Let $\mathbb{N}$ be the set of all natural numbers. We define a groupoid structure on $\mathbb{N}^{\mathbb{N}}$ as follows:
We put $G^1=\mathbb{N}^{\mathbb{N}},\;G^0 =\{(a_n)\in G^1\mid a_{2n-1}=a_{2n},\;\forall n \in \mathbb{n}\}$. We define the range and source maps $r,s:G^1 \to G^0$ as follows:$$r((a_n))=(a_1, a_1, a_3,a_3,a_5,a_5\ldots,a_{2n+1}, a_{2n+1}\ldots)\\s((a_n))=(a_2, a_2,a_4,a_4, a_6, a_6,\ldots,a_{2n}, a_{2n}\ldots)$$
To define the composition map of the groupoid, we assume that $r((b_n))=s((c_n))$. Put $b=(b_n),\;c=(c_n)$. The composition is $$c\circ b=(c_1,b_2,c_3,b_4,\ldots,c_{2n-1},b_{2n},\ldots)$$
The groupoid is equipped with the discrete topology and we consider the associated groupoid $C^* $ algebra.
What is a precise description of this $C^*$ algebra?Is it identical to a well known $C^*$ algebra? What is its $K$-theory?
