Gunnar Restorff

Classification of Cuntz-Krieger algebras up to stable isomorphism

Abstract In this paper we classify all Cuntz-Krieger algebras whose adjacency matrices satisfy condition (II) of Cuntz. The invariant arises naturally from the ideal lattice and the six-term exact sequences from K-theory, while the proof of this invariant being complete depends on recent results on flow equivalence of shifts of finite type by Mike Boyle and Danrun Huang. □

Classification of extensions of classifiable C*-algebras

Abstract For a certain class of extensions e : 0 → B → E → A → 0 of C * -algebras in which B and A belong to classifiable classes of C * -algebras, we show that the functor which sends e to its associated six term exact sequence in K-theory and the positive cones of K 0 ( B ) and K 0 ( A ) is a classification functor. We give two independent applications addressing the classification of a class of C * -algebras arising from substitutional shift spaces on one hand and of graph algebras on the other. The former application leads to the answer of a question of Carlsen and the first named author concerning the completeness of stabilized Matsumoto algebras as an invariant of flow equivalence. The latter leads to the first classification result for nonsimple graph C ∗ -algebras.

The Ordered

Let A be a C*-algebra with real rank zero which has the stable weak cancellation property. Let I be an ideal of A such that I is stable and satisfies the corona factorization property. We prove that 0->I->A->A/I->0 is a full extension if and only if the extension is stenotic and K-lexicographic. As an immediate application, we extend the classification result for graph C*-algebras obtained by Tomforde and the first named author to the general non-unital case. In combination with recent results by Katsura, Tomforde, West and the first author, our result may also be used to give a purely K-theoretical description of when an essential extension of two simple and stable graph C*-algebras is again a graph C*-algebra.

K

In this paper we extend the classification results obtained by Rordam in the paper [16]. We prove a strong classification theorem for the unital essential extensions of Kirchberg algebras, a classification theorem for the non-stable, non-unital essential extensions of Kirchberg algebras, and we characterize the range in both cases. The invariants are cyclic six term exact sequences together with the class of some unit.

-theory of a Full Extension

Abstract We give a classification result for a certain class of C ⁎ -algebras A over a finite topological space X in which there exists an open set U of X such that U separates the finite and infinite subquotients of A . We apply our results to C ⁎ -algebras arising from graphs.

On Rørdam's classification of certain

The smallest primitive ideal spaces for which there exist counterexamples to the classification of non-simple, purely infinite, nuclear, separable C*-algebras using filtrated K-theory, are four-point spaces. In this article, we therefore restrict to real rank zero C*-algebras with four-point primitive ideal spaces. Up to homeomorphism, there are ten different connected T0-spaces with exactly four points. We show that filtrated K-theory classifies real rank zero, tight, stable, purely infinite, nuclear, separable C*-algebras that satisfy that all simple subquotients are in the bootstrap class for eight out of ten of these spaces.

C^*

Abstract We give conditions for when continuous orbit equivalence of one-sided shift spaces implies flow equivalence of the associated two-sided shift spaces. Using groupoid techniques, we prove that this is always the case for shifts of finite type. This generalises a result of Matsumoto and Matui from the irreducible to the general case. We also prove that a pair of one-sided shift spaces of finite type are continuously orbit equivalent if and only if their groupoids are isomorphic, and that the corresponding two-sided shifts are flow equivalent if and only if the groupoids are stably isomorphic. As applications we show that two finite directed graphs with no sinks and no sources are move equivalent if and only if the corresponding graph C ⁎ -algebras are stably isomorphic by a diagonal-preserving isomorphism (if and only if the corresponding Leavitt path algebras are stably isomorphic by a diagonal-preserving isomorphism), and that two topological Markov chains are flow equivalent if and only if there is a diagonal-preserving isomorphism between the stabilisations of the corresponding Cuntz–Krieger algebras (the latter generalises a result of Matsumoto and Matui about irreducible topological Markov chains with no isolated points to a result about general topological Markov chains). We also show that for general shift spaces, strongly continuous orbit equivalence implies two-sided conjugacy.

-algebras with one non-trivial ideal, II

A universal coefficient theorem in the setting of Kirchbergs 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 Bonkats 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.

Classifying C⁎-algebras with both finite and infinite subquotients

We show that the Cuntz splice induces stably isomorphic graph