Robin J. Deeley, Karen R. Strung
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.
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In the first part of the paper, we introduce notions of asymptotic continuous orbit equivalence and asymptotic conjugacy in Smale spaces and characterize them in terms of their asymptotic Ruelle algebras with their dual actions. In the second part, we introduce a groupoid $C^*$-algebra which is an extended version of the asymptotic Ruelle algebra from a Smale space and study the extended Ruelle algebras from the view points of Cuntz--Krieger algebras. As a result, the asymptotic Ruelle algebra is realized as a fixed point algebra of the extended Ruelle algebra under certain circle action.
Group actions on a Smale space and the actions induced on the C*-algebras associated to such a dynamical system are studied. We show that an effective action of a discrete group on a mixing Smale space produces a strongly outer action on the homoclinic algebra. We then show that for irreducible Smale spaces, the property of finite Rokhlin dimension passes from the induced action on the homoclinic algbera to the induced actions on the stable and unstable C*-algebras. In each of these cases, we discuss the preservation of properties---such as finite nuclear dimension, Z-stability, and classifica…
The main result of the present paper is that the stable and unstable C*-algebras associated to a mixing Smale space always contain nonzero projections. This gives a positive answer to a question of the first listed author and Karen Strung and has implications for the structure of these algebras in light of the Elliott program for simple C*-algebras. Using our main result, we also show that the homoclinic, stable, and unstable algebras each have real rank zero.
Trace Asymptotics for C*-Algebras from Smale Spaces
我们考虑在双曲动力系统Smale空间中与稳定和不稳定等价相关的C*-代数。这些系统包括有限型的移位，在这种情况下，这些C*-代数都是AF-代数。这些代数在一个单一的希尔伯特空间上有基本的表示（取决于周期点的选择），它有许多特殊的性质。特别地，第一个代数的任何元素与第二个代数的任何元素之间的乘积是紧的。此外，还有一个单一的酉算子，它实现了对两者的操作。在这里，在系统是混合的假设下，我们证明了这些代数上的（半有限）迹可以通过一个限制过程和通常的算子迹得到。
We study asymptotic continuous orbit equivalence of Smale spaces. We prove that two irreducible Smale spaces are flip conjugate if and only if there exists a periodic point preserving homeomorphism giving an asymptotic continuous orbit equivalence between them. We introduce a notion of asymptotic topological conjugacy and asymptotic flip conjugacy in Smale spaces and characterize them in terms of the associated Ruelle algebras with dual actioons. We finally characterize the flip conjugacy classes of irreducible two-sided topological Markov shifts in terms of the associated Ruelle algebras with…
We prove that all K-homology classes of the stable (and unstable) Ruelle algebra of a Smale space have explicit Fredholm module representatives that are finitely summable on the same smooth subalgebra and with the same degree of summability. The smooth subalgebra is induced by a metric on the underlying Smale space groupoid and fine transversality relations between stable and unstable sets. The degree of summability is related to the fractal dimension of the Smale space. Further, the Fredholm modules are obtained by taking Kasparov products with a fundamental class of the Spanier-Whitehead K-d…
Synchronizing Dynamical Systems: their groupoids and $C^*$-algebras
本文提出同步动力系统以推广 Smale 空间，并研究其相关 $C^\ast$-代数，证明其同宿代数含具有类似 Smale 空间同宿代数行为的理想。
We study invariants associated to Smale spaces obtained from an expanding endomorphism on a (closed connected Riemannian) flat manifold. Specifically, the relevant invariants are the $K$-theory of the associated $C^*$-algebras and Putnam's homology theory for Smale spaces. The latter is isomorphic to the groupoid homology of the groupoids used to construct the $C^*$-algebras.
We consider the operator algebra generated by pseudodifferential operators on a closed smooth surface and shift operator induced by a Morse--Smale diffeomorphism of this surface. Elements in this algebra are considered as operators in the scale of Sobolev spaces and the aim of this paper is to describe how Fredholm property of a given operator depends on the Sobolev smoothness exponent $s$.
