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

TL;DR

This paper constructs and analyzes a $C^*$-algebra associated with a $ ext{Z}^n$-tree, extending graph algebra concepts to ordered group trees and exploring its properties.

Contribution

It introduces a new $C^*$-algebra framework for $ ext{Z}^n$-trees based on groupoid constructions, expanding the scope of graph algebra methods.

Findings

Defined the $C^*$-algebra of the $ ext{Z}^n$-tree

Proved properties of the constructed $C^*$-algebra

Extended graph $C^*$-algebra techniques to ordered group trees

Abstract

Let $\Lambda = \mathbb{Z}^n$ with lexicographic ordering. $\Lambda$ is a totally ordered group. Let $X = \Lambda^+ * \Lambda^+$. Then $X$ is a $\Lambda$-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 $\Lambda$-tree is defined as the $C^*$-algebra of this groupoid. We prove some properties of this $C^*$-algebra.