_{x}$, and decompose the
operator $m_{k_{x}}(\Phi_{x}(T))-m_{\widetilde{k}_{x}}(\Phi_{x}(T))$ as
follows:

$\displaystyle
m_{k_{x}}(\Phi_{x}(T))-m_{\widetilde{k}_{x}}(\Phi_{x}(T))=(\chi_{C_{x}}+\chi_{D_{x}})\left(m_{k_{x}-\widetilde{k}_{x}}(\Phi_{x}(T))\right)(\chi_{C_{x}}+\chi_{D_{x}})$
$\displaystyle=m_{k_{x}-\widetilde{k}_{x}}\left(\chi_{C_{x}}\Phi_{x}(T)\chi_{C_{x}}+\chi_{D_{x}}\Phi_{x}(T)\chi_{D_{x}}+\chi_{C_{x}}\Phi_{x}(T)\chi_{D_{x}}+\chi_{D_{x}}\Phi_{x}(T)\chi_{C_{x}}\right)$
$\displaystyle=m_{k_{x}-\widetilde{k}_{x}}\left(\chi_{D_{x}}\Phi_{x}(T)\chi_{D_{x}}+\chi_{C_{x}}\Phi_{x}(T)\chi_{D_{x}}+\chi_{D_{x}}\Phi_{x}(T)\chi_{C_{x}}\right),$

where we use the fact that $k_{x}$ and $\widetilde{k}_{x}$ coincide on $C_{x}$
in the last equality.

Note that for $\gamma,\beta\in D_{x}$, we have $\gamma\beta^{-1}\notin
K_{\widetilde{n}}$ (otherwise, we would have
$\mathrm{r}(\gamma)=\mathrm{r}(\gamma\beta^{-1})\in\mathrm{r}(K_{\widetilde{n}})$,
which contradicts to that $\gamma\in D_{x}$). Since $\widetilde{n}>n_{0}$, we
have $\|\chi_{D_{x}}\Phi_{x}(T)\chi_{D_{x}}\|<\frac{\varepsilon}{12}$.
Similarly for $\gamma\in D_{x}$ and $\beta\in C_{x}$, we have
$\gamma\beta^{-1}\notin K_{\widetilde{n}}$. This implies that
$\|\chi_{D_{x}}\Phi_{x}(T)\chi_{C_{x}}\|<\frac{\varepsilon}{12}$ and
$\|\chi_{C_{x}}\Phi_{x}(T)\chi_{D_{x}}\|<\frac{\varepsilon}{12}$. Hence

$\|\chi_{D_{x}}\Phi_{x}(T)\chi_{D_{x}}+\chi_{C_{x}}\Phi_{x}(T)\chi_{D_{x}}+\chi_{D_{x}}\Phi_{x}(T)\chi_{C_{x}}\|<\frac{\varepsilon}{4},$

which implies that

(5.7)
$\|m_{k_{x}}(\Phi_{x}(T))-m_{\widetilde{k}_{x}}(\Phi_{x}(T))\|<\frac{\varepsilon}{2}.$

Finally combining (5.5), (5.6) and (5.7), we obtain

$\|\Lambda(h\cdot
f_{T})-T\|=\sup_{x\in\mathcal{G}^{(0)}}\|m_{k_{x}}(\Phi_{x}(T))-\Phi_{x}(T)\|<\varepsilon.$

Note that $h\cdot f_{T}\in C_{c}(\mathcal{G})$, hence we conclude that $T\in
C^{*}_{r}(\mathcal{G})$. ∎

##  6\. Examples

In this section, we apply Theorem 5.1 to the examples mentioned in Section
2.6.

###  6.1. Discrete groups

Let $\Gamma$ be a countable discrete group with unit $1_{\Gamma}$. As
mentioned in Section 2.6.1, the reduced groupoid $C^{*}$-algebra of $\Gamma$
is the same as the reduced group $C^{*}$-algebra $C^{*}_{r}(\Gamma)$, and
$\mathscr{L}(L^{2}(\Gamma))=\mathscr{B}(\ell^{2}(\Gamma))$.

Writing in the matrix form, obviously an operator
$T\in\mathscr{B}(\ell^{2}(\Gamma))$ is $\Gamma$-equivariant _if and only if_
$T_{\alpha\gamma,\beta\gamma}=T_{\alpha,\beta}$ for any
$\alpha,\beta,\gamma\in\Gamma$. Equipping $\Gamma$ with an arbitrary proper
word length metric, it is clear that $T$ is compactly supported _if and only
if_ $T$ has finite propagation in the sense of Definition 2.8. Moreover, the
notion of quasi-locality for $T$ from Definition 2.10 coincides with that from
Definition 4.2(3). Hence in this case, Theorem 5.1 recovers the following
known result:

######  Theorem 6.1.

Let $\Gamma$ be a countable discrete amenable group. Then for an operator
$T\in\mathscr{B}(\ell^{2}(\Gamma))$, the following are equivalent:

  1. (1)

$T$ belongs to $C^{*}_{r}(\Gamma)$;

  2. (2)

$T$ is $\Gamma$-invariant and belongs to $C^{*}_{u}(\Gamma)$;

  3. (3)

$T$ is $\Gamma$-invariant and quasi-local.

######  Remark 6.2.

The above theorem also holds for groups with the _approximation property_ from
[19] (see also [9]). In fact, it was shown in [47] that
$C^{*}_{u}(\Gamma)^{\Gamma}=C^{*}_{r}(\Gamma)$ if $\Gamma$ has the
approximation property. Moreover, it follows from [19, 30] that amenable
groups have the approximation property, and the approximation property implies
Property A. Hence applying [43, Theorem 3.3], we know that
$C^{*}_{u}(\Gamma)=C^{*}_{uq}(\Gamma)$ in this case. Therefore, we obtain that
if $\Gamma$ has the approximation property, then
$C^{*}_{r}(\Gamma)=C^{*}_{u}(\Gamma)^{\Gamma}=C^{*}_{uq}(\Gamma)^{\Gamma}$.

###  6.2. Pair groupoids

Let $X$ be a set and we consider the pair groupoid $X\times X$ from Section
2.6.2. By Theorem 3.12, we know that the map

$\Phi:\mathscr{L}(L^{2}(X\times X))^{X\times X}\longrightarrow\prod_{x\in
X}\mathscr{B}(\ell^{2}(X))$

is a $C^{*}$-monomorphism with image consisting of constant families
$(T^{\prime})_{x\in X}$ which are vector-wise uniformly quasi-local. It is not
hard (similar to, but much easier than, the analysis in Example 3.24) to see
that $(T^{\prime})_{x\in X}$ is vector-wise uniformly quasi-local for any
$T^{\prime}\in\mathscr{B}(\ell^{2}(X))$. Hence similar to Lemma 3.25, we
obtain the following:

######  Lemma 6.3.

Let $X$ be a set and $X\times X$ be the pair groupoid. Then for some fixed
$x_{0}\in X$, the map

(6.1) $\Theta:\mathscr{L}(L^{2}(X\times X))^{X\times
X}\longrightarrow\mathscr{B}(\ell^{2}(X)),\quad T\mapsto\Phi_{x_{0}}(T)$

is a $C^{*}$-isomorphism. Note that $\Theta$ is independent of the choice of
$x_{0}$.

Analogous to the discussion in Example 3.31, the map $\Theta$ in (6.1) extends
the map $\Theta:C^{*}_{r}(X\times X)\cong\mathscr{K}(\ell^{2}(X))$ in (2.6).
Hence we abuse the notation.

Direct calculations show that an operator $T\in\mathscr{L}(L^{2}(X\times
X))^{X\times X}$ is quasi-local in the sense of Definition 4.2(3) if and only
if $\Phi_{0}(T)\in\mathscr{K}(\ell^{2}(X))$, which implies that
$C^{*}_{uq}(X\times X)^{X\times X}=C^{*}_{r}(X\times X)$. Hence Theorem 5.1
holds trivially for pair groupoids.

###  6.3. Coarse groupoids

This example is our motivation of the whole work, and details have already
been spread in Section 3 and Section 4. Here we recall the whole picture
again.

Let $(X,d)$ be a uniformly discrete metric space with bounded geometry, and
$G(X)$ be the associated coarse groupoid. Recall from Lemma 3.25 that for a
fixed $x_{0}\in X$, the map

$\Theta:\mathscr{L}(L^{2}(G(X)))^{G(X)}\longrightarrow\\{T\in\mathscr{B}(\ell^{2}(X)):T\text{
satisfies Equation (\ref{EQ:char for LL2G coarse groupoid})}\\}$

given by $T\mapsto\Phi_{x_{0}}(T)$ is a $C^{*}$-isomorphism. Example 3.31
implies that $\Theta$ extends the isomorphism

$C^{*}_{r}(G(X))\cong C^{*}_{u}(X)$

from (2.7). Furthermore, Example 4.9 implies that the restriction

$\Theta:C^{*}_{uq}(G(X))^{G(X)}\longrightarrow C^{*}_{uq}(X)$

is also an isomorphism. Therefore, our main result recovers [43, Theorem 3.3]
in the Hilbert space case.

