Index maps in the K-theory of graph algebras

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 $E$.

💡 Research Summary

The paper investigates the K‑theoretic structure of graph C⁎‑algebras (C^{}(E)) together with a gauge‑invariant ideal (J). Starting from the well‑known correspondence between gauge‑invariant ideals and saturated hereditary subsets (H\subseteq E^{0}), the authors consider the short exact sequence
(0\to J\to C^{}(E)\to C^{*}(E)/J\to0)
and compute the associated cyclic six‑term exact sequence in K‑theory completely in terms of the adjacency matrix of the underlying directed graph (E).

The key technical step is a block decomposition of the vertex matrix (A_{E}) relative to the partition (E^{0}=H\sqcup (E^{0}\setminus H)). Writing
(A_{E}=\begin{pmatrix} B & X\ 0 & C\end{pmatrix})
the submatrix (B) records edges inside (H), (C) records edges inside the complement, and (X) records edges from the complement into (H). Using the standard description of K‑theory for graph algebras via kernels and cokernels of (I-A_{E}^{t}), the authors obtain explicit formulas:

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