Index maps in the K-theory of graph algebras

Details Index maps in the K-theory of graph algebras Index maps in the K-theory of graph algebras

2010-11-18

arxiv.org/abs/1011.4154v1

Mathematics

Operator Algebras

Scientific paper

10.1017/is011004017jkt156

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 $E$.

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