###  6.4. Transformation groupoids

Let $X$ be a locally compact $\sigma$-compact space, and $\Gamma$ be a
countable discrete group acting on $X$. We consider the transformation
groupoid $X\rtimes\Gamma$ in Section 2.6.4.

First note that $(X\rtimes\Gamma)_{x}$ is identified with $\Gamma$ by $(\alpha
x,\alpha)\mapsto\alpha$ for $x\in X$ and $\alpha\in\Gamma$. Under these
identifications and equipping $\Gamma$ with a proper word length metric, we
have the following. The proof is straightforward, hence omitted.

######  Lemma 6.4.

For a family $(T_{x})_{x\in X}\in\prod_{x\in X}\mathscr{B}(\ell^{2}(\Gamma))$,
we have:

  1. (1)

Writing in the matrix form, $(T_{x})_{x\in X}$ is
$(X\rtimes\Gamma)$-equivariant _if and only if_ the following holds:

(6.2) $(T_{\gamma
x})_{\alpha,\beta}=(T_{x})_{\alpha\gamma,\beta\gamma}\quad\text{for any }x\in
X\text{ and }\alpha,\beta,\gamma\in\Gamma.$

  2. (2)

$(T_{x})_{x\in X}$ belongs to the image of $\iota$ from (3.3) _if and only if_
the map $X\to\mathscr{B}(\ell^{2}(\Gamma)),x\mapsto T_{x}$ is WOT-continuous.

  3. (3)

When $X$ is compact, $(T_{x})_{x\in X}$ is compactly uniformly quasi-local _if
and only if_ $(T_{x})_{x\in X}$ is uniformly quasi-local in the sense of
Definition 5.3(3).

Note that condition (3) above also holds for certain non-compact $X$, while
for simplicity we only focus on the compact case in the sequel. Combining
Corollary 4.6 with Lemma 6.4, we reach the following:

######  Corollary 6.5.

When $X$ is compact, we have a $C^{*}$-monomorphism

$\Phi:C^{*}_{uq}(X\rtimes\Gamma)^{X\rtimes\Gamma}\longrightarrow\prod_{x\in
X}\mathscr{B}(\ell^{2}(\Gamma)),\quad T\mapsto(\Phi_{x}(T))_{x\in X}$

with range consisting of elements $(T_{x})_{x\in X}$ which is uniformly quasi-
local and satisfies (6.2) such that the map $x\mapsto T_{x}$ is WOT-
continuous.

Hence applying Theorem 5.1, we reach the following characterisation for the
reduced crossed product $C(X)\rtimes_{r}\Gamma$:

######  Theorem 6.6.

Let $X$ be a compact space and $\Gamma$ be a countable discrete group with an
amenable action on $X$. Then $C(X)\rtimes_{r}\Gamma$ is $C^{*}$-isomorphic to
the $C^{*}$-subalgebra in $\prod_{x\in X}\mathscr{B}(\ell^{2}(\Gamma))$
consisting of elements $(T_{x})_{x\in X}$ which is uniformly quasi-local and
satisfies (6.2) such that the map $x\mapsto T_{x}$ is WOT-continuous.

##  7\. Beyond equivariance

In Section 5, we study the equivariant parts in uniform Roe and quasi-local
algebras for groupoids and our main result (Theorem 5.1) provides a sufficient
condition to ensure that they are the same. In this section, we will turn to
these $C^{*}$-algebras themselves without the restriction of equivariance. Our
approach is to consider the semi-direct product groupoids and transfer the
general case to the equivariant one, inspired by the discussions in [1,
Section 6].

###  7.1. Semi-direct product groupoids

Here we collect some prelimenary knowledge on groupoid actions and semi-direct
product groupoids, and guide readers to [1] for more details.

Let $X$ be a locally compact space. A _fibre space over $X$_ is a pair
$(Y,p)$, where $Y$ is a locally compact space and $p\colon Y\rightarrow X$ is
a continuous surjective map. For two fibre spaces $(Y_{1},p_{1})$ and
$(Y_{2},p_{2})$ over $X$, we form their _fibred product_ to be

$Y_{1}\prescript{}{p_{1}}{\ast}_{p_{2}}Y_{2}\coloneqq\\{(y_{1},y_{2})\in
Y_{1}\times Y_{2}:p_{1}(y_{1})=p_{2}(y_{2})\\},$

equipped with the topology induced by the product topology.

######  Definition 7.1 ([1, Definition 1.2]).

Let $\mathcal{G}$ be a locally compact groupoid. A left $\mathcal{G}$-space is
a fibre space $(Y,p)$ over $\mathcal{G}^{(0)}$, equipped with a continuous map
$(\gamma,y)\mapsto\gamma y$ from $\mathcal{G}\prescript{}{\mathrm{s}}{*}_{p}Y$
to $Y$ (called a _ $\mathcal{G}$-action on $(Y,p)$_) which satisfies the
following:

  * •

$p(\gamma y)=\mathrm{r}(\gamma)$ for
$(\gamma,y)\in\mathcal{G}\prescript{}{\mathrm{s}}{*}_{p}Y$, and $p(y)y=y$;

  * •

$\gamma_{2}(\gamma_{1}y)=(\gamma_{2}\gamma_{1})y$ for
$(\gamma_{1},y)\in\mathcal{G}\prescript{}{\mathrm{s}}{*}_{p}Y$ and
$\mathrm{s}(\gamma_{2})=\mathrm{r}(\gamma_{1})$.

Given a $\mathcal{G}$-space $(Y,p)$, the associated semi-direct product
groupoid $Y\rtimes\mathcal{G}$ is defined to be
$Y\prescript{}{p}{*}_{\mathrm{r}}\mathcal{G}$ as a topological space. For
$(y,\gamma)\in Y\rtimes\mathcal{G}$, define its range to be
$(y,\mathrm{r}(\gamma))$ and its source to be
$(\gamma^{-1}y,\mathrm{s}(\gamma))$. The product and inverse are given by

$(y,\gamma)(\gamma^{-1}y,\gamma^{\prime})=(y,\gamma\gamma^{\prime})\quad\text{and}\quad(y,\gamma)^{-1}=(\gamma^{-1}y,\gamma^{-1}).$

Clearly $(y,p(y))\mapsto y$ is a homeomorphism from the unit space of
$Y\rtimes\mathcal{G}$ onto $Y$. Hence from now on, we may regard Y as the unit
space of the groupoid $Y\rtimes\mathcal{G}$. As shown in [1, Proposition 1.4
and Proposition 1.6], the range map $\mathrm{r}:Y\rtimes\mathcal{G}\to Y$ is
always open, and $Y\rtimes\mathcal{G}$ is étale when the groupoid
$\mathcal{G}$ itself is étale.

The class of groupoid actions we are interested in come from certain
compactification of the given groupoid. For completeness, let us recall the
following:

######  Definition 7.2.

Let $(Y,p)$ be a fibre space over a locally compact space $X$.

  1. (1)

We say that $(Y,p)$ is fibrewise compact if $p$ is proper, _i.e._ ,
$p^{-1}(K)$ is compact for any compact $K\subseteq X$.

  2. (2)

A fibrewise compactification of $(Y,p)$ is a fibrewise compact fibre space
$(Z,q)$ together with a continuous map $\varphi\colon Y\rightarrow Z$, which
is a homeomorphism onto an open dense subset of $Z$ and satisfies
$q\circ\varphi=p$. Usually we regard $Y$ as a subset of $Z$ in this case, and
the morphism $\varphi$ is just the inclusion.

To introduce concrete fibrewise compactifications, we need refer to the
Gelfand spectra. Let $(Y,p)$ be a fibre space over a locally compact space
$X$. Denote

$p^{*}(C_{0}(X))\coloneqq\\{f\circ p:f\in C_{0}(X)\\}$

and $C_{0}(Y,p)$ to be the closure of the following set in $C_{b}(Y)$:

$C_{c}(Y,p)=\\{g\in C_{b}(Y):\text{ there is a compact }K\subset
X,\,\mathrm{supp}(g)\subset p^{-1}(K)\\}.$

It was shown in [1, Proposition A.2] that the fibrewise compactifications of
$(Y,p)$ are in one-to-one correspondence with the $C^{*}$-subalgebras of
$C_{0}(Y,p)$ containing $p^{*}\left(C_{0}(X)\right)+C_{0}(Y)$. The fibrewise
compactification associated to $C_{0}(Y,p)$ is called the fibrewise Stone-Čech
compactification of $(Y,p)$, denoted by $(\beta_{p}Y,p_{\beta})$.

As a special case, we consider the following: Let $\mathcal{G}$ be a locally
compact étale groupoid. Then the groupoid multiplication of $\mathcal{G}$
provides a $\mathcal{G}$-action on the fibre space $(\mathcal{G},\mathrm{r})$,
_i.e._ ,
$\mathcal{G}\prescript{}{\mathrm{s}}{*}_{\mathrm{r}}\mathcal{G}\to\mathcal{G}$
defined by $(\gamma_{1},\gamma_{2})\mapsto\gamma_{1}\gamma_{2}$. Moreover, it
was shown in [1, Proposition 2.5] that this $\mathcal{G}$-action extends in a
unique way to a $\mathcal{G}$-action on the fibrewise Stone-Čech
compactification
$\left(\beta_{\mathrm{r}}\mathcal{G},\mathrm{r}_{\beta}\right)$. Hence we can
consider the semi-direct product groupoid
$\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}$.

