George A. Elliott, Guihua Gong, Huaxin Lin, Zhuang Niu
We prove that every unital stably finite simple amenable $C^*$-algebra $A$ with finite nuclear dimension and with UCT such that every trace is quasi-diagonal has the property that $A\otimes Q$ has generalized tracial rank at most one, where $Q$ is the universal UHF-algebra. Consequently, $A$ is classifiable in the sense of Elliott.
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A classification theorem is obtained for a class of unital simple separable amenable Z-stable C*-algebras which exhausts all possible values of the Elliott invariant for unital stably finite simple separable amenable Z-stable C*-algebras. Moreover, it contains all unital simple separable amenable C*-algebras which satisfy the UCT and have finite rational tracial rank
Let $A$ be a unital simple separable C*-algebra satisfying the UCT. Assume that $\mathrm{dr}(A)<+\infty$, $A$ is Jiang-Su stable, and $\mathrm{K}_0(A)\otimes \mathbb{Q}\cong \mathbb{Q}$. Then $A$ is an ASH algebra (indeed, $A$ is a rationally AH algebra).
A class of $C^*$-algebras, to be called those of generalized tracial rank one, is introduced, and classified by the Elliott invariant. A second class of unital simple separable amenable $C^*$-algebras, those whose tensor products with UHF-algebras of infinite type are in the first class, to be referred to as those of rational generalized tracial rank one, is proved to exhaust all possible values of the Elliott invariant for unital finite simple separable amenable ${\cal Z}$-stable $C^*$-algebras. An isomorphism theorem for a special sub-class of those $C^*$-algebras are presented. This provide…
We introduce a notion of amenable normal extension S of a unital ring R with a finite approximation system F, encompassing the amenable algebras over a field of Gromov and Elek, the twisted crossed product by an amenable group, and the tensor product with a field extension. It is shown that every Sylvester matrix rank function rk of R preserved by S has a canonical extension to a Sylvester matrix rank function rk_F for S. In the case of twisted crossed product by an amenable group, and the tensor product with a field extension, it is also shown that rk_F depends on rk continuously. When an ame…
When amenable groups have real rank zero $C^*$-algebras
该研究探讨离散顺从群具有实秩零$C^*$-代数的条件，在局部有限群对应实秩零代数的已知结论基础上，证明了实秩零条件下初等顺从且有限Hirsch长度的正规子群必为局部有限群，推进了相关开放问题的研究。
C*-algebras of Essentially Tracially Rank One
本文引入单C*-代数的本质迹秩至多为一的概念，刻画了可分单有限顺从𝒵稳定C*-代数，完成了长期研究项目，还证明了相关拟对角性及分解秩的结论，且不依赖UCT。
We present a classification theorem for a class of unital simple separable amenable ${\cal Z}$-stable $C^*$-algebras by the Elliott invariant. This class of simple $C^*$-algebras exhausts all possible Elliott invariant for unital stably finite simple separable amenable ${\cal Z}$-stable $C^*$-algebras. Moreover, it contains all unital simple separable amenable $C^*$-alegbras which satisfy the UCT and have finite rational tracial rank.
We consider the amenability of groupoids $G$ equipped with a group valued cocycle $c:G\to Q$ with amenable kernel $c^{-1}(e)$. We prove a general result which implies, in particular, that $G$ is amenable whenever $Q$ is amenable and if there is countable set $D\subset G$ such that $c(G^{u})D=Q$ for all $u\in G^{(0)}$. We show that our result is applicable to groupoids arising from partial semigroup actions. We explore these actions in detail and show that these groupoids include those arising from directed graphs, higher rank graphs and even topological higher rank graphs. We believe our metho…
Classification of actions of discrete amenable groups on strongly amenable subfactors of type III$λ$
利用连续分解，我们分类强自由行动的离散顺从集团上的强顺从子因子的III型$_λ, 0 < λ< 1$。温斯洛的基本同态是一个完全不变量。这消除了Loi和Winsløw分类定理中的额外假设，并给出了上循环共轭的完整分类。
An induction proof of strong NF\ for amenable group $C^*$-algebras
本文通过表示论与迹分解方法，归纳证明了可数离散 amenable 群的全群 $C^*$-代数为 strong NF，并提供了直接变体及两证明的共有步骤。
