Zhuang Niu, Qingyun Wang
We show that there is a simple separable unital (non-nuclear) tracially AF algebra $A$ which does not absorb the Jiang-Su algebra $\mathcal Z$ tensorially, i.e., $A \ncong A\otimes\mathcal Z$.
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We define a notion of tracial $\mathcal{Z}$-absorption for simple not necessarily unital C*-algebras, study it systematically, and prove its permanence properties. This extends the notion defined by Hirshberg and Orovitz for unital C*-algebras. The Razak-Jacelon algebra, simple C*-algebras with tracial rank zero, and simple purely infinite C*-algebras are tracially $\mathcal{Z}$-absorbing. We obtain the first purely infinite examples of tracially $\mathcal{Z}$-absorbing C*-algebras which are not $\mathcal{Z}$-absorbing. We use techniques from reduced free products of von~Neumann algebras to co…
In this paper, we introduce a class of non-unital tracial approximation ${\rm C^*}$-algebras. Consider the class of ${\rm C^*}$-algebras which are tracially $\mathcal{Z}$-absorbing (in the sense of Amint, Golestani, Jamali, Phillips's simple tracially $\mathcal{Z}$-absorbing or Castillejos, Li, Szabvo's tracial $\mathcal{Z}$-stability). Then $A$ is tracially $\mathcal{Z}$-absorbing for any simple ${\rm C^*}$-algebra $A$ in the corresponding class of non-unital tracial approximation ${\rm C^*}$-algebras.
We study a tracial notion of Z-absorption for simple, unital C*-algebras. We show that if A is a C*-algebra for which this property holds then A has almost unperforated Cuntz semigroup, and if in addition A is nuclear and separable we show this property is equivalent to having A being Z-absorbing. We furthermore show that this property is preserved under forming certain crossed products by actions satisfying a tracial Rokhlin type property.
In this paper, we introduce some classes of generalized tracial approximation ${\rm C^*}$-algebras. Consider the class of unital ${\rm C^*}$-algebras which are tracially $\mathcal{Z}$-absorbing (or have tracial nuclear dimension at most $n$, or have the property $\rm SP$, or are $m$-almost divisible). Then $A$ is tracially $\mathcal{Z}$-absorbing (respectively, has tracial nuclear dimension at most $n$, has the property $\rm SP$, is weakly ($n, m$)-almost divisible) for any simple unital ${\rm C^*}$-algebra $A$ in the corresponding class of generalized tracial approximation ${\rm C^*}$-algebra…
In this paper, we introduce a class of generalized tracial approximation ${\rm C^*}$-algebras. Let $\mathcal{P}$ be a class of unital ${\rm C^*}$-algebras which have tracially $\mathcal{Z}$-absorbing (tracial nuclear dimension at most $n$, $\rm SP$ property, $m$-almost divisible, weakly $(m, n)$-divisible). Then $A$ has tracially $\mathcal{Z}$-absorbing (tracial nuclear dimension at most $n$, $\rm SP$ property, weakly $m$-almost divisible, secondly weakly $(m, n)$-divisible) for any simple unital ${\rm C^*}$-algebra $A$ in the class of this generalized tracial approximation ${\rm C^*}$-algebra…
We show that if $A$ is a simple (not necessarily unital) tracially $\mathcal{Z}$-absorbing C*-algebra and $α\colon G \to \mathrm{Aut} (A)$ is an action of a finite group $G$ on $A$ with the weak tracial Rokhlin property, then the crossed product $C^*(G, A,α)$ and the fixed point algebra $A^α$ are simple and tracially $\mathcal{Z}$-absorbing, and they are $\mathcal{Z}$-stable if, in addition, $A$ is separable and nuclear. The same conclusion holds for all intermediate C*-algebras of the inclusions $A^α\subseteq A$ and $A \subseteq C^*(G, A,α)$. We prove that if $A$ is a simple tracially $\mathc…
Let $A$ be a simple infinite dimensional stably finite unital C*-algebra, and let $B$ be a centrally large subalgebra of $A$. We prove that if $A$ is tracially ${\mathcal{Z}}$-absorbing if and only if $B$ is tracially ${\mathcal{Z}}$-absorbing. If $A$ and $B$ are also separable and nuclear, we prove that $A$ is ${\mathcal{Z}}$-absorbing if and only if $B$ is ${\mathcal{Z}}$-absorbing.