Following the terminology in [1], a locally compact étale groupoid
$\mathcal{G}$ is said to be _strongly amenable at infinity_ if
$\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}$ is amenable. As shown in
[1], this has a close relation with the notion of exactness for groupoids. By
definition, $\mathcal{G}$ is called _$C^{*}$ -exact_ if the reduced groupoid
$C^{*}$-algebra $C^{*}_{r}(\mathcal{G})$ is exact. Also recall that
$\mathcal{G}$ is said to be _weakly inner amenable_ if for any compact subset
$K\subseteq\mathcal{G}$ and any $\varepsilon>0$, there exists a continuous
bounded positive definite function $f$ on the product groupoid
$\mathcal{G}\times\mathcal{G}$ which is properly supported such that
$|f(\gamma,\gamma)-1|<\varepsilon$ for all $\gamma\in K$.

######  Proposition 7.3 ([1, Corollary 7.4 and Theorem 8.6]).

Let $\mathcal{G}$ be a locally compact, second countable and étale groupoid.
If $\mathcal{G}$ is strongly amenable at infinity, then $\mathcal{G}$ is
$C^{*}$-exact. Conversely, if $\mathcal{G}$ is weakly inner amenable and
$C^{*}$-exact, then $\mathcal{G}$ is strongly amenable at infinity.

Throughout the rest of this section, let $\mathcal{G}$ be a locally compact
étale groupoid and $(\beta_{\mathrm{r}}\mathcal{G},\mathrm{r}_{\beta})$ be the
fibrewise Stone-Čech compactification of $(\mathcal{G},\mathrm{r})$. Consider
the $\mathcal{G}$-action on
$(\beta_{\mathrm{r}}\mathcal{G},\mathrm{r}_{\beta})$ induced by the groupoid
multiplication in $\mathcal{G}$, and the associated semi-direct product
groupoid $\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}$. In the next few
subsections, we will transfer the uniform Roe and quasi-local algebras for
$\mathcal{G}$ to the equivariant part of the corresponding algebras for
$\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}$.

###  7.2. Compactly supported case

Here we deal with the case of compactly supported operators and the uniform
Roe algebra, which was originally studied in [1].

First we show that our notion of the uniform Roe algebra
$C^{*}_{u}(\mathcal{G})$ from Definition 4.10(1) coincides with that of the
uniform $C^{*}$-algebra from [1, Definition 6.1]. To start, let us recall an
extra algebra $C_{t}(\mathcal{G}\ast_{\mathrm{s}}\mathcal{G})$ from [1] (see
also [4]). By definition,

$\mathcal{G}\ast_{\mathrm{s}}\mathcal{G}\coloneqq\mathcal{G}\prescript{}{\mathrm{s}}{*}_{\mathrm{s}}\mathcal{G}=\\{(\gamma,\alpha)\in\mathcal{G}\times\mathcal{G}:\mathrm{s}(\gamma)=\mathrm{s}(\alpha)\\}.$

A _tube_ is a subset of $\mathcal{G}\ast_{\mathrm{s}}\mathcal{G}$ whose image
by the map $(\gamma,\alpha)\mapsto\gamma\alpha^{-1}$ is relatively compact in
$\mathcal{G}$. Denote by $C_{t}(\mathcal{G}\ast_{\mathrm{s}}\mathcal{G})$ the
space of continuous bounded functions on
$\mathcal{G}\ast_{\mathrm{s}}\mathcal{G}$ with support in a tube. Recall from
[1, Section 6] that there is a $\ast$-monomorphism
$\mathcal{T}:C_{t}(\mathcal{G}\ast_{\mathrm{s}}\mathcal{G})\rightarrow\mathscr{L}(L^{2}(\mathcal{G}))$
given by444Note that the formula given in [1] is slightly different from the
above, where the set $\mathcal{G}\ast_{\mathrm{r}}\mathcal{G}$ is considered
instead of $\mathcal{G}\ast_{\mathrm{s}}\mathcal{G}$. Here we follow the
formula from [4, Section 6.5], which is more convenient and compatible to our
setting.:

$((\mathcal{T}f)\xi)(\gamma):=\sum_{\alpha\in\mathcal{G}_{\mathrm{s}(\gamma)}}f(\gamma,\alpha)\xi(\alpha)\quad\mbox{where}\quad\xi\in
L^{2}(\mathcal{G})\mbox{~{}and~{}}\gamma\in\mathcal{G},$

and the _uniform $C^{*}$-algebra of $\mathcal{G}$_ is defined to be the norm
closure of $\mathcal{T}(C_{t}(\mathcal{G}\ast_{\mathrm{s}}\mathcal{G}))$ in
$\mathscr{L}(L^{2}(\mathcal{G}))$.

The following lemma relates the algebra
$C_{t}(\mathcal{G}\ast_{\mathrm{s}}\mathcal{G})$ to our notion of compact
supported operators:

######  Lemma 7.4.

The image $\mathcal{T}(C_{t}(\mathcal{G}\ast_{\mathrm{s}}\mathcal{G}))$
consists of all compactly supported operators in the sense of Definition
4.2(2). Hence the uniform $C^{*}$-algebra of $\mathcal{G}$ from [1, Definition
6.1] coincides with the uniform Roe algebra of $\mathcal{G}$ from Definition
4.10(1).

###### Proof.

First it is clear that for $f\in
C_{t}(\mathcal{G}\ast_{\mathrm{s}}\mathcal{G})$ with support in
$\\{(\gamma,\alpha)\in\mathcal{G}\ast_{\mathrm{s}}\mathcal{G}:\gamma\alpha^{-1}\in
K\\}$ for some compact $K\subseteq\mathcal{G}$, then the operator
$\mathcal{T}f$ has support in $K$. Conversely, for an operator
$T\in\mathscr{L}(L^{2}(\mathcal{G}))$ with support in some compact
$K\subseteq\mathcal{G}$, we define a function
$f:\mathcal{G}\ast_{\mathrm{s}}\mathcal{G}\to\mathbb{C}$ as follows: for
$(\gamma,\alpha)\in\mathcal{G}\ast_{\mathrm{s}}\mathcal{G}$ with
$\mathrm{s}(\gamma)=\mathrm{s}(\alpha)=x$, set
$f(\gamma,\alpha)\coloneqq\langle\delta_{\gamma},\Phi_{x}(T)\delta_{\alpha}\rangle_{\ell^{2}(\mathcal{G}_{x})}$.
Thanks to Corollary 4.6, it is easy to show that $f\in
C_{t}(\mathcal{G}\ast_{\mathrm{s}}\mathcal{G})$ and $\mathcal{T}(f)=T$. Hence
we conclude the proof. ∎

Next, we recall from [1, Lemma 6.3] that there is a $\ast$-algebraic
isomorphism
$\vartheta:C_{c}(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G})\to
C_{t}(\mathcal{G}\ast_{\mathrm{s}}\mathcal{G})$ given by

$\vartheta(f)(\gamma,\alpha):=f(\gamma,\gamma\alpha^{-1})\quad\mbox{for}\quad
f\in C_{c}(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}).$

Hence we obtain a $\ast$-monomorphism

$\theta\coloneqq\mathcal{T}\circ\vartheta:C_{c}(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G})\to\mathscr{L}(L^{2}(\mathcal{G})).$

Direct calculations show that

(7.1)
$\big{(}\theta(f)\xi\big{)}(\gamma)=\sum_{\alpha\in\mathcal{G}_{\mathrm{s}(\gamma)}}f(\gamma,\gamma\alpha^{-1})\xi(\alpha)\text{
for }f\in C_{c}(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G})\text{ and
}\xi\in L^{2}(\mathcal{G}).$

It is proved in [1, Theorem 6.4] that $\theta$ extends to a
$C^{*}$-isomorphism

$\Theta:C^{*}_{r}(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G})\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}C^{*}_{u}(\mathcal{G}).$

Combining with Proposition 4.12, we obtain a $\ast$-isomorphism (still denoted
by $\Theta$):

(7.2)
$\Theta:\overline{\mathbb{C}_{u}[\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}]^{\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}}}^{\mathscr{L}(L^{2}(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}))}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}C^{*}_{u}(\mathcal{G})$

such that $\Theta(\Lambda(f))=\theta(f)$ for $f\in
C_{c}(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G})$.

###  7.3. From equivariance to non-equivariance

Now we aim to extend the map $\Theta$ from (7.2) to all
$(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G})$-equivariant operators in
$\mathscr{L}(L^{2}(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}))$. The
main observation is that the operator $\theta(f)$ for $f\in
C_{c}(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G})$ defined in (7.1) can
be written as a composition of three maps, which will be explained in details.