We demonstrate von Neumann algebra arising from an icc group $Γ$ in Chifan's, Ioana's, and Kida's class of poly-$\mathcal{C}_\text{rss} $, such as a poly-hyperbolic group with no amenable factors in its composition series, satisfies the following rigidity phenomenon discovered in DHI16 (see also CdSS17): every tensor decomposition of the II$_1$ factor $L(Γ) $ must arise from direct product decomposition of $Γ$ by groups which are poly-$ \mathcal{C}_\text{rss}$. Through heavy usage and developments of the techniques in CdSS15, we improve the second author's and their collaborator's work in CKP1…
On the non-commutative Neveu decomposition and stochastic ergodic theorems
该研究证明了半有限及σ-有限von Neumann代数上顺从半群/群作用的Neveu分解，并得到非交换L¹空间上正压缩的随机遍历定理，填补了Danford-Schwartz范畴外的相关空白。
Following Elliott's earlier work, we show that the Elliott invariant of any finite separable simple $C^*$-algebra with finite nuclear dimension can always be described as a scaled simple ordered group pairing together with a countable abelian group which unifies the unital and nonunital, as well as stably projectionless cases. We also show that, for any given such invariant set, there is a finite separable simple $C^*$-algebra, whose Elliott invariant is the given set, a refinement of the range theorem of Elliott in the stable case. In the stably projectionless case, modified model $C^*$-algeb…
Non-Commutative Wiener-Wintner theorem for amenable group actions
本文针对有限von Neumann代数上保迹自同构的可数群作用，建立了Jacobs-de Leeuw-Glicksberg分解，并证明了非交换Wiener-Wintner定理。
We present a stable uniqueness theorem for non-unital C*-algebras. Generalized tracial rank one is defined for stably projectionless simple C*-algebras. Let $A$ and $B$ be two stably projectionless separable simple amenable C*-algebras with $gTR(A)\le 1$and $gTR(B)\le 1.$ Suppose also that $KK(A, D)=KK(B,D)=\{0\}$ for all C*-algebras $D.$ Then $A\cong B$ if and only if they have the same tracial cones with scales. We also show that every separable simple C*-algebra, $A$ with finite nuclear dimension which satisfies the UCT with non-zero traces must have $gTR(A)\le 1$ if $K_0(A)$ is torsion. In…
Operator amenability of Fourier-Stieltjes algebras
算子顺从性的概念是由Z. J. Ruan，1995年。他表明，一个局部紧群G是顺从的当且仅当其傅立叶代数A（G）是经营者顺从。本文研究了Fourier-Stieltjes代数B（G）和约化Fourier-Stieltjes代数B_r（G）的算子顺从性。自然的猜想是，这些代数中的任何一个都是算子顺从的当且仅当G是紧的。我们部分地证明了这个猜想与纯粹的运营商顺从性取代运营商C-顺从性，某些常数C < 5。在这个过程中，我们得到了B（G）的一个新的分解，它可以解释为M（G）分解为离散测度和连续测度的非交换对应。我们进一步引入了算子顺从性的一种变体--算子Connes顺从性，它也考虑了B（G）和B_r（G）上的对偶空间结构.证明了B_r（G）是Connes-affected的当且仅当G是affected的.令人惊讶的是，B（F_2）是算子Connes顺从的，尽管两个生成元中的自由群F_2不顺从.
A Unique Decomposition Result for HT Factors with Torsion Free Core
我们证明了II $_1$因子$M$有一个唯一的（直到酉共轭）围绕"核心子因子“$N \subset M$满足性质HT和一定的”扭自由“条件的交叉积型分解。特别地，这表明，对于无挠的、不顺从子群$Γ\subset SL(2, \Bbb Z)$和$α= e^{2πi t}$，$t \not\in \Bbb Q$，形式为$L_α(\Bbb Z^2) \rtimes Γ$的因子的同构意味着相应群$Γ$的同构。
In this paper, we consider the real rank zero $\mathrm{C}^*$-algebras which can be written as an inductive limit of the Elliott-Thomsen building blocks and prove a decomposition result for the connecting homomorphisms; this technique will be used in the classification theorem.
Regularity properties in the classification program for separable amenable C*-algebras
本文报道了可分顺从C* 代数分类的最新进展。我们的重点是新的明显的作用，如有限的分解秩，严格比较的积极因素，Z-稳定性的规律性，并在Cuntz半群的重要性。我们包括一个简短的历史，该计划的成功，自1989年以来，更详细地看看Villadsen型代数，从而大大改变了景观，并收集公告的结构和性质的Cuntz半群。
It is shown that every Jiang-Su stable approximately subhomogeneous C*-algebra has finite decomposition rank. Previously, it was not even known that such algebras have finite nuclear dimension. A key step in the proof is that subhomogeneous C*-algebra are locally approximated by a certain class of more tractable subhomogeneous algebras, namely, a non-commutative generalization of the class of cell complexes. The result is applied to show that Jiang-Su stable minimal Z-crossed products have finite decomposition rank.
