Index maps in the K-theory of graph algebras

TL;DR

This paper computes the K-theory six-term exact sequence for graph C*-algebras, providing explicit invariants based on adjacency matrices that classify these algebras in many cases.

Contribution

It offers a method to explicitly compute the K-theoretic invariants of graph C*-algebras using adjacency matrices, enhancing classification techniques.

Findings

Explicit formulas for K-theory invariants in terms of kernels and cokernels.

Complete stable isomorphism invariants for certain classes of graph C*-algebras.

Use of six-term exact sequences as comprehensive invariants.

Abstract

Let $C^*(E)$ be the graph $C^*$-algebra associated to a graph E and let J be a gauge invariant ideal in $C^*(E)$. We compute the cyclic six-term exact sequence in $K$-theory of the associated extension in terms of the adjacency matrix associated to $E$. The ordered six-term exact sequence is a complete stable isomorphism invariant for several classes of graph $C^*$-algebras, for instance those containing a unique proper nontrivial ideal. Further, in many other cases, infinite collections of such sequences comprise complete invariants. Our results allow for explicit computation of the invariant, giving an exact sequence in terms of kernels and cokernels of matrices determined by the vertex matrix of $E$.