We show that the following properties of the C*-algebras in a class $\mathcal{P}$ are inherited by simple unital ${\rm C^*}$-algebras in the class of asymptotically tracially in $\mathcal{P}$: $(1)$ $β$-comparison (in the sense of Kirchberg and Rørdam), $(2)$ $n$-comparison (in the sense of Winter).
Divisible properties for asymptotically tracially approximation ${\rm C^*}$-algebras
本文研究渐近迹属于类 P 的单含单位元 C*-代数，证明其继承原类代数的 m-几乎可除与弱 (m,n)-可除性质。
Non unital generalized tracially approximated C*-algebras
本文研究了非单位的广义迹逼近C*-代数类，证明了其与弱迹逼近的等价性，并验证了该类代数具有迹Z-吸收、迹核维数至多n及性质SP等重要性质。
The real and stable rank of tracially complete C*-algebras
本文证明可追溯完备C*-代数在满足CPoU条件下具有零真实秩和一稳定秩，从而完整描述了这些代数的Cuntz半群。
Tracially amenable actions and purely infinite crossed products
本文引入迹顺从作用概念并给出等价刻画，证明含自由群F2的群在简单、单位、Z-稳定C*-代数上的外迹顺从作用具有纯无限交叉积，并提供自由群在AF-代数上的具体例子。
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 study a pair of $C^*$-algebras by associating a $*$-homomorphism from $A$ to $B$ allowing an approximate left-inverse to the sequence algebra of $A$ in a manner reminiscent of several tracial approximation properties. We are particularly interested how regularity properties in the Elliott classification program pass from $B$ to $A$. Among them, we show that the strict comparison property and $\mathcal{Z}$-stability pass from $B$ to $A$ in our setting.
We show that a simple separable unital nuclear nonelementary $C^*$-algebra whose tracial state space has a compact extreme boundary with finite covering dimension admits uniformly tracially large order zero maps from matrix algebras into its central sequence algebra. As a consequence, strict comparison implies $\Z$-stability for these algebras.
We prove that a tracially continuous W$^*$-bundle $\mathcal{M}$ over a compact Hausdorff space $X$ with all fibres isomorphic to the hyperfinite II$_1$-factor $\mathcal{R}$ that is locally trivial already has to be globally trivial. The proof uses the contractibility of the automorphism group $\mathrm{Aut}({\mathcal{R}})$ shown by Popa and Takesaki. There is no restriction on the covering dimension of $X$.
We study the automorphism group of a unital, simple, $\mathcal{Z}$-stable $C^{*}$-algebra. In this paper, we generalize the results by the authors in \cite{pr_auto} to $\mathcal{Z}$-stable $C^{*}$-algebras $\mathfrak{A}$ such that $\mathfrak{A} \otimes \mathfrak{B}$ is a separable, nuclear, simple, tracially AI algebras satisfying the Universal Coefficient Theorem (UCT) of Rosenberg and Schochet \cite{uct}. By the results of Lin in \cite{hl_asyunit} and Winter in \cite{ww_localelliott}, $C^{\ast}$-algebras that satisfies the above condition are classified via $K$-theory and traces.
