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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?}$

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    @Rasmus yes, I do mean that. Corrected in the question. Thanks. I will think about those cases.2012-04-10

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