Spectral triples for hyperbolic dynamical systems
谱三元组被定义为C* 代数与双曲动力系统称为Smale空间。这些光谱三元组之一的光谱维数恢复的Smale空间的拓扑熵。
Homology and K-theory of dynamical systems III. Beyond stably disconnected Smale spaces
本文通过引入空间方向分解，将Smale空间étale群胚的同调不变量研究推广到超越稳定不连通情形，证明Putnam同调与Crainic-Moerdijk群胚同调同构，并给出不稳定C*-代数K群的谱序列计算方法。
A geometric representative for the fundamental class in KK-duality of Smale spaces
该研究针对非交换几何中Smale空间KK-对偶的基本类表示问题，通过推广Bowen马尔可夫划分构造受控利普希茨常数的动力单位分解，结合非周期点采样得到非交换惠特尼嵌入定理类似物，最终构造出表示基本类的θ-可和Fredholm模。
Wieler has shown that every irreducible Smale space with totally disconnected stable sets is a solenoid (i.e., obtained via a stationary inverse limit construction). Using her construction, we show that the associated stable C*-algebra is the stationary inductive limit of a C*-stable Fell algebra that has compact spectrum and trivial Dixmier-Douady invariant. This result applies in particular to Williams solenoids along with other examples. Beyond the structural implications of this inductive limit, one can use this result to in principle compute the K-theory of the stable C*-algebra. A specif…
Wieler solenoids: non-Hausdorff expansiveness, Cuntz-Pimsner models, and functorial properties
本文研究由 Wieler 平稳逆极限诱导的 Fell 代数谱自映射，证明其构成紧致局部 Hausdorff 空间上的扩张满射局部同胚，并据此计算 Smale 空间稳定与不稳定 Ruelle 代数的 K-理论不变量。
We study irreducible Smale spaces with totally disconnected stable sets and their associated $K$-theoretic invariants. Such Smale spaces arise as Wieler solenoids, and we restrict to those arising from open surjections. The paper follows three converging tracks: one dynamical, one operator algebraic and one $K$-theoretic. Using Wieler's Theorem, we characterize the unstable set of a finite set of periodic points as a locally trivial fibre bundle with discrete fibres over a compact space. This characterization gives us the tools to analyze an explicit groupoid Morita equivalence between the gro…
Ring and module structures on dimension groups associated with a shift of finite type
我们研究了与之相关的各种C*-代数的K-理论所得到的有限型移位的不变量。自从Wolfgang Krieger引入这些不变量以来，在过去的30年里，人们对它们进行了深入的研究。它们可以作为单序自由阿尔贝群的归纳极限给予相当具体的描述。移位的有限型的特殊情况下，Smale空间，并在早期的工作中，第二作者已表明，双曲结构的动态在Smale空间诱导自然环和模块结构的某些这些K-群。在这里，我们限制我们的注意力的特殊情况下，移位的有限型，并获得明确的描述方面的归纳极限。
KK-duality for self-similar groupoid actions on graphs
本文将Nekrashevych的KK-对偶从自相似群作用推广到图上自相似群胚作用，去除递归条件并将有限字母表推广为有限图，证明相关C*-代数与Smale空间的Ruelle代数强Morita等价，从而得到Spanier-Whitehead对偶。
We introduce a graph theoretic property called Condition (N) for finitely separated graphs and prove that it is equivalent to both nuclearity and exactness of the associated universal tame graph C*-algebra.
This paper is a continuation of the paper entitled "Subshifts, $λ$-graph bisystems and $C^*$-algebras", arXiv:1904.06464. A $λ$-graph bisystem consists of a pair of two labeled Bratteli diagrams satisfying certain compatibility condition on their edge labeling. For any two-sided subshift $Λ$, there exists a $λ$-graph bisystem satisfying a special property called FPCC. We will construct an AF-algebra ${\mathcal{F}}_{\frak L}$ with shift automorphism $ρ_{\frak L}$ from a $λ$-graph bisystem $({\frak L}^-,{\frak L}^+)$, and define a $C^*$-algebra ${\mathcal R}_{\frak L}$ by the crossed product ${\…
Synchronizing Dynamical Systems: Finitely presented systems and Ruelle algebras
本文确定有限表示系统关联的C*-代数结构，并发展一般同步系统的Ruelle代数基本理论，证明有限表示系统的Ruelle代数与Smale空间情形显式相关。
We provide examples of ambient nuclear C*-algebras of non-nuclear C*-algebras with no proper intermediate C*-algebras. In particular this gives the first examples of minimal ambient nuclear C*-algebras of non-nuclear C*-algebras. For this purpose, we study generic Cantor systems of infinite free product groups.