Tracially lyriform $\mathrm{C}^*$-algebras
本文提出迹性 lyriform $C^*$-代数，用于研究在稳定有限 Elliott 可分类 $C^*$-代数中距离测量的几何与统计。
$\mathcal{W}$-absorbing actions of finite groups on the Razak-Jacelon algebra
本文提出了C$^*$-代数上$\mathcal{W}$-吸收作用的定义，并在共轭与余环共轭意义下完全分类了有限群在Razak-Jacelon代数上的外部$\mathcal{W}$-吸收作用。
Tensorially Absorbing Inclusions of C*-algebras
本文研究C*-代数包含的$\mathcal D$-稳定性，给出超幂刻画，证明稳定性可同时适用于可数个中间代数，且嵌入稠密并近似酉等价于稳定嵌入。
Let $P \subset A$ be a inclusion of unital C*-algebras and $E\colon A \to P$ be a conditional expectation of index finite type. We introduce a Rokhlin property for $E$ and discuss about $\mathcal{D}$-absorbing proeprty, where $\mathcal{D}$ is a separable, unital, strongly self-absorbing C*-algebra. In this paper we consider permanent properties for strongly self-absorbing property under inclusions of unital C*-algebras with a finite Watatani index.
Coronas and strongly self-absorbing C*-algebras
本文研究强自吸收C*-代数与corona代数的关系，证明了可分C*-代数的D-稳定性与其corona代数的D-饱和性、稳定corona的可分D-稳定性的等价性，推广了Calkin代数相关结果并给出应用。
We introduce and study strongly self-absorbing actions of locally compact groups on C*-algebras. This is an equivariant generalization of a strongly self-absorbing C*-algebra to the setting of C*-dynamical systems. The main result is the following equivariant McDuff-type absorption theorem: A cocycle action $(α,u): G\curvearrowright A$ on a separable C*-algebra is cocycle conjugate to its tensorial stabilization with a strongly self-absorbing action $γ: G\curvearrowright\mathcal{D}$, if and only if there exists an equivariant and unital $*$-homomorphism from $\mathcal{D}$ into the central sequ…
In this paper we consider a bootstrap class $\mathfrak C$ of countable discrete groups, which is closed under countable unions and extensions by the integers, and we study actions of such groups on C*-algebras. This class includes all torsion-free abelian groups, poly-$\mathbb Z$-groups, as well as other examples. Using the interplay between relative Rokhlin dimension and semi-strongly self-absorbing actions established in prior work, we obtain the following two main results for any group $Γ\in\mathfrak C$ and any strongly self-absorbing C*-algebra $\mathcal D$: (1) There is a unique strongly …
Calkin algebra, Kazhdan's property (T), strongly self-absorbing C*-algebras
本文证明 Calkin 代数不与 O∞、Kirchberg 代数或任意单位 Z-稳定 C*-代数稳定化后的冠代数同构，关键工具是可分 C*-子代数的相对交换子性质。
In this paper, we establish a connection between Rokhlin dimension and the absorption of certain model actions on strongly self-absorbing C*-algebras. Namely, as to be made precise in the paper, let $G$ be a well-behaved locally compact group. If $\mathcal D$ is a strongly self-absorbing C*-algebra, and $α: G\curvearrowright A$ is an action on a separable, $\mathcal D$-absorbing C*-algebra that has finite Rokhlin dimension with commuting towers, then $α$ tensorially absorbs every semi-strongly self-absorbing $G$-actions on $\mathcal D$. This contains several existing results of similar nature …
Almost elementary groupoid models for $C^*$-algebras
本文证明众多可分类C*-代数的群胚模型为几乎基本的，给出特定Elliott不变量下C*-代数可分类性的群胚理论刻画，并得到Jiang-Su代数变换群胚模型的存在障碍。
Computably strongly self-absorbing C*-algebras
本文引入可计算强自吸收C*-代数的概念，证明了Cuntz代数、特定UHF代数及Jiang-Su代数等满足该性质，并作为Elliott近似交织论证的可计算版本特例。
Endomorphisms of Z-absorbing C*-algebras without conditional expectations
本文构造 Jiang-Su 代数 $\mathcal{Z}$ 上无附条件期望的自同态，回答 Kirchberg 的问题，并据此证明单位可分核 $\mathcal{Z}$-吸收 C*-代数在 $\mathcal{O}_2$ 中不可迁移。
