Exchange rings and real rank zero C*-algebras associated with finitely separated graphs

TL;DR

This paper generalizes Condition (K) for finitely separated graphs and links it to essential freeness and the exchange property in associated algebras, leading to new insights into their structure.

Contribution

It introduces a generalized Condition (K) for finitely separated graphs and establishes its equivalence to essential freeness and the exchange property in related algebras.

Findings

Condition (K) generalization for finitely separated graphs

Equivalence of Condition (K) to essential freeness

Tame separated graph algebras with exchange property are separative

Abstract

We introduce a generalisation of Condition (K) to finitely separated graphs and show that it is equivalent to essential freeness of the associated partial action as well as the exchange property of any of the associated tame algebras. As a consequence, we can show that any tame separated graph algebra with the exchange property is separative.