We show that for any locally compact Hausdorff space $Y$ with finite covering dimension and for any continuous flow $\mathbb{R} \curvearrowright Y$, the resulting crossed product $C^*$-algebra $C_0(Y) \rtimes \mathbb{R}$ has finite nuclear dimension. This generalizes previous results for free flows, where this was proved using Rokhlin dimension techniques. As an application, we obtain bounds for the nuclear dimension of $C^*$-algebras associated to one-dimensional orientable foliations. This result is analogous to the one we obtained earlier for non-free actions of $\mathbb{Z}$. Some novel tec…
For every one-sided shift space $X$ over a finite alphabet, left special elements are those points in $X$ having at least two preimages under the shift operation. In this paper, we show that the Cuntz-Pimsner $C^*$-algebra $\mathcal{O}_X$ has nuclear dimension 1 when $X$ is minimal and the number of left special elements in $X$ is finite. This is done by describing thoroughly the cover of $X$ which also recovers an exact sequence, discovered before by T. Carlsen and S. Eilers.
Under suitable conditions, a substitution tiling gives rise to a Smale space, from which three equivalence relations can be constructed, namely the stable, unstable, and asymptotic equivalence relations. We denote with $S$, $U$, and $A$ their corresponding $C^*$-algebras in the sense of Renault. In this article we show that the $K$-theories of $S$ and $U$ can be computed from the cohomology and homology of a single cochain complex with connecting maps for tilings of the line and of the plane. Moreover, we provide formulas to compute the $K$-theory for these three $C^*$-algebras. Furthermore, w…
Nuclear dimension of subhomogeneous twisted groupoid C*-algebras and dynamic asymptotic dimension
本文刻画了扭曲 étale 群胚 C*-代数的次齐次性并获得其核维数上界，移除了相关结果中的主性假设，并证明了无限二面体群极小作用的动态渐近维数始终为 1。
The multiplier algebra of a nuclear quasidiagonal C*-algebra
拟对角性在很多地方都很有意义--尤其是在单有单位可分核C* 代数的分类程序中。本文给出了单单位可分C* 代数的核性和拟对角性的两个刻画。我们的第一个特征是Dadarlat的精确拟对角C* 代数的特征的核模拟。我们的第二个characterizatiion是“双”的有趣的（和重要的）波帕财产首先研究波帕。
In this paper, we show that a completely positive linear map is weakly nuclear if and only if its complexification is weakly nuclear. It is shown that a real $C^*$-algebra is exact if and only if its complexification is exact and similar case is provided for the quasidiaginality.
We prove that Z-stable, simple, separable, nuclear, non-unital C*-algebras have nuclear dimension at most 1. This completes the equivalence between finite nuclear dimension and Z-stability for simple, separable, nuclear, non-elementary C*-algebras.
We give a framework to produce C*-algebra inclusions with extreme properties. This gives the first constructive nuclear minimal ambient C*-algebras. We further obtain a purely infinite analogue of Dadarlat's modeling theorem on AF-algebras: Every Kirchberg algebra is rigidly and KK-equivalently sandwiched by non-nuclear C*-algebras without intermediate C*-algebras. Finally we reveal a novel property of Kirchberg algebras: They embed into arbitrarily wild C*-algebras as rigid maximal C*-subalgebras.
We obtain an improved upper bound for the nuclear dimension of extensions of $\mathcal{O}_\infty$-stable $\rm{C}^*$-algebras. In particular, we prove that the nuclear dimension of a full extension of an $\mathcal{O}_\infty$-stable $\rm{C}^*$-algebra by a stable AF algebra is one.
