Wilhelm Winter
We prove the title. This characterizes the Jiang-Su algebra Z as the uniquely determined initial object in the category of strongly self-absorbing C*-algebras.
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We show that the Fraïssé limit of a category of unital separable $C^*$-algebras which is sufficiently closed under tensor products of its objects and morphisms is strongly self-absorbing, given that it has approximate inner half-flip. We use this connection between Fraïssé limits and strongly self-absorbing $C^*$-algebras to give a self-contained and rather elementary proof for the well known fact that the Jiang-Su algebra is strongly self-absorbing.
Strongly Self-Absorbing C*-algebras which contain a nontrivial projection
证明了强自吸收C* 代数-代数是实秩零的，并且如果它包含一个非平凡的投影，则它吸收Jiang-Su代数.我们还考虑的情况下，UCT是自动的强自吸收C* 代数-代数，和K-理论的方式来表征时，Kirchberg代数是强自吸收。
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.
We give a shorter proof of the fact that the Jiang-Su algebra is strongly self-absorbing. This is achieved by introducing and studying so-called unitarily suspended endomorphisms of generalized dimension drop algebras. Along the way we prove uniqueness and existence results for maps between dimension drop algebras and UHF-algebras.
Suppose that $A,B$ are nuclear, separable ${\rm C}^*$-algebras of stable rank one and real rank zero, $A$ is unital simple, $B$ is stable and $({\rm K}_0(B),{\rm K}_0^+(B))$ is weakly unperforated in the sense of Elliott \cite{Ell}. We show that any unital extension with trivial index maps of $A$ by $B$ is absorbing.
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…
Trivialization of C(X)-algebras with strongly self-absorbing fibres
设$A$是一个可分单位$C(X)$ -代数，它的每根纤维同构于同一个强自吸收和$K_{1}$ -内射$C^{*}$ -代数$D$ .我们证明了在紧T2空间$X$是有限维的条件下，$A$和$C(X) \otimes D$同构于$C(X)$ -代数.这个陈述是已知的不扩展到无限维的情况下。
This is a continuation of the study of strongly self-absorbing actions of locally compact groups on C*-algebras. Given a strongly self-absorbing action $γ: G\curvearrowright\mathcal{D}$, we establish permanence properties for the class of separable C*-dynamical systems absorbing $γ$ tensorially up to cocycle conjugacy. Generalizing results of both Toms-Winter and Dadarlat-Winter, it is proved that the desirable equivariant analogues of the classical permanence properties hold in this context. For the permanence with regard to equivariant extensions, we need to require a mild extra condition on…
Localizing the Elliott Conjecture at Strongly Self-absorbing C*-algebras --An Appendix
本文对W. Winter证明了两个具有局部有限分解性质且投影分离迹的单位可分简单顺从${\cal Z}$ -吸收C* 代数-代数是同构的，如果它们的$K$ -理论是逆生成的且它们的Elliott不变量相同.
In this paper we define a Rokhlin property for automorphisms of non-unital C*-algebras and for endomorphisms. We show that the crossed product of a C*-algebra by a Rokhlin automorphism preserves absorption of a strongly self-absorbing C*-algebra, and use this result to deduce that the same result holds for crossed products by endomorphisms in the sense of Stacey. This generalizes earlier results of the second named author and W. Winter.
Permutations of Strongly Self-Absorbing C*-algebras
设G是作用于{1，.的有限群，n}。对于任何C* 代数A，这定义了G的α在A^{ \otimes n}上的作用。证明了如果A张量吸收无限型UHF代数Jiang-Su代数或A近似可除，则A \times_α G也具有相应的性质.
