Automorphisms of Cuntz-Krieger algebras

We prove that the natural homomorphism from Kirchberg's ideal-related KK-theory, KKE(e, e'), with one specified ideal, into Hom_{\Lambda} (\underline{K}_{E} (e), \underline{K}_{E} (e')) is an isomorphism for all extensions e and e' of separable, nuclear C*-algebras in the bootstrap category N with the K-groups of the associated cyclic six term exact sequence being finitely generated, having zero exponential map and with the K_{1}-groups of the quotients being free abelian groups. This class includes all Cuntz-Krieger algebras with exactly one non-trivial ideal. Combining our results with the results of Kirchberg, we classify automorphisms of the stabilized purely infinite Cuntz-Krieger algebras with exactly one non-trivial ideal modulo asymptotically unitary equivalence. We also get a classification result modulo approximately unitary equivalence. The results in this paper also apply to certain graph algebras.Comment: 26 page

On the classification of nonsimple graph C*-algebras

We prove that a graph C*-algebra with exactly one proper nontrivial ideal is classified up to stable isomorphism by its associated six-term exact sequence in K-theory. We prove that a similar classification also holds for a graph C*-algebra with a largest proper ideal that is an AF-algebra. Our results are based on a general method developed by the first named author with Restorff and Ruiz. As a key step in the argument, we show how to produce stability for certain full hereditary subalgebras associated to such graph C*-algebras. We further prove that, except under trivial circumstances, a unique proper nontrivial ideal in a graph C*-algebra is stable.Comment: 27 pages, uses XY-pic; Version II comments: A few minor typos correcte

Ideal related K-theory with coefficients

In this paper, we define an invariant, which we believe should be the substitute for total K-theory in the case when there is one distinguished ideal. Moreover, some diagrams relating the new groups to the ordinary K-groups with coefficients are constructed. These diagrams will in most cases help to determine the new groups, and will in a companion paper be used to prove a universal multi-coefficient theorem for the one distinguished ideal case for a large class of algebras

Nonsplitting in Kirchberg's ideal-related KK-theory

A universal coefficient theorem in the setting of Kirchberg's ideal-related KK-theory was obtained in the fundamental case of a C*-algebra with one specified ideal by Bonkat and proved there to split, unnaturally, under certain conditions. Employing certain K-theoretical information derivable from the given operator algebras in a way introduced here, we shall demonstrate that Bonkat's UCT does not split in general. Related methods lead to information on the complexity of the K-theory which must be used to classify *-isomorphisms for purely infinite C*-algebras with one non-trivial ideal.Comment: 14 pages, minor typos fixed, 5 figures adde

The Ranges of K-theoretic Invariants for Nonsimple Graph Algebras

There are many classes of nonsimple graph C*-algebras that are classified by the six-term exact sequence in K-theory. In this paper we consider the range of this invariant and determine which cyclic six-term exact sequences can be obtained by various classes of graph C*-algebras. To accomplish this, we establish a general method that allows us to form a graph with a given six-term exact sequence of K-groups by splicing together smaller graphs whose C*-algebras realize portions of the six-term exact sequence. As rather immediate consequences, we obtain the first permanence results for extensions of graph C*-algebras. We are hopeful that the results and methods presented here will also prove useful in more general cases, such as situations where the C*-algebras under investigations have more than one ideal and where there are currently no relevant classification theories available.Comment: 40 page