First recall that the unit space of
$\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}$ is
$\beta_{\mathrm{r}}\mathcal{G}$ and note that the source fibre at
$z\in\beta_{\mathrm{r}}\mathcal{G}$ is

$(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G})_{z}=\\{(\gamma
z,\gamma):\gamma\in\mathcal{G}\text{ such that
}\mathrm{s}(\gamma)=\mathrm{r}_{\beta}(z)\\},$

which is bijectively mapped onto $\mathcal{G}_{\mathrm{r}_{\beta}(z)}$ via the
map

$p_{z}:(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G})_{z}\longrightarrow\mathcal{G}_{\mathrm{r}_{\beta}(z)},\quad(\gamma
z,\gamma)\mapsto\gamma.$

Hence in the following, we will write $(\gamma
z,\gamma)\in\beta_{\mathrm{r}}\mathcal{G}\times\mathcal{G}$ such that
$\mathrm{s}(\gamma)=\mathrm{r}_{\beta}(z)$ for a general element in
$\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}$. Furthermore, $p_{z}$
induces a unitary

$P_{z}:\ell^{2}((\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G})_{z})\longrightarrow\ell^{2}(\mathcal{G}_{\mathrm{r}_{\beta}(z)})\quad\text{by}\quad
P_{z}(\delta_{(\gamma z,\gamma)})=\delta_{\gamma},$

which further induces a $\ast$-isomorphism

(7.3)
$\mathrm{Ad}_{z}:\mathscr{B}(\ell^{2}((\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G})_{z}))\longrightarrow\mathscr{B}(\ell^{2}(\mathcal{G}_{\mathrm{r}_{\beta}(z)})),\quad
T\mapsto P_{z}TP_{z}^{*}.$

Let us consider the map $\iota:C_{c}(\mathcal{G})\to
C_{c}(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G})$ defined by

$(\iota\xi)(\gamma z,\gamma)\coloneqq\xi(\gamma)\quad\text{for}\quad(\gamma
z,\gamma)\in\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}\quad\text{and}\quad\xi\in
C_{c}(\mathcal{G}).$

This map is well-defined since for $\xi\in C_{c}(\mathcal{G})$ with support in
a compact set $K\subseteq\mathcal{G}$, then $\iota\xi$ has support in
$\mathrm{r}_{\beta}^{-1}(\mathrm{r}(K))\times K$, which is compact in
$\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}$. Moreover for $\xi\in
C_{c}(\mathcal{G})$, we have

$\|\iota\xi\|^{2}_{L^{2}(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G})}=\sup_{z\in\beta_{\mathrm{r}}\mathcal{G}}\|(\iota\xi)|_{(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G})_{z}}\|^{2}=\sup_{z\in\beta_{\mathrm{r}}\mathcal{G}}\sum_{\gamma\in\mathcal{G}_{\mathrm{r}_{\beta}(z)}}|\xi(\gamma)|^{2}=\|\xi\|^{2}_{L^{2}(\mathcal{G})}.$

Hence $\iota$ can be extended to an isometry (with the same notation):

$\iota:L^{2}(\mathcal{G})\longrightarrow
L^{2}(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}).$

On the other hand, we define another map

$\kappa:C_{c}(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G})\longrightarrow
C_{c}(\mathcal{G})\quad\text{by}\quad(\kappa\eta)(\gamma)\coloneqq\eta(\gamma,\gamma)$

for $\eta\in C_{c}(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G})$ and
$\gamma\in\mathcal{G}$. It is clear that $\kappa$ is well-defined. Moreover,
we have:

$\displaystyle\|\kappa\eta\|^{2}_{L^{2}(\mathcal{G})}$
$\displaystyle=\sup_{x\in\mathcal{G}^{(0)}}\sum_{\gamma\in\mathcal{G}_{x}}|(\kappa\eta)(\gamma)|^{2}=\sup_{x\in\mathcal{G}^{(0)}}\sum_{\gamma\in\mathcal{G}_{x}}|\eta(\gamma,\gamma)|^{2}$
$\displaystyle\leq\sup_{z\in\beta_{\mathrm{r}}\mathcal{G}}\sum_{\gamma\in\mathcal{G}_{\mathrm{r}_{\beta}(z)}}|\eta(\gamma
z,\gamma)|^{2}=\|\eta\|^{2}_{L^{2}(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G})}.$

Hence $\kappa$ can be extended to a contractive map (with the same notation):

$\kappa:L^{2}(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G})\longrightarrow
L^{2}(\mathcal{G}).$

Now we are in the position to extend the map $\Theta$ from (7.2) to all
$(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G})$-equivariant operators in
$\mathscr{L}(L^{2}(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}))$. More
precisely, we define

$\Theta:\mathscr{L}(L^{2}(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}))^{\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}}\to\mathscr{L}(L^{2}(\mathcal{G}))$

by

(7.4) $\Theta(T):=\kappa\circ T\circ\iota\quad\text{for}\quad
T\in\mathscr{L}(L^{2}(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}))^{\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}}.$

In other words, the operator $\Theta(T)$ makes the following diagram commutes:

${L^{2}(\mathcal{G})}$${L^{2}(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G})}$${L^{2}(\mathcal{G})}$${L^{2}(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}).}$$\scriptstyle{\Theta(T)}$$\scriptstyle{\iota}$$\scriptstyle{T}$$\scriptstyle{\kappa}$

We will show in Lemma 7.5 below that the image of $\Theta$ is indeed contained
in $\mathscr{L}(L^{2}(\mathcal{G}))$. First, note that Proposition 3.30
implies that
$T\in\mathscr{L}(L^{2}(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}))^{\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}}$
can be written as $T=\Lambda(f_{T})$ where $f_{T}$ is the associated left
convolver. Direct calculations show that $\Theta(T)$ satisfies the following:

(7.5)
$\big{(}\Theta(T)\xi\big{)}(\gamma)=\sum_{\alpha\in\mathcal{G}_{\mathrm{s}(\gamma)}}f_{T}(\gamma,\gamma\alpha^{-1})\xi(\alpha)\quad\text{
for }\quad\xi\in L^{2}(\mathcal{G}).$

Comparing (7.1) with (7.5), it is clear that $\Theta$ in (7.4) indeed extends
the isomorphism
$\Theta:C^{*}_{r}(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G})\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}C^{*}_{u}(\mathcal{G})$
in (7.2). Hence again we abuse the notation.

On the other hand, using (7.5) it is straightforward to prove:

######  Lemma 7.5.

With the same notation as above, we have:

  1. (1)

for
$T\in\mathscr{L}(L^{2}(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}))^{\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}}$,
the operator $\Theta(T)$ is an adjointable operator on $L^{2}(\mathcal{G})$
with adjoint $\Theta(T^{*})$;

  2. (2)

for
$T,S\in\mathscr{L}(L^{2}(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}))^{\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}}$,
we have $\Theta(TS)=\Theta(T)\Theta(S)$;

  3. (3)

$\Theta$ is injective.

Combining the analysis above, we reach the following:

######  Corollary 7.6.

The map
$\Theta:\mathscr{L}(L^{2}(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}))^{\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}}\to\mathscr{L}(L^{2}(\mathcal{G}))$
given by (7.4) is a $C^{*}$-monomorphism, which extends the isomorphism in
(7.2).

######  Remark 7.7.

A careful reader might already notice that the map $\Theta$ can be further
extended to
$\mathscr{L}(L^{2}(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}))$ using
the same formula (7.4). However in this case, it is unclear whether Lemma 7.5
still holds since (7.5) is no longer available.

Finally, we provide an extra viewpoint on $\Theta$ in terms of the slicing
maps from Section 3.2, which will be used later to characterise the quasi-
local algebra. Recall from (3.1) that we have the slicing maps:

(7.6)
$\Phi^{\rtimes}\colon\mathscr{L}(L^{2}(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}))^{\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}}\longrightarrow\prod_{z\in\beta_{\mathrm{r}}\mathcal{G}}\mathscr{B}(\ell^{2}((\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G})_{z})),\quad
T\mapsto(\Phi^{\rtimes}_{z}(T))_{z\in\beta_{\mathrm{r}}\mathcal{G}}$

and

(7.7)
$\Phi\colon\mathscr{L}(L^{2}(\mathcal{G}))\longrightarrow\prod_{x\in\mathcal{G}^{(0)}}\mathscr{B}(\ell^{2}(\mathcal{G}_{x})),\quad
T^{\prime}\mapsto(\Phi_{x}(T^{\prime}))_{x\in\mathcal{G}^{(0)}}.$

For
$T\in\mathscr{L}(L^{2}(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}))^{\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}}$
and $x\in\mathcal{G}^{(0)}$, we consider the operator
$\mathrm{Ad}_{x}(\Phi^{\rtimes}_{x}(T))\in\mathscr{B}(\ell^{2}(\mathcal{G}_{x}))$
where $\mathrm{Ad}_{x}$ is from (7.3). Using (7.5), it is straightforward to
prove:

######  Lemma 7.8.

With the same notation as above, for
$T\in\mathscr{L}(L^{2}(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}))^{\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}}$
and $x\in\mathcal{G}^{(0)}$ we have

$\Phi_{x}(\Theta(T))=\mathrm{Ad}_{x}(\Phi^{\rtimes}_{x}(T)).$

######  Remark 7.9.

In fact, the map $\Theta$ can be alternatively defined using the slicing maps
above. More precisely, consider the map

(7.8)
$\mathfrak{p}:\prod_{z\in\beta_{\mathrm{r}}\mathcal{G}}\mathscr{B}(\ell^{2}((\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G})_{z}))\longrightarrow\prod_{x\in\mathcal{G}^{(0)}}\mathscr{B}(\ell^{2}(\mathcal{G}_{x})),\quad(T_{z})_{z\in\beta_{\mathrm{r}}\mathcal{G}}\mapsto(\mathrm{Ad}_{x}(T_{x}))_{x\in\mathcal{G}^{(0)}}.$

Given
$T\in\mathscr{L}(L^{2}(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}))^{\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}}$,
we consider the family $\mathfrak{p}\circ\Phi^{\rtimes}(T)$. Then we can apply
Theorem 3.12 to verify that $\mathfrak{p}\circ\Phi^{\rtimes}(T)$ belongs to
$\Phi(\mathscr{L}(L^{2}(\mathcal{G})))$, and hence it provides an operator
$\Phi^{-1}(\mathfrak{p}\circ\Phi^{\rtimes}(T))$, which is defined to be
$\Theta(T)$. More details will be provided in the next subsection. Lemma 7.8
shows that these two definitions are the same.