We give two characterizations of tracially nuclear C*-algebras. The first is that the finite summand of the second dual is hyperfinite. The second is in terms of a variant of the weak* uniqueness property. The necessary condition holds for all tracially nuclear C*-algebras. When the algebra is separable, we prove the sufficiency.
We exhibit examples of simple separable nuclear C*-algebras, along with actions of the circle group and outer actions of the integers, which are not equivariantly isomorphic to their opposite algebras. In fact, the fixed point subalgebras are not isomorphic to their opposites. The C*-algebras we exhibit are well behaved from the perspective of structure and classification of nuclear C*-algebras: they are unital C*-algebras in the UCT class, with finite nuclear dimension. One is an AH-algebra with unique tracial state and absorbs the CAR algebra tensorially. The other is a Kirchberg algebra.
Strict comparison and Z-absorption of nuclear C*-algebras
对于任意具有n个极值迹的单可分无限维核C* 代数-代数，证明了Z-吸收、严格比较和性质（SI）是等价的.我们还证明了迹秩为零的单位可分单核C* 代数-代数是近似可分的，因而是Z-吸收的.
We show that every nuclear $\mathcal O_\infty$-stable *-homomorphism with a separable exact domain has nuclear dimension at most 1. In particular separable, nuclear, $\mathcal O_\infty$-stable C*-algebras have nuclear dimension 1. We also characterise when $\mathcal O_\infty$-stable C*-algebras have finite decomposition rank in terms of quasidiagonality and primitive-ideal structure, and determine when full $\mathcal O_2$-stable *-homomorphisms have nuclear dimension 0.
On the nuclear dimension of certain UCT-Kirchberg algebras
本文部分回答了W.温特和J·撒迦利亚证明了具有无挠的$K_1$ -群的UCT-类中的每个Kirchberg代数的核维数为1。
The class of simple separable KK-contractible (KK-equivalent to $\{0\}$) C*-algebras which have finite nuclear dimension is shown to classified by the Elliott invariant. In particular, the class of C*-algebras $A\otimes \mathcal W$ is classifiable, where $A$ is a simple separable C*-algebra with finite nuclear dimension and $\mathcal W$ is the simple inductive limit of Razak algebras with unique trace, which is bounded.
Decomposable approximations of nuclear C*-algebras
我们证明了核C* 代数-代数具有一个改进的完全正逼近性质，其中通过有限维代数近似分解的映射是零阶映射的凸组合.我们利用这一点证明了紧包含在C* 代数-代数B中的可分核C* 代数-代数A嵌入到B中。
Nuclear C*-algebras: 99 problems
本文探讨核C*-代数的结构与分类，提出99个问题以推动相关研究。
On ${\OL}_{\infty}$ structure of nuclear $C^*$-algebras
研究了核$C^*$ -代数的局部算子空间结构。证明了$C^*$ -代数是核的当且仅当它对于某些（实际上对于每个）$\la > 6$是$\OL_{\infty, \la}$空间. $\OL_\infty$常数$λ$为核$C^*$ -代数提供了一个有趣的不变量\[ \OL_\infty (\A) = \inf\{\la: ~ \A ~{\rm is ~ an} ~ {\OL}_{\infty, \la} ~ {\rm space}\} \].如果$\A$是一个核$C^*$ -代数，那么我们有$1\le \OL_\infty (\A) \le 6$，如果$\A$是一个有$\OL_{\infty} (\A) \le (\frac {1+{\sqrt 5}}2)^{\frac 12}$的酉核$C^*$ -代数，我们证明了$\A$一定是稳定有限的.我们还研究了$C^*$ -代数上的刚性$\OL_{\infty, 1^+}$结构和刚性完全序$\OL_{\infty, 1^+}$结构之间的联系，其中后者的结构已经由Blackadar和Kirchberg在强NF $C^*$ -代数的刻画中研究过.本文的另一个主要结果是证明了在单位核$C^*$ -代数上这两个局部结构实际上是等价的。我们通过证明如果一个单位（核）$C^*$ -代数是刚性${\OL}_{\infty, 1^+}$空间，则它是内拟对角的，…
We prove the title by constructing 2-colourable completely positive approximations for the Toeplitz algebra. Besides results about nuclear dimension and completely positive contractive order zero maps, our argument involves projectivity of the cone over a finite dimensional C*-algebra and Lin's theorem on almost normal matrices.