We investigate symmetries on unital Kirchberg algebras with respect to the Rokhlin property and finite Rokhlin dimension. In stark contrast to the restrictiveness of the Rokhlin property, every such outer action has Rokhlin dimension at most 1. A consequence of these observations is a relationship between the nuclear dimension of an $\mathcal{O}_\infty$-absorbing C*-algebra and its $\mathcal{O}_2$-stabilization. A more direct and alternative approach to this is given as well. Several applications of this relationship are discussed to cover a fairly large class of $\mathcal{O}_\infty$-absorbing…
The relative commutant $A'\cap A^{\mathcal{U}}$ of a strongly self-absorbing algebra $A$ is indistinguishable from its ultrapower $A^{\mathcal{U}}$. This applies both to the case when $A$ is the hyperfinite II$_1$ factor and to the case when it is a strongly self-absorbing C*-algebra. In the latter case we prove analogous results for $\ell_\infty(A)/c_0(A)$ and reduced powers corresponding to other filters on $\bf N$. Examples of algebras with approximately inner flip and approximately inner half-flip are provided, showing the optimality of our results. We also prove that strongly self-absorbi…
Computing cohomology groups that classify bundles of strongly self-absorbing $C^*$-algebras
本文研究了强自吸收C*代数丛的上同调群，通过普通上同调和连通K-理论计算，并开发了复杂周期拓扑K-理论单位谱的唯一性结果。
Infinite Loop Spaces and Group Actions on Strongly Self-Absorbing C*-Algebras
研究强自吸收C* - 代数上群作用等的提升障碍，证明相关交叉模分类空间有张量积诱导的无限循环空间结构，证实猜想并扩展与稳定同伦理论联系，通过构造\(\mathbb{I}\) - FCPs等方法证明，确定自然变换取值于上同调群。
Actions on classifiable C*-algebras without equivariant property (SI)
该研究构造了可分类C*-代数上迹顺从但非顺从的群作用例子，证明严格比较下顺从等价于迹顺从加等变Matui--Sato性质（SI），且等变$\u2124$-稳定性不蕴含等变性质（SI）。
Say that a separable, unital C*-algebra D is strongly self-absorbing if there exists an isomorphism $ϕ: D \to D \otimes D$ such that $ϕ$ and $id_D \otimes 1_D$ are approximately unitarily equivalent $*$-homomorphisms. We study this class of algebras, which includes the Cuntz algebras $\mathcal{O}_2$, $\mathcal{O}_{\infty}$, the UHF algebras of infinite type, the Jiang--Su algebra Z and tensor products of $\Oh_{\infty}$ with UHF algebras of infinite type. Given a strongly self-absorbing C*-algebra D we characterise when a separable C*-algebra absorbs D tensorially (i.e., is D-stable), and prove…
Bundles of strongly self-absorbing $C^*$-algebras with a Clifford grading
本文研究了带Clifford格的强自吸收C*代数束的分类，建立了一个上同调理论并证明了相关同构关系。
Classification of equivariantly $\mathcal{O}_2$-stable amenable actions on nuclear $\mathrm{C}^\ast$-algebras
本文研究可分稳定核$\text{C}^*$-代数上满足等距移位吸收、张量吸收$\bigcirc_2$平凡作用的顺从$G$-作用，证明其可由本原理想空间上的诱导$G$-作用在循环共轭下分类，还给出正合群的幺元版本与紧群的共轭分类结果。
We show that a separable, nuclear C*-algebra satisfies the UCT if it has a Cartan subalgebra. Furthermore, we prove that the UCT is closed under crossed products by group actions which respect Cartan subalgebras. This observation allows us to deduce, among other things, that a crossed product $\mathcal O_2\rtimes_α\mathbb Z_p $ satisfies the UCT if there is some automorphism $γ$ of $\mathcal O_2$ with the property that $γ(\mathcal D_2)\subseteq \mathcal O_2\rtimes_α\mathbb Z_p$ is regular, where $\mathcal D_2$ denotes the canonical masa of $\mathcal O_2$. We prove that this condition is automa…