###  7.4. Quasi-local case

In this subsection, we consider the quasi-local algebras and the main result
is the following:

######  Proposition 7.10.

The restriction of the map $\Theta$ given by (7.4) provides a
$C^{*}$-isomorphism:

$\Theta:C^{*}_{uq}(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G})^{\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}C^{*}_{uq}(\mathcal{G}).$

The proof is divided into several parts, making use of the slicing maps at the
end of Section 7.3. First recall from Section 3.2 and 3.3, the operator fibre
spaces are given by

$E^{\rtimes}\coloneqq\bigsqcup_{z\in\beta_{\mathrm{r}}\mathcal{G}}\mathscr{B}(\ell^{2}((\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G})_{z})),\quad
E^{\rtimes}_{\mathcal{G}}\coloneqq\bigsqcup_{z\in\mathcal{G}}\mathscr{B}(\ell^{2}((\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G})_{z}))$

and

$E\coloneqq\bigsqcup_{x\in\mathcal{G}^{(0)}}\mathscr{B}(\ell^{2}(\mathcal{G}_{x})),$

equipped with the topology from Definition 3.6 (note that $\mathcal{G}$ is a
dense subset in $\beta_{\mathrm{r}}\mathcal{G}$). Also denote by
$\Gamma_{b}(E^{\rtimes}),\Gamma_{b}(E^{\rtimes}_{\mathcal{G}})$ and
$\Gamma_{b}(E)$ the associated algebras of continuous bounded sections.
Similar to Lemma 3.15, we have the following:

######  Lemma 7.11.

Every section $\sigma\in\Gamma_{b}(E^{\rtimes}_{\mathcal{G}})$ is extendable.

The proof is almost the same as the one for Lemma 3.15. In fact, borrowing the
notation therein, we can shrink the neighbourhoods $U_{\gamma^{\prime}}$ and
$U_{\gamma^{\prime\prime}}$ to ensure that the function
$f_{\gamma^{\prime},\gamma^{\prime\prime}}\in C_{0}(\mathcal{G},r)$ (see
Section 7.1 to recall the definition). Hence we omit the details.

On the other hand, note that for the slicing maps (7.6) and (7.7), given
$T\in\mathscr{L}(L^{2}(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}))^{\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}}$
the property of equivariance means the following:

$V_{(\gamma z,\gamma)}\Phi^{\rtimes}_{z}(T)=\Phi^{\rtimes}_{\gamma
z}(T)V_{(\gamma z,\gamma)}\quad\text{for}\quad(\gamma
z,\gamma)\in\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}$

where $V_{(\gamma
z,\gamma)}:\ell^{2}((\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G})_{z})\to\ell^{2}((\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G})_{\gamma
z})$ is given by $V_{(\gamma z,\gamma)}(\delta_{(\alpha
z,\alpha)})=\delta_{(\alpha z,\alpha\gamma^{-1})}$ (see Section 3.4). In the
following, we will consider families $(T_{z})_{z\in\mathcal{G}}$ satisfying:

(7.9) $V_{(\gamma z,\gamma)}T_{z}=T_{\gamma z}V_{(\gamma
z,\gamma)}\quad\text{for}\quad(\gamma
z,\gamma)\in\mathcal{G}\rtimes\mathcal{G}.$

Applying Corollary 4.8 together with Lemma 7.11, we obtain the following
characterisations for the equivariant quasi-local algebras:

######  Corollary 7.12.

The map

$\mathrm{Res}\circ\Phi^{\rtimes}:C^{*}_{uq}(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G})^{\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}}\longrightarrow\prod_{z\in\mathcal{G}}\mathscr{B}(\ell^{2}((\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G})_{z})),\quad
T\mapsto(\Phi^{\rtimes}_{z}(T))_{z\in\mathcal{G}}$

is a $C^{*}$-isomorphism with image consisting of $(T_{z})_{z\in\mathcal{G}}$
such that $z\mapsto T_{z}$ is a continuous section of
$E_{\mathcal{G}}^{\rtimes}$, $(T_{z})_{z\in\mathcal{G}}$ is compactly
uniformly quasi-local and satisfies (7.9).

######  Corollary 7.13.

The map

$\Phi:C^{*}_{uq}(\mathcal{G})\longrightarrow\prod_{x\in\mathcal{G}^{(0)}}\mathscr{B}(\ell^{2}(\mathcal{G}_{x})),\quad
T\mapsto(\Phi_{x}(T))_{x\in\mathcal{G}^{(0)}}$

is a $C^{*}$-isomorphism with image consisting of
$(T_{x})_{x\in\mathcal{G}^{(0)}}$ such that $x\mapsto T_{x}$ is a continuous
section of $E$ and $(T_{x})_{x\in\mathcal{G}^{(0)}}$ is compactly uniformly
quasi-local.

Recall from Lemma 7.8 and Remark 7.9 that we have the following commutative
diagram:

${\mathscr{L}(L^{2}(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}))^{\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}}}$${\prod_{z\in\mathcal{G}}\mathscr{B}(\ell^{2}((\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G})_{z}))}$${\mathscr{L}(L^{2}(\mathcal{G}))}$${\prod_{x\in\mathcal{G}^{(0)}}\mathscr{B}(\ell^{2}(\mathcal{G}_{x}))}$$\scriptstyle{\Theta}$$\scriptstyle{\mathrm{Res}\circ\Phi^{\rtimes}}$$\scriptstyle{\mathfrak{p}}$$\scriptstyle{\Phi}$

Hence to prove Proposition 7.10, it suffices to verify whether the map
$\mathfrak{p}$ provides a bijection between the images of quasi-local algebras
characterised by Corollary 7.12 and Corollary 7.13. First note that the
equivariance condition (7.9) directly implies the following:

######  Lemma 7.14.

The following restriction of $\mathfrak{p}$ is a bijection:

$\mathfrak{p}:\\{(T_{z})_{z\in\mathcal{G}}\in\prod_{z\in\mathcal{G}}\mathscr{B}(\ell^{2}((\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G})_{z}))\text{
satisfying }(\ref{EQ:equiv for
Phi})\\}\longrightarrow\prod_{x\in\mathcal{G}^{(0)}}\mathscr{B}(\ell^{2}(\mathcal{G}_{x})).$

Now we turn to the condition of continuity for sections:

######  Lemma 7.15.

For a family
$(T_{z})_{z\in\mathcal{G}}\in\prod_{z\in\mathcal{G}}\mathscr{B}(\ell^{2}((\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G})_{z}))$
satisfying (7.9), the map $z\mapsto T_{z}$ is a continuous section of
$E^{\rtimes}_{\mathcal{G}}$ _if and only if_ the map
$x\mapsto\mathrm{Ad}_{x}(T_{x})$ is a continuous section of $E$.