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…
In this paper, we accomplish two objectives. Firstly, we extend and improve some results in the theory of (semi-)strongly self-absorbing C*-dynamical systems, which was introduced and studied in previous work. In particular, this concerns the theory when restricted to the case where all the semi-strongly self-absorbing actions are assumed to be unitarily regular, which is a mild technical condition. The central result in the first part is a strengthened version of the equivariant McDuff-type theorem, where equivariant tensorial absorption can be achieved with respect to so-called very strong c…
Toward the classification of strongly self-absorbing $\mathrm{C}^*$-dynamical systems of compact groups
本文提出关于强自吸收C*-动力系统在等价bootstrap范畴中equivariant KK理论的猜想，证明该猜想对所有有限EPPO群成立，建立了一种K理论公式。
Strongly self-absorbing C*-algebras
设可分有单位元的C*-代数D是强自吸收的，如果存在同构$ϕ: D \to D \otimes D$使得$ϕ$和$id_D \otimes 1_D$是近似酉等价的$*$ -同态.我们研究了这类代数，包括Cuntz代数$\mathcal{O}_2$，$\mathcal{O}_{\infty}$，无限型UHF代数，Jiang-Su代数Z以及$\Oh_{\infty}$与无限型UHF代数的张量积.给定一个强自吸收C*-代数D，我们证明了当一个可分C*-代数张量地吸收D时（即，是D-稳定的），并证明了这类可分D-稳定C* 代数的封闭性.最后，我们计算了可能的K-群，并证明了一些分类结果，这些结果表明上面列出的例子是唯一的强自吸收C* 代数。
C_0(X)-algebras, stability and strongly self-absorbing C*-algebras
我们分别研究了稳定的和所谓的D-稳定的C* 代数的持久性。更精确地说，我们证明了C_0（X）-代数A是稳定的，如果它的所有纤维都是稳定的，只要它的紧可度量化空间X具有有限覆盖维数或A的Cuntz半群几乎是无孔的（这是C* 代数-代数张量吸收Jiang-Su代数Z时自动满足的条件）.进一步证明了：如果D是K_1-内射强自吸收C* 代数-代数，则A张量吸收D当且仅当它的所有纤维张量吸收D，同样假设X是有限维的.后一种说法推广了Blanchard和Kirchberg的结果。我们还证明了关于X的维数的条件不能被丢弃。在此基础上，我们得到了具有弱无孔Cuntz半群的C*-代数何时稳定的一个有用的刻画，这使得我们能够证明稳定性传递到Z-吸收C* 代数的扩张.
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。
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。
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。
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。
We have previously shown that the isomorphism classes of orientable locally trivial fields of $C^*$-algebras over a compact metrizable space $X$ with fiber $D\otimes \mathbb{K}$, where $D$ is a strongly self-absorbing $C^*$-algebra, form an abelian group under the operation of tensor product. Moreover this group is isomorphic to the first group $\bar{E}^1_D(X)$ of the (reduced) generalized cohomology theory associated to the unit spectrum of topological K-theory with coefficients in $D$. Here we show that all the torsion elements of the group $\bar{E}^1_D(X)$ arise from locally trivial fields …
We show that the Dixmier-Douady theory of continuous field $C^*$-algebras with compact operators $\mathbb{K}$ as fibers extends significantly to a more general theory of fields with fibers $A\otimes \mathbb{K}$ where $A$ is a strongly self-absorbing C*-algebra. The classification of the corresponding locally trivial fields involves a generalized cohomology theory which is computable via the Atiyah-Hirzebruch spectral sequence. An important feature of the general theory is the appearance of characteristic classes in higher dimensions. We also give a necessary and sufficient $K$-theoretical cond…
Corona algebras and strongly self-absorbing $\mathrm{C}^{\ast}$-dynamics
该研究探讨局部紧群上σ-单位C*-动力系统在冠代数上诱导的群作用的结构，证明其满足动力学折叠性质，还建立了冠作用的γ-饱和性与原作用γ-吸收性的等价关系，是相关工作的动力学版本。
