$C^*$-algebra of the $\mathds{Z}^n$-tree

Abstract

Let $Λ= \mathbb{Z}^n$ with lexicographic ordering. $Λ$ is a totally ordered group. Let $X = Λ^+ * Λ^+$. Then $X$ is a $Λ$-tree. Analogous to the construction of graph $C^*$-algebras, we form a groupoid whose unit space is the space of ends of the tree. The $C^*$-algebra of the $Λ$-tree is defined as the $C^*$-algebra of this groupoid. We prove some properties of this $C^*$-algebra.