Minimal Dynamics and K-Theoretic Rigidity: Elliott’s Conjecture

We characterize subhomogeneity for twisted étale groupoid $\text{C}^{*}$-algebras and obtain an upper bound on their nuclear dimension. As an application, we remove the principality assumption in recent results on upper bounds on the nuclear dimension of a twisted étale groupoid $\text{C}^{*}$-algebra in terms of the dynamic asymptotic dimension of the groupoid and the covering dimension of its unit space. As a non-principal example, we show that the dynamic asymptotic dimension of any minimal (not necessarily free) action of the infinite dihedral group $D_{\infty }$ on an infinite compact Hausdorff space $X$ is always one. So if we further assume that $X$ is second-countable and has finite covering dimension, then $C(X)\rtimes _{r} D_{\infty }$ has finite nuclear dimension and is classifiable by its Elliott invariant.

The principal aim of the present 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 * {\mathrm{C}^{*}} -algebras. Here, we exhibit an étale equivalence relation whose groupoid C * {\mathrm{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 Lefschetz–Hopf Theorem would imply that it does not admit a minimal homeomorphism.

We introduce a notion of Rokhlin dimension for one parameter automorphism groups of C*-algebras. This generalizes Kishimoto's Rokhlin property for flows, and is analogous to the notion of Rokhlin dimension for actions of the integers and other discrete groups introduced by the authors and Zacharias in previous papers. We show that finite nuclear dimension and absorption of a strongly self-absorbing C*-algebra are preserved under forming crossed products by flows with finite Rokhlin dimension, and that these crossed products are stable. Furthermore, we show that a flow on a commutative C*-algebra arising from a free topological flow has finite Rokhlin dimension, whenever the spectrum is a locally compact metrizable space with finite covering dimension. For flows that are both free and minimal, this has strong consequences for the associated crossed product C*-algebras: Those containing a non-zero projection are classified by the Elliott invariant (for compact manifolds this consists of topological K-theory together with the space of invariant probability measures and a natural pairing given by the Ruelle-Sullivan map).

We study the topological variant of Rokhlin dimension for topological dynamical systems (X,{\alpha},Z^m) in the case where X is assumed to have finite covering dimension. Finite Rokhlin dimension in this sense is a property that implies finite Rokhlin dimension of the induced action on C*-algebraic level, as was discussed in a recent paper by Ilan Hirshberg, Wilhelm Winter and Joachim Zacharias. In particular, it implies under these conditions that the transformation group C*-algebra has finite nuclear dimension. Generalizing a result of Yonatan Gutman, we show that free Z^m-actions on finite dimensional spaces satisfy a strengthened version of the so-called marker property, which yields finite Rokhlin dimension for said actions.

Let $X$ be an infinite compact metrizable space, and let $\sigma: X\to X$ be a minimal homeomorphism. Suppose that $(X, \sigma)$ has zero mean topological dimension. The associated C*-algebra $A=\mathrm{C}(X)\rtimes_\sigma\mathbb Z$ is shown to absorb the Jiang-Su algebra $\mathcal Z$ tensorially, i.e., $A\cong A\otimes\mathcal Z$. This implies that $A$ is classifiable when $(X, \sigma)$ is uniquely ergodic. Moreover, without any assumption on the mean dimension, it is shown that $A\otimes A$ always absorbs the Jiang-Su algebra.

Chapter 52 - The interplay between compact spaces and the Banach spaces of their continuous functions

For a compact space K we denote by $$C_w(K)$$ ( $$C_p(K)$$ ) the space of continuous real-valued functions on K endowed with the weak (pointwise) topology. In this paper we address the following basic question which seems to be open: Suppose that K is an infinite (metrizable) compact space. Can $$C_w(K)$$ and $$C_p(K)$$ be homeomorphic? We show that the answer is “no”, provided K is an infinite compact metrizable C-space. In particular our proof works for any infinite compact metrizable finite-dimensional space K.

Compact spaces of diversity two

This paper considers fuzzifying topologies, a special case of I-fuzzy topologies (bifuzzy topologies), introduced by Ying [1]. It investigates topological notions defined by means of α-open sets when these are planted into the framework of Ying’s fuzzifying topological spaces (by Łukasiewicz logic in [0,1]) . The concept of α-irresolute functions and α-compactness in the framework of fuzzifying topology are introduced and some of their properties are obtained. We use the finite intersection property to give a characterization of fuzzifying α-compact spaces. Furthermore, we study the image of fuzzifying α-compact spaces under fuzzifying α-continuity and fuzzifying α-irresolute maps.

On Efimov spaces and Radon measures

On the weak and pointwise topologies in function spaces II

In this paper, we study the properties of Q* compact space and generate new results in the space. In particular, we investigate the Q*-compactness of topological spaces with separable, Q*-metrizable, Q*-Hausdorff, homeomorphic, connected and finite intersection properties

The theory of discrete groups acting on finite dimensional Euclidean open balls by hyperbolic isometries was borne around the end of 19th century within the works of Fuchs, Klein and Poincaré. We develop the theory of discrete groups acting by hyperbolic isometries on the open unit ball of an infinite dimensional separable Hilbert space. We present our investigations on the geometry of limit sets at the sphere at infinity with an attempt to highlight the differences between the finite and infinite dimensional theories. We discuss the existence of fixed points of isometries and the classification of isometries. Various notions of discreteness that were equivalent in finite dimensions, no longer turn out to be in our setting. In this regard, the robust notion of strong discreteness is introduced and we study limit sets for properly discontinuous actions. We go on to prove a generalization of the Bishop-Jones formula for strongly discrete groups, equating the Hausdorff dimension of the radial limit set with the Poincaré exponent of the group. We end with a short discussion on conformal measures and their relation with Hausdorff and packing measures on the limit set.

A resolution for the product of a compactum with a polyhedron

A pair of spaces of upper semi-continuous maps and continuous maps