###### Proof.

By definition, the map $z\mapsto T_{z}$ is continuous if and only if for any
$z_{i}\to z$, $\gamma_{i}^{\prime}\to\gamma^{\prime}$ and
$\gamma^{\prime\prime}_{i}\to\gamma^{\prime\prime}$ in $\mathcal{G}$ with
$\mathrm{r}(z_{i})=\mathrm{s}(\gamma^{\prime}_{i})=\mathrm{s}(\gamma^{\prime\prime}_{i})$
and
$\mathrm{r}(z)=\mathrm{s}(\gamma^{\prime})=\mathrm{s}(\gamma^{\prime\prime})$,
then

(7.10)
$\langle\delta_{(\gamma^{\prime\prime}_{i}z_{i},\gamma^{\prime\prime}_{i})},T_{z_{i}}(\delta_{(\gamma^{\prime}_{i}z_{i},\gamma^{\prime}_{i})})\rangle\to\langle\delta_{(\gamma^{\prime\prime}z,\gamma^{\prime\prime})},T_{z}(\delta_{(\gamma^{\prime}z,\gamma^{\prime})})\rangle$

Setting $\mathrm{s}(z_{i})=x_{i}$ and $\mathrm{s}(z)=x$, then (7.9) implies
that

$V_{(z_{i},z_{i})}T_{x_{i}}V_{(z_{i},z_{i})}^{*}=T_{z_{i}}\quad\text{and}\quad
V_{(z,z)}T_{x}V_{(z,z)}^{*}=T_{z}.$

Hence (7.10) can be rewritten as:

$\langle\delta_{(\gamma^{\prime\prime}_{i}z_{i},\gamma^{\prime\prime}_{i}z_{i})},T_{x_{i}}(\delta_{(\gamma^{\prime}_{i}z_{i},\gamma^{\prime}_{i}z_{i})})\rangle\to\langle\delta_{(\gamma^{\prime\prime}z,\gamma^{\prime\prime}z)},T_{x}(\delta_{(\gamma^{\prime}z,\gamma^{\prime}z)})\rangle.$

Applying the isomorphisms $\mathrm{Ad}_{x}$, the above is also equivalent to:

$\langle\delta_{\gamma^{\prime\prime}_{i}z_{i}},\mathrm{Ad}_{x_{i}}(T_{x_{i}})(\delta_{\gamma^{\prime}_{i}z_{i}})\rangle\to\langle\delta_{\gamma^{\prime\prime}z},\mathrm{Ad}_{x}(T_{x})(\delta_{\gamma^{\prime}z})\rangle,$

which is (by definition) nothing but the continuity of the map
$x\mapsto\mathrm{Ad}_{x}(T_{x})$. ∎

Finally, we consider the condition of compactly uniform quasi-locality.

######  Lemma 7.16.

A family
$(T_{z})_{z\in\mathcal{G}}\in\prod_{z\in\mathcal{G}}\mathscr{B}(\ell^{2}((\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G})_{z}))$
satisfying (7.9) is compactly uniformly quasi-local _if and only if_ the
family $(\mathrm{Ad}_{x}(T_{x}))_{x\in\mathcal{G}^{(0)}}$ is compactly
uniformly quasi-local.

###### Proof.

_Necessity_ : By definition, given $\varepsilon>0$ there exists a compact
subset $\tilde{K}\subseteq\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}$
such that for any $z\in\mathcal{G}$ and
$\tilde{A}_{z},\tilde{B}_{z}\subseteq(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G})_{z}$
with $\tilde{A}_{z}\cap(\tilde{K}\cdot\tilde{B}_{z})=\emptyset$ and
$(\tilde{K}\cdot\tilde{A}_{z})\cap\tilde{B}_{z}=\emptyset$, we have
$\|\chi_{\tilde{A}_{z}}T_{z}\chi_{\tilde{B}_{z}}\|<\varepsilon$. Take

$K\coloneqq\\{\gamma\in\mathcal{G}:\text{ there exists
}\gamma^{\prime}\in\mathcal{G}\text{ such that
}(\gamma^{\prime},\gamma)\in\tilde{K}\\},$

which is compact. Then for any $x\in\mathcal{G}^{(0)}$ and
$A_{x},B_{x}\subseteq\mathcal{G}_{x}$ with $A_{x}\cap(K\cdot B_{x})=\emptyset$
and $(K\cdot A_{x})\cap B_{x}=\emptyset$, then we have
$p_{x}^{-1}(A_{x})\cap(\tilde{K}\cdot p_{x}^{-1}(B_{x}))=\emptyset$ and
$(\tilde{K}\cdot p_{x}^{-1}(A_{x}))\cap p_{x}^{-1}(B_{x})=\emptyset$. Hence

$\varepsilon\geq\|\chi_{p_{x}^{-1}(A_{x})}T_{x}\chi_{p_{x}^{-1}(B_{x})}\|=\|\mathrm{Ad}_{x}(\chi_{p_{x}^{-1}(A_{x})}T_{x}\chi_{p_{x}^{-1}(B_{x})})\|=\|\chi_{A_{x}}\mathrm{Ad}_{x}(T_{x})\chi_{B_{x}}\|,$

which shows that the family $(\mathrm{Ad}_{x}(T_{x}))_{x\in\mathcal{G}^{(0)}}$
is compactly uniformly quasi-local.

_Sufficiency_ : Given $\varepsilon>0$ there exists a compact subset
$K\subseteq\mathcal{G}$ such that for any $x\in\mathcal{G}^{(0)}$ and
$A_{x},B_{x}\subseteq\mathcal{G}_{x}$ with $A_{x}\cap(K\cdot B_{x})=\emptyset$
and $(K\cdot A_{x})\cap B_{x}=\emptyset$, we have
$\|\chi_{A_{x}}\mathrm{Ad}_{x}(T_{x})\chi_{B_{x}}\|<\varepsilon$. Take

$\tilde{K}\coloneqq\\{(\gamma^{\prime},\gamma)\in\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}:\gamma\in
K\\},$

which is compact. Then for any $z\in\mathcal{G}$ and
$\tilde{A}_{z},\tilde{B}_{z}\subseteq(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G})_{z}$
with $\tilde{A}_{z}\cap(\tilde{K}\cdot\tilde{B}_{z})=\emptyset$ and
$(\tilde{K}\cdot\tilde{A}_{z})\cap\tilde{B}_{z}=\emptyset$, we set
$x=\mathrm{s}(z)$ and note that $V_{(z,z)}T_{x}V^{*}_{(z,z)}=T_{z}$. Hence we
have

$\displaystyle\|\chi_{\tilde{A}_{z}}T_{z}\chi_{\tilde{B}_{z}}\|$
$\displaystyle=\|(V^{*}_{(z,z)}\chi_{\tilde{A}_{z}}V_{(z,z)})\cdot
T_{x}\cdot(V^{*}_{(z,z)}\chi_{\tilde{B}_{z}}V_{(z,z)})\|$
$\displaystyle=\|\chi_{\tilde{A}_{z}\cdot(z,z)}\cdot
T_{x}\cdot\chi_{\tilde{B}_{z}\cdot(z,z)}\|$ (7.11)
$\displaystyle=\|\mathrm{Ad}_{x}(\chi_{\tilde{A}_{z}\cdot(z,z)})\cdot\mathrm{Ad}_{x}(T_{x})\cdot\mathrm{Ad}_{x}(\chi_{\tilde{B}_{z}\cdot(z,z)})\|.$

It is straightforward to check that
$\mathrm{Ad}_{x}(\chi_{\tilde{A}_{z}\cdot(z,z)})=\chi_{A_{x}}$ and
$\mathrm{Ad}_{x}(\chi_{\tilde{B}_{z}\cdot(z,z)})=\chi_{B_{x}}$, where

$A_{x}\coloneqq\\{\alpha\in\mathcal{G}_{x}:(\alpha,\alpha)\in\tilde{A}_{z}\cdot(z,z)\\}=\\{\alpha\in\mathcal{G}_{x}:(\alpha,\alpha
z^{-1})\in\tilde{A}_{z}\\}$

and similarly, $B_{x}\coloneqq\\{\beta\in\mathcal{G}_{x}:(\beta,\beta
z^{-1})\in\tilde{A}_{z}\\}$. If $A_{x}\cap(K\cdot B_{x})\neq\emptyset$, then
there exists $\alpha\in A_{x}$, $\beta\in B_{x}$ and $\gamma\in K$ such that
$\alpha=\gamma\beta$. Then we have

$(\alpha,\alpha z^{-1})=(\alpha,\gamma)\cdot(\beta,\beta z^{-1}),$

which is a contradiction to that
$\tilde{A}_{z}\cap(\tilde{K}\cdot\tilde{B}_{z})=\emptyset$. Similarly, we have
$(K\cdot A_{x})\cap B_{x}=\emptyset$. Hence combining with (7.4), we obtain:

$\|\chi_{\tilde{A}_{z}}T_{z}\chi_{\tilde{B}_{z}}\|=\|\mathrm{Ad}_{x}(\chi_{\tilde{A}_{z}\cdot(z,z)})\cdot\mathrm{Ad}_{x}(T_{x})\cdot\mathrm{Ad}_{x}(\chi_{\tilde{B}_{z}\cdot(z,z)})\|=\|\chi_{A_{x}}\mathrm{Ad}_{x}(T_{x})\chi_{B_{x}}\|\leq\varepsilon,$

which concludes the proof. ∎

###### Proof of Proposition 7.10.

Combining Corollary 7.12, Corollary 7.13, Lemma 7.14, Lemma 7.15 and Lemma
7.16, we conclude the proof. ∎

######  Remark 7.17.

In fact, the above procedure is also available for the case of compactly
supported operators. Hence combining with Lemma 7.4, we obtain an alternative
proof for [1, Theorem 6.4].