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…
On the KK-theory of strongly self-absorbing C*-algebras
设$\Dh$和$A$是酉可分的$C^{*}$ -代数，$\Dh$是强自吸收的.已知从$\Dh$到$A \otimes \Dh$的任何两个有单位的$^*$ -同态是近似酉等价的.我们表明，如果$\Dh$也是$K_{1}$ -内射，他们甚至是渐近酉等价的。这特别意味着$\Dh$的任何单位自同态都是渐近内的。此外，$\Dh$的自同构空间是紧可收缩的（在点范数拓扑中），在这个意义上，对于任何紧Hausdorff空间$X$，同伦类的集合$[X,\Aut(\Dh)]$约化为一个点。相应的陈述适用于$\Dh$的酉自同态空间。作为应用，我们用$^*$ -同态和渐近酉等价刻画了Kasparov群$KK(\Dh, A\ot \Dh)$.一路上，我们证明了卡斯帕罗夫群$KK(\Dh, A\ot \Dh)$同构于$K_0(A\ot \Dh)$。
The stable and the real rank of Z-absorbing C*-algebras
设A是一个C* 代数，A同构于A张量Z，其中Z是Jiang-Su代数：一个有单位的、单的、稳定有限的、可分的、核的、无限维的C* 代数，它具有与复数相同的Elliott不变量.证明了：（i）A上矩阵代数的正元素等价类的Cuntz半群W（A）是弱无穿孔的; (ii)如果A是正合的，则A是纯无限的当且仅当A是无迹的。(iii)如果A是可分的且有核的，则A同构于A张量O_infty当且仅当A是无迹的。(iv)如果A是单幺元的，则A的稳定秩是1当且仅当A是有限的。当A的秩为零时，我们也可以证明它是零。
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 …
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。
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 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 $\mathc…
Rokhlin actions and self-absorbing C*-algebras
设A是一个单位可分的C* 代数，D是一个K_1-内射强自吸收的C* 代数。我们证明了如果A是D-吸收的，那么A与紧的第二可数群或Z或R的交叉积也是D-吸收的，假设作用满足Rokhlin性质。在紧Rokhlin作用的情况下，我们证明了一个类似的陈述近似整除。
Topological T-duality for bundles of strongly self-absorbing C*-algebras and some physical applications
本文将拓扑 T-对偶性的 C*-代数框架推广至强自吸收 C*-代数丛，建立 K-理论映射，用以描述 II 型弦论通量背景中的 Fermionic、Timelike 等树级对偶。
Localizing the Elliott conjecture at strongly self-absorbing C*-algebras
我们正式引入了在给定的强自吸收C*-代数$D$上局部化Elliott猜想的概念;我们还解释了核C* 代数的已知分类定理如何适合于这个概念。作为这方面的一个新结果，我们利用Lin最近的结果证明了（在一个较弱的K-理论条件下）具有局部有限分解秩和UCT且投影分离迹的可分、有单位元的单C* 代数-代数类满足Elliott猜想。我们的主要结果是制定在一个更一般的方式，这使我们能够勾勒出一个战略，可能会删除的迹空间条件以及K-理论的限制完全。当把我们的结果和Elliott，Gong和Li最近的分类定理都看作是实秩零情形的推广时，这两种方法在某种意义上是垂直的.攻击一般情况的策略旨在将这两种方法结合起来。我们的分类定理涵盖了简单的ASH代数，它的投影分离迹（和K-群的k-生成的扭转部分）;然而，它并不依赖于一个归纳极限结构。此外，在单家族的情况下，它并不以任何方式依赖于是否存在预测。事实上，这是第一个这样的结果，它以一种自然的方式，涵盖了所有已知的实秩零的有单位的、可分的、单的、核的和稳定有限的C* 代数-代数（其K-群具有k-生成的扭部分）以及（无投影的）Jiang-Su代数本身。
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。
Elliott and Kucerovsky stated that a non-unital extension of separable $C^\ast$-algebras with a stable ideal, is nuclearly absorbing if and only if the extension is purely large. However, their proof was flawed. We give a counter example to their theorem as stated, but establish an equivalent formulation of nuclear absorption under a very mild additional assumption to being purely large. In particular, if the quotient algebra is non-unital, then we show that the original theorem applies. We also examine how this effects results in classification theory.
A universal property for the Jiang-Su algebra
证明了有单位的可分C* 代数的无穷张量幂幂张量地吸收Jiang-Su代数Z当且仅当它有单位地包含一个无特征标的次齐次代数.这就产生了Z在一个范畴中的一个简洁的泛性质，这个范畴如此之大，以至于不存在没有已知位于它之外的特征标的单位可分C* 代数。而且，这个范畴包含了我们的绝大多数常用的可分顺从C* 代数，并且在通向可分超代数和可分代数，从而通向单位张量积，单位直限，和可数离散群的交叉积我们主要结果的一个推论是强自吸收ASH代数是Z-稳定的，因此满足W.冬天得出了Z是唯一的无投影强自吸收ASH代数的结论，完成了强自吸收ASH代数的分类。
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。
It is determined exactly which classifiable C*-algebras have approximately inner flip. The answer includes a number of C*-algebras with torsion in their K-theory, and a number of C*-algebras that are self-absorbing but not strongly self-absorbing.
