Leavitt path algebras and direct limits

TL;DR

This paper introduces Leavitt path algebras for arbitrary graphs and develops direct limit methods to extend key algebraic properties from countable to uncountable graphs.

Contribution

It presents a framework using direct limits to generalize properties of Leavitt path algebras from countable to uncountable graphs.

Findings

Characterization of simplicity for uncountable graphs

Extension of the exchange property to uncountable graphs

Cancellation conditions for K-theoretic monoids

Abstract

An introduction to Leavitt path algebras of arbitrary directed graphs is presented, and direct limit techniques are developed, with which many results that had previously been proved for countable graphs can be extended to uncountable ones. Such results include characterizations of simplicity, characterizations of the exchange property, and cancellation conditions for the K-theoretic monoid of equivalence classes of idempotent matrices.