######  Remark 7.18.

Readers might wonder whether Lemma 7.16 still holds for vector-wise uniform
quasi-locality. If fact using a similar idea in the proof of Lemma 7.16, we
can show that if a family
$(T_{z})_{z\in\mathcal{G}}\in\prod_{z\in\mathcal{G}}\mathscr{B}(\ell^{2}((\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G})_{z}))$
satisfying (7.9) is vector-wise uniformly quasi-local, then the family
$(\mathrm{Ad}_{x}(T_{x}))_{x\in\mathcal{G}^{(0)}}$ is vector-wise uniformly
quasi-local. Hence applying Theorem 3.12 together with Lemma 7.15, we obtain
that $\mathfrak{p}\circ\Phi^{\rtimes}(T)$ belongs to
$\Phi(\mathscr{L}(L^{2}(\mathcal{G})))$ for
$T\in\mathscr{L}(L^{2}(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}))^{\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}}$.
As already mentioned in Remark 7.9, this provides an alternative approach to
define the map $\Theta$.

However, it is unclear whether the opposite direction still holds for vector-
wise uniform quasi-locality. If it is true, then $\Theta$ will provide a
$C^{*}$-isomorphism between
$\mathscr{L}(L^{2}(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}))^{\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}}$
and $\mathscr{L}(L^{2}(\mathcal{G}))$.

###  7.5. Conclusion

Combining the discussions in the previous subsections, we obtain that the
$C^{*}$-monomorphism

$\Theta:\mathscr{L}(L^{2}(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}))^{\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}}\to\mathscr{L}(L^{2}(\mathcal{G}))$

provides $C^{*}$-isomorphisms

$\overline{\mathbb{C}_{u}[\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}]^{\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}}}^{\mathscr{L}(L^{2}(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}))}\big{(}\cong
C^{*}_{r}(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G})\big{)}\cong
C^{*}_{u}(\mathcal{G})$

and

$C^{*}_{uq}(\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G})^{\beta_{\mathrm{r}}\mathcal{G}\rtimes\mathcal{G}}\cong
C^{*}_{uq}(\mathcal{G}).$

Hence applying Theorem 5.1, we reach the following quasi-local
characterisation for the uniform Roe algebra $C^{*}_{u}(\mathcal{G})$:

######  Theorem 7.19.

Let $\mathcal{G}$ be a locally compact, $\sigma$-compact and étale groupoid.
Suppose $\mathcal{G}$ is either strongly amenable at infinity, or secondly
countable weakly inner amenable and $C^{*}$-exact. Then we have
$C^{*}_{u}(\mathcal{G})=C^{*}_{uq}(\mathcal{G})$.

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# Duplicate Detection with GenAI

Ian Ormesher
ZEISS Digital Partners
Munich
<EMAIL_ADDRESS>
0009-0008-4041-9443

###### Abstract

Customer data is often stored as records in Customer Relations Management
systems (CRMs). Data which is manually entered into such systems by one of
more users over time leads to data replication, partial duplication or fuzzy
duplication. This in turn means that there no longer a single source of truth
for customers, contacts, accounts, etc. Downstream business processes become
increasing complex and contrived without a unique mapping between a record in
a CRM and the target customer. Current methods to detect and de-duplicate
records use traditional Natural Language Processing techniques known as
“Entity Matching”. In this paper we show how using the latest advancements in
Large Language Models and Generative AI can vastly improve the identification
and repair of duplicated records. On common benchmark datasets we find an
improvement in the accuracy of data de-duplication rates from 30% using NLP
techniques to almost 60% using our proposed method.

##  1 Introduction

The task of identifying duplicate records is often done by pairwise record
comparisons and is referred to as “Entity Matching” (EM). Whilst EM is broader
in it’s scope and definition, we describe a technique here that can be used to
identify duplicate records within a table using embedding vectors and distance
metrics.

##  2 Traditional Approach

The five steps of a typical duplicate detection pipeline based on pairwise
record comparisons (EM) is described by Panse et al. [1].

This detection pipeline consists of several steps which include the following:

  * •

Candidate Generation

  * •

Blocking

  * •

Matching

  * •

Clustering

These steps are illustrated by their diagram shown in Figure 1.

Figure 1: The five steps of a typical duplicate detection pipeline based on
pairwise record comparisons

###  2.1 Candidate Generation

In the usual EM method, we would produce candidate records by combining all
the records in the table with themselves to produce a cartesian product. For
example, $row_{i}$ is combined with $row_{j}$ and produces a candidate record
of $(row_{i}+row_{j})$. This process is repeated for all the rows with all the
other rows. For a table of N rows this would produce (N x N) – N candidate
records. We subtract N from the total because we never combine a row with
itself. As an example, a table of 200 records would produce (200 x 200) – 200
= 39,800 candidate records.

###  2.2 Blocking

The idea of blocking is to eliminate those records that we know could not be
duplicates of each other because they have different values for the “blocked”
column. As an example, If we were considering customer records, a potential
column to block on could be something like “City”. This is because we know
that even if all the other details of the record are similar enough, they
cannot be the same customer if they’re located in different cities. Once we
have generated our candidate records, we then use blocking to eliminate those
records that have different values for the blocked column.

###  2.3 Matching

Following on from blocking we now examine all the candidate records and
calculate traditional NLP similarity-based attribute value metrics with the
fields from $row_{i}$ and $row_{j}$. Using these metrics, we can determine if
we have a potential match or un-match.

###  2.4 Clustering

Now that we have a list of candidate records that match, we can then group
them into clusters.

##  3 Proposed Method

There are several steps to the proposed method, but the most important thing
to note is that we no longer need to perform the “Candidate Generation” step
of the traditional method. The new steps become:

  * •

Create Match Sentences

  * •

Create Embedding Vectors

  * •

Clustering

###  3.1 Create Match Sentences

First a “Match Sentence” is created by concatenating the attributes we are
interested in and separating them with spaces.

As an example, let’s say that we have a customer record which looks like the
one shown in Table 1.

Table 1: Example Customer Record name1 | name2 | name3 | address | city
---|---|---|---|---
John | Hartley | Smith | 20 Main Street | London

We would create a "Match Sentence" by concatenating with spaces the name1,
name2, name3, address and city attributes to make the following:

John Hartley Smith 20 Main Street London

###  3.2 Create Embedding Vectors

Once our “Match Sentence” has been created it is then encoded into vector
space using our chosen embedding model. This is achieved by using “Sentence
Transformers” [2]. The output of this encoding will be a floating-point vector
of pre-defined dimensions. These dimensions relate to the embedding model that
is used. We used the all-mpnet-base-v2[3] embedding model which has a vector
space of 768 dimensions. This embedding vector is then appended to the record.
This is done for all the records, as shown in Figure 2.

Figure 2: The steps to creating the embedding vector

###  3.3 Clustering

Once embedding vectors have been calculated for all the records, the next step
is to create clusters of similar records. To do this we use the DBSCAN
technique. DBSCAN works by first selecting a random record and finding records
that are close to it using a distance metric. There are 2 different kinds of
distance metrics that were used:

  * •

L2 Norm distance

  * •

Cosine Similarity

For each of these metrics an epsilon value was used as a threshold value. All
records that were within the epsilon distance and had the same value for the
“blocked” column were then added to this cluster. Once that cluster was
complete another random record was selected from the unvisited records and a
cluster then created around it. This then continues until all the records have
been visited. See the description of this algorithm using pseudo code 1.

Algorithm 1 DBSCAN

1:$epsilon\geq 0$, $distancemetric$, $blockedcolumn(s)$

2:Pick a random unvisited record

3:Set this record state to visited and look for another unvisited record with
the same value for the $blockedcolumn(s)$ and within $epsilon$ distance using
the $distancemetric$

4:if no point is found then

5: go back to step 1

6:else

7: designate this a cluster group and add the record to it

8: go back to step 2

9:end if

##  4 Experiments

There were two sets of experiments that were performed:

  * •

with our existing customer data

  * •

with the musicbrainz data

###  4.1 Customer Data Experiments

For each of the 2 distance measures various epsilon values were tried to see
how this affected the clustering. They were also compared against the base
traditional Entity Matching. See Table 2.

Table 2: Customer Data Experiments ID | Distance Metric | Epsilon
---|---|---
1 | Traditional Entity Matching |
2 | Cosine similarity | 0.1
3 | Cosine similarity | 0.05
4 | L2 norm | 0.5
5 | L2 norm | 0.25
6 | L2 norm | 0.125
7 | Cosine similarity | 0.025

The embedding model used was the all-mpnet-base-v2[3] which encodes text into
an embedding vector of 768 dimensions.

####  4.1.1 Results

The values of epsilon for each of the different experiments were chosen based
on the number of matching groups it produced and the size of the largest
group. Since there is initially no ground truth with which to compare these
records to (until the users label the data) we cannot show precision, accuracy
or F-scores. Bearing that in mind, we can show the max group size and the
nunber of match groups. These were good indicators of how good the resulting
proposed duplicates were.

These results are given in Table 3

Table 3: Size of Matching Groups Type | Epsilon | Max Group Size | No. Match Groups
---|---|---|---
Traditional Entity Matching |  | 739 | 4,122
Cosine similarity | 0.1 | 206 | 2,422
Cosine similarity | 0.05 | 72 | 1,161
L2 norm | 0.5 | 240 | 2,959
L2 norm | 0.25 | 51 | 735
L2 norm | 0.125 | 51 | 156
Cosine similarity | 0.025 | 51 | 589

###  4.2 Musicbrainz Experiments

A publicly available dataset that was generated for testing Entity Matching
was used. These datasets are provided by the University of Leipzig and can be
seen at their website[4]. We used the musicbrainz-200K dataset for our
experiments. This dataset contains 193,750 tuples, where each tuple represents
a certain audio recording. The size of the generated duplicate clusters ranges
from 1 to 5.

