Index maps in the \(K\)-theory of graph algebras
The cyclic six-term exact sequence associated to a gauge-invariant ideal \(J\) of a graph \(C^\ast\)-algebra \(C^\ast(E)\) is a complete stable isomorphism invariant of \(C^\ast(E)\) in several important cases. The combination of different such six-term exact sequences provides a complete invariant in more general cases. Taking this into account, the authors provide concrete formulae to compute the groups and homomorphisms appearing in a six-term exact sequence associated to a pair \((J, C^\ast(E))\) as before. In particular, they show that the index map \(\partial_0 : K_0(C^\ast(E)/J)\rightarrow K_1(J)\) is always the zero map, regardless of whether \(C^\ast(E)\) has real rank zero or not. The harder part of the computation concerns the description of the other index map \(\partial _1\).
scientific article; zbMATH DE number 6033589
scientific article