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)…
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…
The homotopy groups of the equivariant automorphism group of Kirchberg algebras with compact group actions and equivariant Dadarlat-Pennig theory
本文前半部分用等变KK-理论描述了具有等距平移吸收紧群作用的Kirchberg代数的等变自同构群的同伦群，给出了Dadarlat结果的等变版本；后半部分统一处理了强自吸收作用的等变Dadarlat-Pennig理论。
$G$-kernels of Kirchberg algebras
本文利用$K$-理论为Kirchberg代数的$G$-核引入新不变量，推导了纯无限情形的阻碍类约束，并基于该不变量与Dadarlat-Pennig理论分类了强自吸收Kirchberg代数的$\mathbb{Z}^n$-核。
We initiate the study of compact group actions on C*-algebras from the perspective of model theory, and present several applications to C*-dynamics. Firstly, we prove that the continuous part of the central sequence algebra of a strongly self-absorbing action is indistinguishable from the continuous part of the sequence algebra, and in fact equivariantly isomorphic under the Continuum Hypothesis. As another application, we present a unified approach to several dimensional inequalities in C*-algebras, which is done through the notion of order zero dimension for an (equivariant) *-homomorphism. …
Purely infinite C*-algebras: ideal-preserving zero homotopies
本文证明了：如果A是可分的、核的、O_infty吸收的（或强纯无限的）C*-代数，且A在理想系统保持下同伦于零，则A是M_k（C_0（G，v））形式的C* 代数的归纳极限，其中G是一个有限图（C_0（G，v）是G上在G的一个区别点v处为零的连续函数的代数）.进一步证明了任何可分的、核的、稳定的、O_2吸收的C ~*-代数同构于C ~*-代数D与整数通过作用α的交叉积，其中D是M_k（C_0（G，v））形式的C* 代数的归纳极限（且D是O_2吸收的且以理想系统保持的方式同伦到零）.
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…
The generator problem was posed by Kadison in 1967, and it remains open until today. We provide a solution for the class of C*-algebras absorbing the Jiang-Su algebra Z tensorially. More precisely, we show that every unital, separable, Z-stable C*-algebras A is singly generated, which means that there exists an element x in A that is not contained in any proper sub-C*-algebra of A. To give applications of our result, we observe that Z can be embedded into the reduced group C*-algebra of a discrete group that contains a non-cyclic, free subgroup. It follows that certain tensor products with red…
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。
Let $Λ$ be a countably infinite property (T) group, and let $D$ be UHF-algebra of infinite type. We prove that there exists a continuum of pairwise non (weakly) cocycle conjugate, strongly outer actions of $Λ$ on $D$. The proof consists in assigning, to any second countable abelian pro-$p$ group $G$, a strongly outer action of $Λ$ on $D$ whose (weak) cocycle conjugacy class completely remembers the group $G$. The group $G$ is reconstructed from the action through its (weak) 1-cohomology set endowed with a canonical pairing function. Our construction also shows the following stronger statement:…
We prove that a wreath product of an abelian group and a non-amenable group is not strongly Ulam stable. Previously this was known for groups containing free subgroups, due to work of Burger, Ozawa and Thom, and for some surface groups, due to work by Kazhdan. We also show that these wreath products are not deformation rigid.
Strongly outer actions of certain torsion-free amenable groups on the Razak-Jacelon algebra
本文证明了对于特定无挠顺从群类$\mathfrak{C}$中的群$Γ$，其在Razak-Jacelon代数$\mathcal{W}$上的所有强外作用均彼此上闭链共轭，这是Szabó关于强自吸收C$^*$-代数结果的类似结论。
Equivariant $KK$-theory of Bernoulli shifts on $C^*$-algebras with approximately inner flip
本文基于 Enders--Schemeitat--Tikuisis 分类，证明了 UCT 类中具有近似内翻转的可分 $C^*$-代数是 $K$-理论自吸收的当且仅当其 Bernoulli 平移 $KK^G$-等价于平凡作用，并推广了相关 $K$-理论公式。
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