Decomposition rank of UHF-absorbing C*-algebras
设A是具有唯一迹态的有单位可分单C* 代数-代数.证明了若A是核拟对角的，则A在泛UHF-代数张量下的分解秩至多为1。然后证明了A是核的、拟对角的和有严格比较的当且仅当A有有限分解秩。对于这样的A，我们也给出了一个直接的证明，证明了张量为UHF-代数的A的迹秩为零。应用这一特征，我们得到了一个反例的Powers-Sakai猜想。
Rank conditions and amenability for rings associated to graphs
研究有向图对应的路径环、Cohn路径环和Leavitt路径环，给出当系数环R满足秩条件或强秩条件时，图需满足的充要条件，并应用于刻画可均路径代数的图。
Quantitative decomposition and approximation for quasi-local operators
本文提出定量方法，将拟局部算子分解为块秩一算子并给出有限传播逼近，从而定量证明Roe与拟局部代数的有界块秩刚性结果。
We construct two types of unital separable simple $C^*$-alebras $A_z^{C_1}$ and $A_z^{C_2},$ one is exact but not amenable, and the other is non-exact. Both have the same Elliott invariant as the Jiang-Su algebra, namely, $A_z^{C_i}$ has a unique tracial state, $$(K_0(A_z^{C_i}), K_0(A_z^{C_i})_+, [1_{A_z^{C_i}} ])=(\mathbb Z, \mathbb Z_+,1)$$ and $K_{1}(A_z^{C_i})=\{0\}$ ($i=1,2$). We show that $A_z^{C_i}$ ($i=1,2$) is essentially tracially in the class of separable ${\cal Z}$-stable $C^*$-alebras of nuclear dimension 1. $A_z^{C_i}$ has stable rank one, strict comparison for positive elements…
Uniqueness theorems for topological higher-rank graph C*-algebras
本文研究了拓扑高秩图的边界路群胚。我们证明了所有这样的群胚都是拓扑顺从的。我们证明了拓扑高秩图的C* 代数-代数是核的，并证明了规范不变唯一性定理和Cuntz-Krieger唯一性定理.然后给出了拓扑高秩图C* 代数简单的充要条件，以及它是纯无限的条件.
We study a class of stably projectionless simple C*-algebras which may be viewed as having generalized tracial rank one in analogy with the unital case. Some structural question concerning these simple C*-algebras are studied. The paper also serves as a technical support for the classification of separable stably projectionless simple amenable Jiang-Su stable C*-algebras.
We introduce an invariant, called mean rank, for any module M of the integral group ring of a discrete amenable group $Γ$, as an analogue of the rank of an abelian group. It is shown that the mean dimension of the induced $Γ$-action on the Pontryagin dual of M, the mean rank of M, and the von Neumann-Lück rank of M all coincide. As applications, we establish an addition formula for mean dimension of algebraic actions, prove the analogue of the Pontryagin-Schnirelmnn theorem for algebraic actions, and show that for elementary amenable groups with an upper bound on the orders of finite subgroups…
Asymptotically Unitary Equivalence and Classification of Simple Amenable C*-algebras
设$C$和$A$是两个迹秩不大于1的酉可分顺从单C* 代数-代数.设$C$满足泛系数定理，并设$ϕ_1, ϕ_2: C\to A$是两个单位单态.我们证明了存在$A$的酉$\{u_t: t\in [0, \infty)\}$的连续路径使得$$ \lim_{t\to\infty}u_t^*ϕ_1(c)u_t=ϕ_2(c)\tforal c\in C $$当且仅当$KK(C,A),$ $ϕ_1^‡=ϕ_2^‡,$ $(ϕ_1)_T=(ϕ_2)_T$中的$[ϕ_1]=[ϕ_2]$和与$ϕ_1$和$ϕ_2$相关联的旋转相关映射$\bar{R}_{ϕ_1,ϕ_2}$为零.将此结果与W. Winter，我们给出了一类严格大于迹秩为0或1的可分\CA类的酉可分单顺从\CA类的${\cal A}$的分类定理.这类代数包含了$K_0$的状态空间与迹状态空间相同的所有有单位元的单ASH-代数以及降维圈代数的单归纳极限。此外，它还包含一些$K_0$ -群不是Riesz的有单位的单ASH-代数。主要结果的一个推论是：所有${\cal Z}$ -稳定的单AH-代数同构于没有维数增长的单AH-代数。
Amenable traces and Følner C*-Algebras
