Constructing minimal homeomorphisms on point-like spaces and a dynamical presentation of the Jiang–Su algebra

Abstract

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,

Cartan subalgebras in C*-algebras. Existence and uniqueness

“…The fifth section illustrates various techniques to construct infinitely many Cartan subalgebras in a given C*-algebra. The first one uses the recent construction of Deeley, Putnam and Strung [9] of the Jiang-Su algebra Z as a groupoid C*-algebra. It applies to unital C*-algebras which have a Cartan subalgebra whose spectrum has finite covering dimension and which are Z-stable.…”

Some classifiable groupoid $$C^{*}$$ C ∗ -algebras with prescribed K-theory

“…Another extension of the results in [4] is as follows. The key idea in [4] is to alter the sphere and the dynamics so as to insert a 'tube' which is invariant under the homeomorphism. If one instead inserted k of these tubes, one obtains a space Z with a minimal homeomorphism ζ such that…”

Every classifiable simple C*-algebra has a Cartan subalgebra

“…In the purely infinite case, groupoid models and hence Cartan subalgebras have been constructed in [41] (see also [29, § 5]). For special classes of stably finite unital C*-algebras, groupoid models have been constructed in [8,35] using topological dynamical systems. Using a new approach, the goal of this paper is to answer Question 1.1 by constructing Cartan subalgebras in all the C*-algebra models constructed in [11,19,20], covering all classifiable stably finite C*-algebras, in particular in all classifiable unital C*algebras.…”

“…8. It is worth pointing out that a groupoid model has been constructed for Z in [8] using a different construction (but the precise dimension of the unit space has not been determined in [8]). Moreover, G. Szabó and S. Vaes independently found groupoid models for W, again using constructions different from ours.…”