Partial Translation Algebras for Trees

TL;DR

This paper explores how partial translation algebras associated with subspaces of trees can generate important classical $C^*$-algebras and their extensions, linking geometric structures with operator algebras.

Contribution

It shows that key $C^*$-algebras and extensions can be realized through partial translation algebras on subspaces of trees, connecting geometric and algebraic frameworks.

Findings

Classical $C^*$-algebras arise from partial translation algebras on trees

Extensions of $C^*$-algebras are naturally obtained from subspace structures

The approach unifies geometric and algebraic perspectives in operator algebras

Abstract

In arXiv:math/0603621 we introduced the notion of a partial translation $C^*$-algebra for a discrete metric space. Here we demonstrate that several important classical $C^*$-algebras and extensions arise naturally by considering partial translation algebras associated with subspaces of trees.