This paper serves as a source of examples of Rokhlin actions or locally representable actions of finite groups on C*-algebras satisfying a certain UHF-absorption condition. We show that given any finite group $G$ and a separable, unital C*-algebra $A$ that absorbs $M_{|G|^\infty}$ tensorially, one can lift any group homomorphism $G\to\operatorname{Aut}(A)/{\approx_u}$ to an honest Rokhlin action $γ$ of $G$ on $A$. Unitality may be dropped in favour of stable rank one or being stable. If $A$ belongs to a certain class of C*-algebras that is classifiable by a suitable invariant (e.g. $K$-theory)…
Continuing our project on noncommutative (stable) homotopy we construct symmetric monoidal $\infty$-categorical models for separable $C^*$-algebras $\mathtt{SC^*_\infty}$ and noncommutative spectra $\mathtt{NSp}$ using the framework of Higher Algebra due to Lurie. We study smashing (co)localizations of $\mathtt{SC^*_\infty}$ and $\mathtt{NSp}$ with respect to strongly self-absorbing $C^*$-algebras. We analyse the homotopy categories of the localizations of $\mathtt{SC^*_\infty}$ and give universal characterizations thereof. We construct a stable $\infty$-categorical model for bivariant connect…
We study circle actions with the Rokhlin property, in relation to their restrictions to finite subgroups. We construct examples showing the following: the restriction of a circle action with the Rokhlin property (even on a real rank zero $C^*$-algebra), need not have the Rokhlin property; and even if every restriction of a given circle action has the Rokhlin property, the circle action itself need not have it. As a positive result, we show that the restriction of a circle action with the Rokhlin property to the subgroup $\mathbb{Z}_n$ has the Rokhlin property if the underlying algebra absorbs …
We give an operator algebraic model for the first group of the unit spectrum $gl_1(KU)$ of complex topological K-theory, i.e. $[X, BGL_1(KU)]$, by bundles of stabilized infinite Cuntz C*-algebras $O_{\infty} \otimes \K$. We develop similar models for the localizations of $KU$ at a prime $p$ and away from $p$. Our work is based on the $\mathcal{I}$-monoid model for the units of $K$-theory by Sagave and Schlichtkrull and it was motivated by the goal of finding connections between the infinite loop space structure of the classifying space of the automorphism group of stabilized strongly self-abso…
For a locally compact group $G$ and a strongly self-absorbing $G$-algebra $(\mathcal{D},δ)$, we obtain a new characterization of absorption of a strongly self-absorbing action using almost equivariant completely positive maps into the underlying algebra. The main technical tool to obtain this characterization is the existence of almost equivariant lifts for equivariant completely positive maps, proved in recent work of the authors. This characterization is then used to show that an equivariant $C_0(X)$-algebra with $\mathrm{dim}_{\mathrm{cov}}(X)<\infty$ is $(\mathcal{D},δ)$-stable if and only…
Equivariant $\mathcal{D}$-stability for Actions of Tensor Categories
本文为酉张量范畴在 C$^*$-代数上的作用引入等变 $\mathcal{D}$-稳定性概念，证明其等价于 $\mathcal{D}$ 到 Kirchberg 中心序列代数某子代数的单位嵌入，并据此证明一大类 AF-作用具有 $\mathcal{Z}$-稳定性。
Satisfiability problems and algebras of boolean constraint system games
本文研究布尔约束系统的量子可满足性理论，通过联合谱投影给出BCS代数新等价表述，构造C*-可满足但非迹可满足的约束系统，利用约简关系的*-同态强化量子版Schaefer二分定理相关结果，并揭示其与非超线性群问题的关联。
KK-rigidity of simple nuclear C*-algebras
研究单的核 Z-稳定 C*-代数的 KK-刚性，证明在迹与 KK-理论上可逆的单位嵌入可推出同构，并给出强自吸收情形的单位保持 KK-等价判据。
Almost elementary étale groupoids
