Computing K-theory and Ext for graph C*-algebras

TL;DR

This paper computes the K-theory and Ext groups for graph C*-algebras associated with countable directed graphs, extending previous results and revealing new relationships with Exel-Laca algebras.

Contribution

It generalizes existing K-theory and Ext computations to all countable graphs, and clarifies the Morita equivalence relations with Exel-Laca algebras.

Findings

K-theory and Ext are computed for all countable graph C*-algebras.

The results extend previous computations for row-finite graphs.

C*(E_A) is not necessarily Morita equivalent to O_A for certain matrices.

Abstract

K-theory and Ext are computed for the C*-algebra C*(E) of any countable directed graph E. The results generalize the K-theory computations of Raeburn and Szymanski and the Ext computations of Tomforde for row-finite graphs. As a consequence, it is shown that if A is a countable {0,1} matrix and E_A is the graph obtained by viewing A as a vertex matrix, then C*(E_A) is not necessarily Morita equivalent to the Exel-Laca algebra O_A.