Nuclear dimension of extensions of commutative C*-algebras by Kirchberg algebras
本文研究含交换单位商和稳定 Kirchberg 理想的 C*-代数扩张，给出其核维数，并确定适用该结论的有限有向图。
Subtriviality of continuous fields of nuclear C*--algebras
本文推广了Kirchberg给出的可分核\cst -代数的特征，证明了给定紧致可度量化空间X上的核C* 代数-代数A的一个有单位的可分连续域，C（X）-代数A同构于平凡连续域C（X; O_2）的一个有单位的C（X）-子代数，C（X;O_2）的模一投影象.
On Nuclearity of $C^*$-algebras of Fell bundles over étale groupoids
本文证明了：如果E是可转移的局部紧Hausdorff广群上的Fell丛，使得单位空间上的每个纤维都是核的，则$C?^*r(E)$也是核的.为了证明这个结果，我们引入了Fell丛与固定$C^*$ -代数的（极小）张量积.
The class of simple separable KK-contractible (KK-equivalent to $\{0\}$) C*-algebras which have finite nuclear dimension is shown to be classified by the Elliott invariant. In particular, the class of C*-algebras $A\otimes \mathcal W$ is classifiable, where $A$ is a simple separable C*-algebra with finite nuclear dimension and $\mathcal W$ is the simple inductive limit of Razak algebras with unique trace, which is bounded.
We compute the nuclear dimension of separable, simple, unital, nuclear, Z-stable C*-algebras. This makes classification accessible from Z-stability and in particular brings large classes of C*-algebras associated to free and minimal actions of amenable groups on finite dimensional spaces within the scope of the Elliott classification programme.
We prove that faithful traces on separable and nuclear C*-algebras in the UCT class are quasidiagonal. This has a number of consequences. Firstly, by results of many hands, the classification of unital, separable, simple and nuclear C*-algebras of finite nuclear dimension which satisfy the UCT is now complete. Secondly, our result links the finite to the general version of the Toms-Winter conjecture in the expected way and hence clarifies the relation between decomposition rank and nuclear dimension. Finally, we confirm the Rosenberg conjecture: discrete, amenable groups have quasidiagonal C*-…
We review the notion of nuclear dimension for C*-algebras introduced by Winter and Zacharias. We explain why it is a non-commutative version of topological dimension. After presenting several examples, we give a brief overview of the state of the art.
Homogeneity of the pure state space for the separable nuclear C*-algebras
证明了所有可分单核C* 代数的纯状态空间在渐近内自同构群的作用下是齐次的。如果对C* 代数不假设简单性，则对于上述作用，其GNS表示忠实的纯态集合是齐次的。
Nuclear dimension and Z-stability of non-simple C*-algebras
我们研究了非简单C* 代数的以下正则性性质之间的相互作用：有限核维度、Z稳定性和Cuntz半群中的代数正则性。我们表明，有限的核维数意味着代数正则性的Cuntz半群，提供已知的I型障碍是避免的。我们演示了如何有限的核维数可以用来研究的结构的中心序列代数，通过因式分解的身份映射的中心序列代数，在类似的因式分解中产生的定义核维数的方式。利用中心序列代数的有关结果，证明了对于充分非I型可分C* 代数，有限核维数蕴涵Z-稳定性的猜想.证明了C* 代数没有纯无限子代数，其本原理想空间有紧开集基; C* 代数没有纯无限子代数，其本原理想空间是Hausdorff空间.特别地，这涵盖了具有有限分解秩和实秩为零的C* 代数-代数。我们的结果对于具有局部有限核维数的（M，N）-纯（Cuntz半群的一个正则性条件）C* 代数-代数更一般地成立.
We show that separable, nuclear and strongly purely infinite C*-algebras have finite nuclear dimension. In fact, the value is at most three. This exploits a deep structural result of Kirchberg and Rørdam on strongly purely infinite C*-algebras that are homotopic to zero in an ideal-system preserving way.
