The ordered K-theory of a full extension

TL;DR

This paper characterizes when a full extension of certain C*-algebras is stenotic and K-lexicographic, extending classification results for graph C*-algebras to non-unital cases using K-theory.

Contribution

It provides a K-theoretical criterion for full extensions of C*-algebras with real rank zero and applies this to classify and describe graph C*-algebras.

Findings

Extension is full iff it is stenotic and K-lexicographic

Extended classification to non-unital graph C*-algebras

Provided K-theoretical conditions for extensions of simple stable graph C*-algebras

Abstract

Let A be a C*-algebra with real rank zero which has the stable weak cancellation property. Let I be an ideal of A such that I is stable and satisfies the corona factorization property. We prove that 0->I->A->A/I->0 is a full extension if and only if the extension is stenotic and K-lexicographic. As an immediate application, we extend the classification result for graph C*-algebras obtained by Tomforde and the first named author to the general non-unital case. In combination with recent results by Katsura, Tomforde, West and the first author, our result may also be used to give a purely K-theoretical description of when an essential extension of two simple and stable graph C*-algebras is again a graph C*-algebra.