In the post " Reduced groupoid $C^*$-algebra " I discuss two possible ways to construct the reduced groupoid $C^* $-algebra $C_\lambda^* (G)$ of an étale locally compact Hausdorff groupoid $G$.
However, I came across a new one in Aidan Sims' notes " https://www.aidansims.com/files/GroupoidsNotes-2017.pdf ". Here, to each $x \in G^{(0)}$ we associate the left regular representation of $C_c(G)$ on $B(\ell^2(G_x))$ given by $\lambda_x(f)g = (f*g)|_{G_x}$. We thus define the reduced group $C^* $-algebra by the closure of
$$ \left( \bigoplus\limits_{x \in G^{(0)}} \lambda_x \right)(C_c(G)) \subseteq \bigoplus\limits_{x \in G^{(0)}} B(\ell^2(G_x)). $$
The norm, which for distinction we will call $\| \cdot \|_\oplus$ is given by
$$ \|f\|_\oplus = \sup\limits_{x \in G^{(0)}} \|\lambda_x(f)\| = \sup\limits_{x \in G^{(0)}} \sup\limits_{\|g\|_{\ell^2(G_x)} = 1} \sum\limits_{\gamma \in G_x} |f*g(\gamma)|^2, $$
On the other hand, if we use the Hilbert Modules approach (can be found here " Reduced groupoid $C^*$-algebra ") the norm is
$$ \|f\|_\lambda = \sup\limits_{x \in G^{(0)}} \sup\limits_{\|g\|_{L^2(G)} = 1} \sum\limits_{\gamma \in G_x} |f*g(\gamma)|^2 $$
The goal of course, is to show that the norms are equal, and this almost proves it, however we need to compare $\|g\|_{L^2(G)} = 1$ with $\|g\|_{\ell^2(G_x)} = 1$.
In Aidan's notes its written that it "is a fairly straightforward exercise, if you are familiar with Hilbert modules" (p. 23). I am very interested in knowing what he means by this since my knowledge of Hilbert Modules is rudimentary and I feel it would be a good way to avoid the technical detail I was having above.
Showing $\|f\|_\lambda=\|f\|_\oplus$ is handled similarly to your previous post. In this situation, rather than considering a regular Borel measure $\mu$ on $G^{(0)}$ with full support, we consider the counting measure on $G^{(0)}$. There is still a unitary $U:L^2(G)\otimes_{C_0(G^{(0)})}\ell^2(G^{(0)})\to\oplus_{x\in G^{(0)}}\ell^2(G_x)$ such that for all $\xi\in C_c(G)$, $\eta\in\ell^2(G^{(0)})$, and $\gamma\in G$, we have \begin{align*} U(\xi\otimes\eta)(\gamma)=\xi(\gamma)\eta(s(\gamma)). \end{align*} Moreover, the same calculation as in the answer to your previous post shows us that for all $f\in C_c(G)$ we have \begin{align*} \lambda(f)\otimes1=U^*\lambda_\oplus(f)U, \end{align*} where $\lambda_\oplus=\oplus_{x\in G^{(0)}}\lambda_x$. Hence $\|\lambda(f)\otimes1\|=\|f\|_\oplus$. Since the representation of $C_0(G^{(0)})$ on $\ell^2(G^{(0)})$ is faithful, the $*$-homomorphism $\mathbb{B}(L^2(G))\to\mathbb{B}(L^2(G)\otimes_{C_0(G^{(0)})}\ell^2(G^{(0)}))$ given by $T\mapsto T\otimes 1$ is injective. Hence $\|f\|_\lambda=\|\lambda(f)\otimes1\|$, and therefore $\|f\|_\lambda=\|f\|_\oplus$.