####  4.2.1 Match Sentence

The match sentences were created by concatenating the following fields with
spaces:

  * •

title

  * •

length

  * •

artist

  * •

album

  * •

year

  * •

language

An example record is shown in Table 4

Table 4: Example Record title | length | artist | album | year | language
---|---|---|---|---|---
009-Ballade a donner | 4m 2sec | Luce Dufault | Luce Dufault (1996) | 96 | French

With this example record the "Match Sentence" becomes:

009-Ballade a donner 4m 2sec Luce Dufault Luce Dufault (1996) 96 French

####  4.2.2 Success Metric

In order to be able to evaluate the success or otherwise of the generated
match groups we used the following metrics:

  * •

True positives (TP): Correctly declared duplicates

  * •

False positives (FP): Incorrectly declared duplicates

  * •

True negatives (TN): Correctly avoided pairs

  * •

False negatives (FN): Missed duplicates

  * •

precision = TP / (TP + FP)

  * •

recall = TP / (TP + FN)

  * •

F-score = 2 * (precision * recall) / (precision + recall)

An initial baseline was run against the dataset using the traditional NLP
approach. The records were first sorted on the ’artist’ attribute which was
then used for blocking. The similarity metric used was the Levenshtein
Distance algorithm on the ’title’ attribute, with a similarity threshold value
of 0.9. This method is referred to as the ’NBA1’ when showing the results
later.

Various experiments were then run using the proposed technique and different
values of epsilon.

####  4.2.3 Results

It was found that actually the cosine similarity distance metric produced good
match groups. The l2 distance metric did not yield any good results with the
Musicbrainz dataset. Using the cosine similarity distance metric with varying
epsilon values produced the following best matching results.

Table 5: Clustering Results Epsilon | Dup | TP | FP | FN | F-score
---|---|---|---|---|---
0.175 | 143,750 | 43,759 | 54,318 | 45,673 | 0.47
0.245 | 143,750 | 59,272 | 42,261 | 42,261 | 0.58
0.265 | 143,750 | 59,029 | 37,361 | 47,360 | 0.58
0.3250 | 143,750 | 42,890 | 25,422 | 75,438 | 0.46

When a graph of the epsilon values against the F-scores was produced, it was
possible to see a curve with an obvious maximum value. See Figure 3

Figure 3: Experimental results epsilon against F-score

Comparing the results from the traditional NLP method (with the name NBA1 in
the table) with the proposed method we can see a massive improvement in the
accuracy of the clustering groups that are produced. There is an interesting
article that was produced in June 2023 [5] that talks about different
techniques that can be used to improve the clustering capabilities of the
traditional NLP method. These improved results have also been included in this
table to show that even using some or all of these techniques, our proposed
method is still producing comparably good clusters.

See Table 6.

Table 6: Clustering Methods Results Clustering Method | Dup | TP | FP | FN | F-score
---|---|---|---|---|---
NBA1 | 1,201,486 | 29,357 | 1,172,129 | 114,393 | 0.04
SNM1 | 59,757 | 30,737 | 29,020 | 113,013 | 0.3
NBA2 | 66,052 | 53,029 | 13,023 | 90,721 | 0.5
SNM2 | 79,134 | 60,438 | 18,696 | 83,312 | 0.54
ProposedMethod | 143,750 | 59,272 | 42,261 | 42,261 | 0.58

The description of the various Clustering Methods are: NBA1: Naive Blocking
Algorithm; SNM1: Sorted Neighbourhood Method; NBA2: Naive Blocking Algorithm
with pre-processing; SNM2: Sorted Neighbourhood Method with pre-processing.

Example cluster groups are shown in Figure 4

Figure 4: Match Groups for the Musicbrainz 200K dataset (epsilon=0.245)

###  4.3 Visualising Clustering

A useful way to visual the way the records are clustered is to first calculate
for each record the distance to it’s nearest neighbour. This was done using
the cosine similarity distance metric. Once this was completed the embedding
vectors of 768 dimensions were then reduced to just 2 dimensions by using a
UMAP [6] algorithm provided by the umap-learn[7] PyPi package. These then
became [X, Y] co-ordinates on a scatter plot with each point coloured using
the distance metric. The darker the colouring the closer the point is to
another. See Figure 5

Figure 5: 2D UMAP Musicbrainz 200K nearest neighbour plot

##  5 Benefits of the proposed approach

There are a number of benefits that the proposed approach has over the
traditional approach. These are the following:

###  5.1 No training of a model required

Because we are using a pretrained Large Language Model to produce our vectors,
there is no need to do further training as we proceed. We already start with
good potential match groups.

###  5.2 No pre-processing of the data required

In order to improve the identification of potential matches, pre-processing of
the data is often performed. These include filling in missing values, sorted
with nearest neighbours and other such techniques. All of these techniques can
improve the matching, but also introduce errors. With the proposed approach no
pre-processing of the data is required, which in turn means that no errors are
introduced.

###  5.3 User Engagement

Because our initial proposed match groups seem already reasonable to the
users, this engages them from the beginning. Using the traditional approach
means that the initial proposed match groups aren’t very good and until the
users label them we aren’t then able to train a model and propose better
candidates. By this point user engagement and confidence is low.

###  5.4 Adaptable

As bigger and better LLMs are produced, these can easily be swapped with the
current LLM being used. There will be no need to change the underlying code.

###  5.5 Understands Language

Because of the way that the LLMs have been trained, they understand language.
Things such as abbreviations and substitutions are easily picked up, without
any pre-processing of the data required. Things such as "and" and "&" in
English are known to be equivalent. The same in other languages such as German
where "und", "U." and "&" are known to be equivalent.

##  6 Future Work

As more and more embedding models are brought out which work over an
increasingly larger vector space, it would be good to see how this improves
our ability to cluster similar records together. The assumption would seem to
be that a larger vector space would ensure a greater ability to cluster
similar records together. But it would be good to check that this did indeed
turn out to be the case.

It will be beneficial to research an approach to calculate the optimal epsilon
based on the distribution of the matching records.

##  7 Conclusion

We have introduced an innovative approach to identifying duplicate records
that is comparatively better than the traditional NLP approach, even with
various improvements and pre-processing of data that can be done. The proposed
approach has the advantage of not requiring any pre-processing of the data to
be done before it can be used. Pre-processing of the data can improve accuracy
but also introduces its own errors. Another advantage of the proposed approach
is that it does not require labelled data to work better, or the training of a
model. It also understands differences in language. As bigger and better LLMs
are created these can easily be swapped for the current LLM without any
changes to underlying technique.

##  8 Resources

Useful documentation and Jupyter notebooks have been made available that
explain this approach in detail using the Musicbrainz test data. This is
available at [8].

## Acknowledgments

I would like to thank Dr. Ben Hoyle, a fellow worker at Zeiss Digital
Partners, who has helped me with this paper in many different ways. He has
been involved in the production of over 50 papers and his experience and
guidance has been invaluable. I really couldn’t have done this without his
input. Thank you Ben.

## References

  * [1] Fabian Panse and Felix Naumann.  Evaluation of duplicate detection algorithms: From quality measures to test data generation.  2021 IEEE 37th International Conference on Data Engineering (ICDE), pages 2373–2376, 2021.
  * [2] Sentence transformers.  https://huggingface.co/sentence-transformers.  Accessed: 2024-06-17.
  * [3] all-mpnet-base-v2 embedding model.  https://huggingface.co/sentence-transformers/all-mpnet-base-v2.  Accessed: 2024-06-17.
  * [4] Benchmark datasets for entity resolution.  https://dbs.uni-leipzig.de/research/projects/benchmark-datasets-for-entity-resolution.  Accessed: 2024-06-17.
  * [5] Abdulrazzak Ali, Nurul Emran, Safiza Kamal, Zahriah Othman, Awsan Salem, Maslita Abd Aziz, Nor Bohari, and Noraswaliza Abdullah.  Improving the efficiency of clustering algorithm for duplicates detection.  Indonesian Journal of Electrical Engineering and Computer Science, 30:1586, 06 2023.
  * [6] L. McInnes, J. Healy, and J. Melville.  UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction.  ArXiv e-prints, February 2018.
  * [7] umap-learn pypi package.  https://pypi.org/project/umap-learn.  Accessed: 2024-06-17.
  * [8] Genai duplication detection resources.  https://github.com/ianormy/genai_duplicate_detection_paper.  Accessed: 2024-06-17.

## Biography Of Author

Ian Ormesher is a seasoned full-stack Data Scientist with a robust background
in training and deploying AI models in production environments. With a career
spanning over four decades in the computer industry, he has honed his skills
in Machine Learning, Deep Neural Networks, NLP, GenAI, Reinforcement Learning,
and Computer Vision. He is proficient in a wide array of programming languages
and data analysis tools.