本文讨论了作用在希尔伯特空间上的酉和可分C* 代数上的顺从迹的一种逼近方法，它是用非零有限秩投影的Følner序列表示的.我们应用这种方法来改善由于Arveson和Bédos的谱逼近结果。我们还提出了一个抽象的特征，在单位可分C* 代数代数的单位完全正映射，允许一个非退化表示，它有一个Følner序列，或等价地，一个顺从的迹。这类似于Voiculescu对拟对角C* 代数的抽象刻画。我们定义Følner C* 代数-代数为满足这些等价条件的有单位可分C* 代数-代数。最后，我们也提到了一些持久性的性质，这些代数。
Multiply generated dynamical systems and the duality of higher rank graph algebras
定义了局部同胚系统的半直积广群$T=(T_{1},..., T_{r})$。我们证明，这种建设产生顺从广群。关联代数是类Cuntz代数。我们使用这种结构的高阶图代数，以给出一个拓扑解释$E$ -理论之间的$C^{*}(Λ)$和$C^{*}(Λ^{op})$的对偶。
Decomposition rank of subhomogeneous $C^*$-algebras
本文分析了E. Kirchberg和作者）的次齐次$C^*$ -代数。特别地，我们证明了一个次齐次$C^*$ -代数有分解秩$n$当且仅当它是拓扑维数$n$的递归次齐次，并且$n$由本原理想空间决定.作为应用，我们使用Q. Lin和N. C. Phillips证明了：设$A$是来自紧致光滑流形和极小自同态的交叉积$C^*$ -代数.则$A$的分解秩由底层流形的覆盖维数决定。
We consider 2-positive almost order zero (disjointness preserving) maps on C*-algebras. Generalizing the argument of M. Choi for multiplicative domains, we give an internal characterization of almost order zero for 2-positive maps. It is also shown that complete positivity can be reduced to 2-positivity in the definition of decomposition rank for unital separable C*-algebras.
Decomposition rank and Z-stability
证明了有限分解秩的可分单有单位的C* 代数-代数张量吸收Jiang-Su代数Z.这对艾略特的计划有许多后果，即通过它们的K-理论数据对核C* 代数进行分类。特别地，它完成了与唯一遍历的，光滑的，最小的动力系统的C* 代数的分类，通过他们的有序K-群。
In this paper we consider the decomposition of positive semidefinite matrices as a sum of rank one matrices. We introduce and investigate the properties of various measures of optimality of such decompositions. For some classes of positive semidefinite matrices we give explicitly these optimal decompositions. These classes include diagonally dominant matrices and certain of their generalizations, $2\times 2$, and a class of $3\times 3$ matrices.
Group von Neumann algebras, inner amenability, and unit groups of continuous rings
本文证明非内顺从离散群 $G$ 的群冯诺依曼代数相伴算子环的单位群在 rank 度量拓扑下非顺从，由此构造出非顺从单位群的非离散不可约连续环实例，并建立与 $\mathrm{II}_{1}$ 因子酉群 Eymard--Greenleaf 顺从性的联系。
On the classification of simple Z-stable C*-algebras with real rank zero and finite decomposition rank
证明了：若A是可分单有单位C* 代数，它张量地吸收Jiang-Su代数Z，且实秩为零，分解秩有限，则A是Lin意义下的迹AF，对迹状态空间没有任何限制.因此，艾略特猜想对于上述满足泛系数定理的C* 代数类是成立的。特别地，这样的代数完全由它们的有序K-理论决定。它们是拓扑维数小于或等于3的近似齐次的，拓扑维数至多为2的近似次齐次的，并且它们的分解秩也不大于大于2。
Decomposition rank of Z-stable C*-algebras
本文证明了C（X）\otimes Z形式的C* 代数的分解秩至多为2，其中X是紧的，Hausdorff和Z表示Jiang-Su代数.这相当于具有足够规则纤维的C*-丛的降维结果。它建立了一个重要的情况下猜想的精细结构核C* 代数的汤姆斯和第二命名的作者，即使在一个非简单的设置，并提供证据表明，拓扑维数的非交换空间是由纤维，而不是底空间。
Simple C*-algebras with locally finite decomposition rank
我们引入了局部有限分解秩的概念，这是许多稳定有限核C* 代数所共有的一个结构性质。这一概念与Elliott利用K-理论数据对核C* 代数进行分类的计划特别相关。本文研究了它的一些性质，证明了具有局部有限分解秩、实秩为零的单单位C* 代数-代数在张量上吸收Jiang-Su代数Z后，在Lin意义下迹秩为零。因此，任何这样的C* 代数，如果它另外满足泛系数定理，则它是拓扑维数至多为3的近似齐次的。我们的结果特别证实了实秩为零的单单位Z-稳定ASH代数类的Elliott猜想。此外，这意味着实秩为零的单单位Z-稳定AH代数不仅在ASH意义下维数增长缓慢，而且在AH意义下维数增长也缓慢。
PHP decompositions and primitivity of group rings for linear groups
本文证明具有小泽性质PHP且含非阿贝尔自由子群的可数群的群环对任意域本原，并推论非平凡可数线性群若具有平凡可和根则也满足此性质，同时建立了PHP类的一些永久性质。
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