There are two $C^\ast$-algebras associated to the $\ast$-algebra (under a convolution and the usual involution) $$C_c(G) := \{ f:G\longrightarrow \mathbb C :\:f \text{ has compact support}\}$$ of compactly supported continuous functions on a locally compact groupoid $G$: its completions in the norms $$ \|f\|_{\max} := \sup \{\|\pi(f)\|:\:\pi \text{ is a bounded involutive representation of C_c(G)}\}, $$ $$ \|f\|_r := \sup \{\|\pi_x(f)\|:\: x\in G^0 \}, $$ where $\pi_x$ is a regular representation in the fiber of $x$ under the target map.
Many references on groupoid $C^\ast$-algebras give the following example. Let $G$ be the groupoid of the equivalence relation on the set $$X:= [0,1]\times\{0,1\}$$ given by $(x,0)\sim (x,1)$ for $x \in (0,1).$ Marcolli writes in her ``Lectures on Arithmetic Noncommutative Geometry'' that the corresponding quotient space has no interesting functions, while the convolution algebra on $G$ is $$C_c(G) = \{f:G\longrightarrow C([0,1])\otimes M_{2}(\mathbb C):\: f(0),\,f(1) \text{ are diagonal} \}.$$ None of the sources I've found explain this description of $C_c(G)$ in detail sufficient for my understanding. So I ask $$\fbox{Question: Why does $C_c(G)$ have the above form for this groupoid?}$$