Index maps in the K-theory of graph algebras

TY - JOUR

T1 - Index maps in the K-theory of graph algebras

AU - Meier Carlsen, Toke

AU - Eilers, Søren

AU - Tomforde, Mark

PY - 2012/4

Y1 - 2012/4

N2 - 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 associated to the extension 0 → J → C *(E)/J → 0 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, finite collections of such sequences constitute 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.

AB - 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 associated to the extension 0 → J → C *(E)/J → 0 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, finite collections of such sequences constitute 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.

U2 - 10.1017/is011004017jkt156

DO - 10.1017/is011004017jkt156

M3 - Journal article

SN - 1865-2433

VL - 9

SP - 385

EP - 406

JO - Journal of K-Theory

JF - Journal of K-Theory

IS - 2

ER -