Abstract. The principal aim of this paper is to give a dynamical presentation of the Jiang-Su algebra. Originally constructed as an inductive limit of prime dimension drop algebras, the Jiang-Su algebra has gone from being a poorly understood oddity to having a prominent positive role in George Elliott's classification programme for separable, nuclear C * -algebras. Here, we exhibit anétale equivalence relation whose groupoid C * -algebra is isomorphic to the Jiang-Su algebra. The main ingredient is the construction of minimal homeomorphisms on infinite, compact metric spaces, each having the same cohomology as a point. This construction is also of interest in dynamical systems. Any self-map of an infinite, compact space with the same cohomology as a point has Lefschetz number one. Thus, if such a space were also to satisfy some regularity hypothesis (which our examples do not), then the LefschetzHopf Theorem would imply that it does not admit a minimal homeomorphism. IntroductionThe fields of operator algebras and dynamical systems have a long history of mutual influence. On the one hand, dynamical systems provide interesting examples of operator algebras and have often provided techniques which are successfully imported into the operator algebra framework. On the other hand, results in operator algebras are often of interest to those in dynamical systems. In ideal situations, significant information is retained when passing from dynamics to operator algebras, and vice versa. This relationship has been particularly interesting for the classification of C * -algebras. An extraordinary result in this setting is the classification, up to strong orbit equivalence, of the minimal dynamical systems on a Cantor set and the corresponding K-theoretical classification of the associated crossed product C * -algebras [9,24]. Classification for separable, simple, nuclear C * -algebras remains an interesting open problem. To every simple separable nuclear C * -algebra one assigns a computable set of invariants involving K-theory, tracial state spaces, and the pairing between these objects. George Elliott conjectured that for all such C * -algebras,

We show that the homoclinic C * -algebras of mixing Smale spaces are classifiable by the Elliott invariant. To obtain this result, we prove that the stable, unstable, and homoclinic C * -algebras associated to such Smale spaces have finite nuclear dimension. Our proof of finite nuclear dimension relies on Guentner, Willett, and Yu's notion of dynamic asymptotic dimension.2000 Mathematics Subject Classification. 46L35, 37D20.

No abstract

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Let X be an infinite compact metric space, α : X → X a minimal homeomorphism, u the unitary implementing α in the transformation group C * -algebra C(X) ⋊α Z, and S a class of separable nuclear C * -algebras that contains all unital hereditary C * -subalgebras of C * -algebras in S. Motivated by the success of tracial approximation by finite dimensional C * -algebras as an abstract characterization of classifiable C * -algebras and the idea that classification results for C * -algebras tensored with UHF algebras can be used to derive classification results up to tensoring with the Jiang-Su algebra Z, we prove that (C(X) ⋊α Z) ⊗ M q ∞ is tracially approximately S if there exists a y ∈ X such that the C * -subalgebra (C * (C(X), uC 0 (X \ {y}))) ⊗ M q ∞ is tracially approximately S. If the class S consists of finite dimensional C * -algebras, this can be used to deduce classification up to tensoring with Z for C * -algebras associated to minimal dynamical systems where projections separate tracial states. This is done without making any assumptions on the real rank or stable rank of either C(X)⋊α Z or C * (C(X), uC 0 (X \{y})), nor on the dimension of X. The result is a key step in the classification of C * -algebras associated to uniquely ergodic minimal dynamical systems by their ordered K-groups. It also sets the stage to provide further classification results for those C * -algebras of minimal dynamical systems where projections do not necessarily separate traces.

Abstract. We prove that in a simple, unital, exact, Z-stable C * -algebra of stable rank one, the distance between the unitary orbits of self-adjoint elements with connected spectrum is completely determined by spectral data. This fails without the assumption of Z-stability.

Let β : S n → S n , for n = 2k + 1, k ≥ 1, be one of the known examples of a nonuniquely ergodic minimal diffeomorphism of an odd dimensional sphere. For every such minimal dynamical system (S n , β) there is a Cantor minimal system (X, α) such that the corresponding product system (X × S n , α × β) is minimal and the resulting crossed product C * -algebra C(X ×S n )⋊ α×β Z is tracially approximately an interval algebra (TAI). This entails classification for such C * -algebras. Moreover, the minimal Cantor system (X, α) is such that each tracial state on C(X × S n ) ⋊ α×β Z induces the same state on the K 0 -group and such that the embedding of C(S n ) ⋊ β Z into C(X × S n ) ⋊ α×β Z preserves the tracial state space. This implies C(S n ) ⋊ β Z is TAI after tensoring with the universal UHF algebra, which in turn shows that the C * -algebras of these examples of minimal diffeomorphisms of odd dimensional spheres are classified by their tracial state spaces.

Floyd gave an example of a minimal dynamical system which was an extension of an odometer and the fibres of the associated factor map were either singletons or intervals. Gjerde and Johansen showed that the odometer could be replaced by any Cantor minimal system. Here, we show further that the intervals can be generalized to cubes of arbitrary dimension and to attractors of certain iterated function systems. We discuss applications.