本研究引入几乎基本性作为平展群胚的新有限逼近性质，证明极小几乎基本群胚的约化C*-代数为$\boldsymbol{\text{Z}}$-稳定，还探讨了其与群胚相关性质的联系。
We study strongly outer actions of discrete groups on C*-algebras in relation to (non)amenability. In contrast to related results for amenable groups, where uniqueness of strongly outer actions on the Jiang-Su algebra is expected, we show that uniqueness fails for all nonamenable groups, and that the failure is drastic. Our main result implies that if $G$ contains a copy of the free group, then there exist uncountable many, non-cocycle conjugate strongly outer actions of $G$ on any Jiang-Su stable tracial C*-algebra. Similar conclusions apply for outer actions on McDuff tracial von Neumann alg…
We prove that for every exact discrete group $Γ$, there is an intermediate C*-algebra between the reduced group C*-algebra and the intersection of the group von Neumann algebra and the uniform Roe algebra which is realized as the intersection of a decreasing sequence of isomorphs of the Cuntz algebra $\mathcal{O}_2$. In particular, when $Γ$ has the AP (approximation property), the reduced group C*-algebra is realized in this way. We also study extensions of the reduced free group C*-algebras and show that any exact absorbing or unital absorbing extension of it by any stable separable nuclear C…
We develop an algebraic formalism for topological $\mathbb{T}$-duality. More precisely, we show that topological $\mathbb{T}$-duality actually induces an isomorphism between noncommutative motives that in turn implements the well-known isomorphism between twisted K-theories (up to a shift). In order to establish this result we model topological K-theory by algebraic K-theory. We also construct an $E_\infty$-operad starting from any strongly self-absorbing $C^*$-algebra $\mathcal{D}$. Then we show that there is a functorial topological K-theory symmetric spectrum construction ${\bf K}_Σ^{top}(-…
We prove lifting theorems for completely positive maps going out of exact $C^\ast$-algebras, where we remain in control of which ideals are mapped into which. A consequence is, that if $\mathsf X$ is a second countable topological space, $\mathfrak A$ and $\mathfrak B$ are separable, nuclear $C^\ast$-algebras over $\mathsf X$, and the action of $\mathsf X$ on $\mathfrak A$ is continuous, then $E(\mathsf X; \mathfrak A, \mathfrak B) \cong KK(\mathsf X; \mathfrak A, \mathfrak B)$ naturally. As an application, we show that a separable, nuclear, strongly purely infinite $C^\ast$-algebra $\mathfrak…
Let $P \subset A$ be an inclusion of $σ$-unital C*-algebras with a finite index in the sense of Izumi. Then we introduce the Rokhlin property for a conditional expectation $E$ from $A$ onto $P$ and show that if $A$ is simple and satisfies any of the property $(1) \sim (12)$ listed in the below, and $E$ has the Rokhlin property, then so does $P$. (1) Simplicity;(2) Nuclearity;(3) C*-algebras that absorb a given strongly self-absorbing C*-algebra $\mathcal{D}$; (4)C*-algebras of stable rank one; (5) C*-algebras of real rank zero;(6) C*-algebras of nuclear dimension at most $n$, where $n \in Z^+$…
The uniqueness theorem for Kasparov theory
本文证明了Kasparov理论中的一般唯一性定理，改进了之前的稳定唯一性定理，并扩展到其他$KK$-理论变种。
Strict comparison holds in the uniform Roe algebra of a discrete amenable group
本文证明离散可逆群的一致Roe代数中，正元素迹态下的严格不等式蕴含Cuntz次等价，并证明其通用极小集交叉积C*-代数是强AH代数。
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