Topological Free Entropy Dimensions in Nuclear C$^*$-algebras and in Full Free Products of C$^*$-algebras
本文引入了单位C $^*$代数中元素元组$n$的拓扑轨道维数的概念。利用这一概念，我们得到核C $^*$代数的任意一族自伴生成元的Voiculescu拓扑自由熵维数小于或等于1.我们还证明了单位C $^*$代数的全自由积的拓扑自由熵维数是可加的。在附录中，我们证明了Blackadar和Kirchberg的单位MF代数的单位全自由积也是MF代数。
Nuclearity of semigroup C*-algebras and the connection to amenability
研究了与群的子半群相关联的C* 代数。对于一类包含拟格序群中的正锥的半群和左Ore半群，我们将相应的半群C* 代数-代数刻画为逆半群的C* 代数-代数、群胚C* 代数-代数和群交叉积的全角.这些描述使我们能够用左正则表示的忠实性和群作用的顺从性来刻画半群C* 代数的核性。此外，我们还确定了半群C* 代数的边界元何时是UCT Kirchberg代数.这导致了Cuntz代数和环C* 代数的统一方法。
I give an overview of recent developments in the structure and classification theory of separable, simple, nuclear C*-algebras. I will in particular focus on the role of quasidiagonality and amenability for classification, and on the regularity conjecture and its interplay with internal and external approximation properties.
Trace spaces of simple nuclear C*-algebras with finite-dimensional extreme boundary
设A是至少具有一个迹态的有单位可分的单无限维核C * 代数-代数.证明了：若A的迹空间具有紧的有限维极边界，则存在矩阵代数到A的某个中心序列代数的酉嵌入，该中心序列代数由A的迹空间上的一致拓扑所确定.作为应用，证明了若A有严格比较，则A张量吸收Jiang-Su代数。
Consider a graph C*-algebra C*(E) with a purely infinite ideal I (possibly all of C*(E)) such that I has only finitely many ideals and C*(E)/I is approximately finite dimensional. We prove that the nuclear dimension of C*(E) is 1. If I has infinitely many ideals, then the nuclear dimension of C*(E) is either 1 or 2.
Perturbations of nuclear C*-algebras
Kadison和Kastler在可分希尔伯特空间上的所有有界算子的C*-子代数的集合上引入了一个自然度量。他们证明了充分闭代数是酉共轭的。当一个代数是可分的且有核的时，我们建立了这个猜想。我们也考虑这些概念的片面版本，我们从涉及可分核C* 代数的某些近包含中得到嵌入。在本文的最后，我们展示了我们的方法如何导致改进的一些类型的代数，目前感兴趣的分类程序的特征。
Nuclear dimension for an inclusion of unital C*-algebras
设$P \subset A$是具有有限Watatani指标的可分单位C* 代数-代数的包含.假设$E \colon A \rightarrow P$具有Rokhlin性质，即存在投影$e \in A' \cap A^\infty$使得$E^\infty(e) = ({\rm Index}E)^{-1}1$。证明了如果$A$的核维数为$n$，则$P$的核维数小于或等于$n$ .特别地，如果有限群$G$在$A$上的作用$α$具有Rokhlin性质，则交叉积代数$A \rtimes_αG$的核维数小于或等于$A$的核维数。
The nuclear dimension of extensions of commutative C*-algebras by the compact operators
本文将 Brake 与 Winter 的 Toeplitz 代数结果推广到紧算子对 C(X) 的 C*-代数扩张，证明其核维数恰为空间 X 的维数。
We study Z-actions on unital simple separable stably finite C*-algebras of finite nuclear dimension. Assuming that the extreme boundary of the trace space is compact and finite dimensional, and that the induced action on the trace space is trivial, we show that strongly outer Z-actions have finite Rokhlin dimension in the sense of Hirshberg, Winter and Zacharias.
We prove that every Kirchberg algebra in the UCT class has nuclear dimension 1. We first show that Kirchberg 2-graph algebras with trivial $K_0$ and finite $K_1$ have nuclear dimension 1 by adapting a technique developed by Winter and Zacharias for Cuntz algebras. We then prove that every Kirchberg algebra in the UCT class is a direct limit of 2-graph algebras to obtain our main theorem.
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