The C*-algebra of a minimal homeomorphism with finite mean dimension has finite radius of comparison

The Stable Rank of Diagonal ASH Algebras and Crossed Products by Minimal Homeomorphisms

We introduce a subclass of recursive subhomogeneous algebras, in which each of the pullback maps is diagonal in a suitable sense. We define the notion of a diagonal map between two such algebras and show that every simple inductive limit of these algebras with diagonal bonding maps has stable rank one. As an application, we prove that for any infinite compact metric space $T$ and minimal homeomorphism $h\colon T\to T$, the associated dynamical crossed product $\mathrm{C^*}(\mathbb{Z},T,h)$ has stable rank one. This affirms a conjecture of Archey, Niu, and Phillips. We also show that the Toms-W…

Minimal Dynamics and K-theoretic Rigidity: Elliott's Conjecture

Let X be an infinite, compact, metrizable space of finite covering dimension and h a minimal homeomorphism of X. We prove that the crossed product of C(X) by h absorbs the Jiang-Su algebra tensorially and has finite nuclear dimension. As a consequence, these algebras are determined up to isomorphism by their graded ordered K-theory under the necessary condition that their projections separate traces. This result applies, in particular, to those crossed products arising from uniquely ergodic homeomorphisms.

Crossed products by minimal homeomorphisms

Let X be an infinite compact metric space with finite covering dimension and let h be a minimal homeomorphism of X. Let A be the associated crossed product C*-algebra. We show that A has tracial rank zero whenever the image of K_0 (A) in the affine functions on the tracial state space of A is dense. As a consequence, we show that these crossed product C*-algebras are in fact simple AH algebras with real rank zero. When X is connected and h is further assumed to be uniquely ergodic, then the above happens if and only if the rotation number associated to h has irrational values. By applying the …

$\mathrm{C}^*$-algebras associated to homeomorphisms twisted by vector bundles over finite dimensional spaces

In this paper we study Cuntz--Pimsner algebras associated to $\mathrm{C}^*$-correspondences over commutative $\mathrm{C}^*$-algebras from the point of view of the $\mathrm{C}^*$-algebra classification programme. We show that when the correspondence comes from an aperiodic homeomorphism of a finite-dimensional infinite compact metric space $X$ twisted by a vector bundle, the resulting Cuntz--Pimsner algebras have finite nuclear dimension. When the homeomorphism is minimal, this entails classification of these $\mathrm{C}^*$-algebras by the Elliott invariant. This establishes a dichotomy: when t…

Permanence of stable rank one for centrally large subalgebras and crossed products by minimal homeomorphisms

We define centrally large subalgebras of simple unital C*-algebras, strengthening the definition of large subalgebras in previous work. We prove that if A is any infinite dimensional simple separable unital C*-algebra which contains a centrally large subalgebra with stable rank one, then A has stable rank one. We also prove that large subalgebras of crossed product type are automatically centrally large. We use these results to prove that if X is a compact metric space which has a surjective continuous map to the Cantor set, and h is a minimal homeomorphism of X, then C* (Z, X, h) has stable r…

局部同胚的动力学比较

Dynamical comparison for local homeomorphisms

本文证明了紧可度量化空间上极小满射非单射局部同胚对应的Deaconu–Renault群胚的动力学比较性质，由此得到相关C*-代数的性质、验证了Matui AH猜想，还证明了极小群作用的经典小边界性质。

Minimal dynamical systems on connected odd dimensional spaces

Let $β: S^{2n+1}\to S^{2n+1}$ be a minimal homeomorphism ($n\ge 1$). We show that the crossed product $C(S^{2n+1})\rtimes_β \Z$ has rational tracial rank at most one. More generally, let $Ω$ be a connected compact metric space with finite covering dimension and with $H^1(Ω, \Z)=\{0\}.$ Suppose that $K_i(C(Ω))=\Z\oplus G_i$ for some finite abelian group $G_i,$ $i=0,1.$ Let $β: Ω\toΩ$ be a minimal homeomorphism. We also show that $A=C(Ω)\rtimes_β\Z$ has rational tracial rank at most one and is classifiable. In particular, this applies to the minimal dynamical systems on odd dimensional real proj…

Minimal dynamical systems on the product of the Cantor set and the circle II

Let $X$ be the Cantor set and $ϕ$ be a minimal homeomorphism on $X\times\T$. We show that the crossed product $C^*$-algebra $C^*(X\times\T,ϕ)$ is a simple $A\T$-algebra provided that the associated cocycle takes its values in rotations on $\T$. Given two minimal systems $(X\times\T,ϕ)$ and $(Y\times\T,ψ)$ such that $ϕ$ and $ψ$ arise from cocycles with values in isometric homeomorphisms on $\T$, we show that two systems are approximately $K$-conjugate when they have the same $K$-theoretical information.

