Index maps in theK-theory of graph algebras

Abstract

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 associated to the extensionin terms of the adjacency matrix associated to E. The ordered sixterm 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 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.

Semiprojectivity and properly infinite projections in graph C⁎-algebras

“…These have finitely generated K-theory, and had C * (E, S) been semiprojective, we would have that K * (C * (E, S)) K * (C * (F i , S F i )) for some i, which is a contradiction. The second claim is noted, e.g., in [CET12].…”

Strong classification of purely infinite Cuntz-Krieger algebras

“…We may also without loss of generality assume that j is the only component of E. Now let x = (x i ) andS x , the projections P x = S * x S x = S x S * x ,P x =S * xSx =S xS * x and the unitaries U x = S x + 1 − P x andŨ x =S x + 1 −P x in matrix algebras over C * (S) = C * (T ) as in [CET12], where the ones without a tilde correspond to the family S while the ones with a tilde correspond to the family T . From the matrix (U …”

The K-theoretical range of Cuntz–Krieger algebras

“…First, we recall relevant definitions and properties of graph C * -algebras, and explain how one can determine, for a graph E, whether the graph C * -algebra C * (E) can be regarded as a (tight) C * -algebra over a given finite space X. We also introduce a formula from [6] for calculating reduced filtered K-theory of a graph C * -algebra using the adjacency matrix of its defining graph. Finally, Proposition 4.3 in conjunction with Theorem 4.4 consitutes the desired range-of-invariant result.…”

“…Let Y ∈ LC(X) and U ∈ O(Y ) be given, and define C = Y \ U . Then by [6], the six-term exact sequence induced by C * (E)…”