Søren Eilers

COMPUTING CONTINGENCIES FOR STABLE RELATIONS

Close ties between approximation properties for relations on C*-algebra elements and lifting results for the universal C*-algebras the relations generate is a very widespread and useful phenomenon in C*-algebra theory. In this paper, we explore how to achieve results of this kind when the approximation properties and the lifting results are true only in special cases determined by K-theoretical contingencies. To interpolate between properties of these two basic types, we must investigate C*-algebras given by softened relations, in particular with emphasis on their K-theory. A surprisingly weak correlation between the K-theory of the C*-algebras given by exact and softened relations leads to delicate problems which must be treated with care. As an example of a set of relations which are prone to an analysis of this kind we study the pairs of unitaries commuting up to rational rotation.

Asymptotic Unitary Equivalence in KK-Theory

A description of the Kasparov group KK(A,B) is given in terms of Cuntz pairs of representations and the notion of proper asymptotic unitary equivalence that we introduce here. The use of the word ‘proper’ reflects the crucial fact that all unitaries implementing the equivalence can be chosen to be compact perturbations of identity. The result has significant applications to the classification theory of nuclear C∗-algebras. Mathematics Subject Classifications (2000): 19K35, 46L80, 46L35.

Augmenting dimension group invariants for substitution dynamics

We present new invariants for substitutional dynamical systems. Our main contribution is a flow invariant which is strictly finer than, but related and akin to, the dimension groups of Herman, Putnam and Skau. We present this group as a stationary inductive limit of a system associated to an integer matrix defined from combinatorial data based on the class of special words of the dynamical system.

Approximate homogeneity is not a local property

Abstract It is shown that the AH algebras satisfy a certain splitting property at the level of K-theory with torsion coefficients. The splitting property is used to prove the following: There are locally homogeneous C*-algebras which are not AH algebras. The class of AH algebras is not closed under countable inductive limits. There are real rank zero split quasidiagonal extensions of AH algebras which are not AH algebras.

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

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.

-theory of a Full Extension

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 extension 0 ! J ! C � (E) ! C � (E)/J ! 0 in terms of the adjacency matrix associated to E. The ordered six- term exact sequence is a complete stable isomorphism invariant for se- veral 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.

Classifying C⁎-algebras with both finite and infinite subquotients

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.

INDEX MAPS IN THE K-THEORY OF GRAPH ALGEBRAS

We present an algorithm which for any aperiodic and primitive substitution outputs a finite representation of each special word in the shift space associated to that substitution, and determines when such representations are equivalent under orbit and shift tail equivalence. The algorithm has been implemented and applied in the study of certain new invariants for flow equivalence of substitutional dynamical systems.