Constructions in minimal amenable dynamics and applications to the classification of $\mathrm{C}^*$-algebras

We study the existence of minimal dynamical systems, their orbit and minimal orbit-breaking equivalence relations, and their applications to C*-algebras and K-theory. We show that given any finite CW-complex there exists a space with the same K-theory and cohomology that admits a minimal homeomorphism. The proof relies on the existence of homeomorphisms on point-like spaces constructed by the authors in previous work, together with existence results for skew product systems due to Glasner and Weiss. To any minimal dynamical system one can associate minimal equivalence relations by breaking orb…

Crossed products and minimal dynamical systems

Let $X$ be an infinite compact metric space with finite covering dimension and let $α, β: X\to X$ be two minimal homeomorphisms. We prove that the crossed product $C^*$-algebras $C(X)\rtimes_α\Z$ and $C(X)\rtimes_\belta\Z$ are isomorphic if and only if they have isomorphic Elliott invariant. In a more general setting, we show that if $X$ is an infinite compact metric space and if $α: X\to X$ is a minimal homeomorphism such that $(X, α)$ has mean dimension zero, then the tensor product of the crossed product with a UHF-algebra of infinite type has generalized tracial rank at most one. This impl…

Local trivializations of suspended minimal Cantor systems and the stable orbit-breaking subalgebra

It is introduced an analogue of the orbit-breaking subalgebra for the case of free flows on locally compact metric spaces, which has a natural approximate structure in terms of a fixed point and any nested sequence of central slices around this point. It is shown that in the case of minimal flows admitting a compact Cantor central slice, the resulting $C^*$-algebra is the stabilization of the Putnam orbit-breaking subalgebra associated to the induced homeomorphism on the central slice.

C*-algebras of a Cantor system with finitely many minimal subsets: structures, K-theories, and the index map

We study homeomorphisms of a Cantor set with $k$ ($k < +\infty$) minimal invariant closed (but not open) subsets; we also study crossed product C*-algebras associated to these Cantor systems and their certain orbit-cut sub-C*-algebras. In the case that $k\geq 2$, the crossed product C*-algebra is stably finite, has stable rank 2, and has real rank zero if in addition $(X, σ)$ is aperiodic. The image of the index map is connected to certain directed graphs arising from the Bratteli-Vershik-Kakutani model of the Cantor system. Using this, it is shown that the ideal of the Bratteli diagram (of th…

三元算子环的最小张量积的理想结构

The ideal structure of the minimal Tensor Product of Ternary Rings of Operators

研究三元算子环与C*-代数的算子空间内射张量积的理想空间结构，通过两个映射刻画其原理想与素理想，证明在核-核拓扑下连续并诱导同胚，并给出最小素理想空间的乘积分解。

AT-algebras from fiberwise essentially minimal zero-dimensional dynamical systems

We introduce a type of zero-dimensional dynamical system (a pair consisting of a totally disconnected compact metrizable space along with a homeomorphism of that space), which we call "fiberwise essentially minimal", and we prove that the associated crossed product $ C^* $-algebra of such a system is an AT-algebra. Under the additional assumption that the system has no periodic points, we prove that the associated crossed product $ C^* $-algebra has real rank zero, which tells us that such $ C^* $-algebras are classifiable by $ K $-theory. We show that the definition of fiberwise essentially m…

The space of ideals in the minimal tensor product of $C^*$-algebras

For $C^*$-algebras $A_1, A_2$ the map $(I_1,I_2)\to ker(q_{I_1}\otimes q_{I_2})$ from $Id^{\prime}(A_1)\times Id^{\prime}(A_2)$ into $Id^{\prime}(A_1\otimes_{\mathrm{min}} A_2) is a homeomorphism onto its image which is dense in the range. Here, for a $C^*$-algebra $A$, the space of all proper closed two sided ideals endowed with an adequate topology is denoted $Id^{\prime}(A)$ and $q_I$ is the quotient map of $A$ onto $A/I$. New proofs of the equivalence of the property (F) of Tomiyama for $A_1\otimes_{\mathrm{min}} A_2$ with certain other properties are presented.