Z^2-actions on Kirchberg algebras

A weak homotopy equivalence type result related to Kirchberg algebras

We obtain a weak homotopy equivalence type result between two topological groups associated with a Kirchberg algebra: the unitary group of the continuous asymptotic centralizer and the loop group of the automorphism group of the stabilization. This result plays a crucial role in our subsequent work on the classification of poly-$\mathbb{Z}$ group actions on Kirchberg algebras. As a special case, we show that the $K$-groups of the continuous asymptotic centralizer are isomorphic to the $KK$-groups of the Kirchberg algebra.

Kirchberg代数上的遍历自同构

Ergodic automorphisms on Kirchberg algebras

本文结合C*-代数扩张理论与Pimsner构造，证明了任意可数无限离散群在幺正Kirchberg代数上存在遍历作用，并利用Gabe--Szabó定理与Baum--Connes猜想证明了无核无限离散群逐点外作用具有遍历上圈扰动。

Equivariant Kirchberg-Phillips-type absorption for amenable group actions

We show an equivariant Kirchberg-Phillips-type absorption theorem for pointwise outer actions of discrete amenable groups on Kirchberg algebras with respect to natural model actions on the Cuntz algebras $\mathcal{O}_\infty$ and $\mathcal{O}_2$. This generalizes results known for finite groups and poly-$\mathbb{Z}$ groups. The model actions are shown to be determined, up to strong cocycle conjugacy, by natural abstract properties, which are verified for some examples of actions arising from tensorial shifts. We also show the following homotopy rigidity result, which may be understood as a prec…

交换 C*-代数被 Kirchberg 代数扩张的核维数

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。

带紧群作用的Kirchberg代数的等变自同构群的同伦群与等变Dadarlat-Pennig理论

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。

关于互反 Kirchberg 代数的万有性质与唯一遍历自同构

On universal property of reciprocal Kirchberg algebras and uniquely ergodic automorphisms

本文证明互反 Kirchberg 代数的万有性质，并借此在任意有限生成 K 群的含单位 Kirchberg 代数上构造具有唯一不变纯态的非周期遍历自同构。

UCT-Kirchberg algebras have nuclear dimension one

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.

稳定UCT Kirchberg代数的主群胚模型

Principal groupoid models for stable UCT Kirchberg algebras

本文证明每个稳定UCT Kirchberg代数都有一个主étale群胚模型，从而包含一个C$^*$-对角元；对于单UCT Kirchberg代数$A$且$[1_A]_0$在$K_0(A)$中具有无限阶的情况，该方法同样适用，特别地得到了Cuntz代数$\mathcal{O}_\infty$的主étale群胚模型。

与有向群图相关的$C^*$-代数，以及Kirchberg代数的模型

$C^*$-algebras associated to directed graphs of groups, and models of Kirchberg algebras

本研究引入与有向群图相关的$C^*$-代数，证明无源行有限有向群图对应的组合$C^*$-代数与有向树边界群作用交叉积Morita等价，且这类代数包含稳定UCT Kirchberg代数类，为Kirchberg代数提供了新的组合模型。

Kirchberg 关于中心序列代数的引理在近似内自同构群中的应用

An application of Kirchberg's lemma on central sequence algebras to groups of approximately inner automorphisms

本文利用 Kirchberg 关于中心序列代数的满射引理，证明稳定秩一可分单位 C*-代数的近似内自同构到商满射结果，并部分推广至不可分情形。

The Reciprocal Kirchberg Algebras

For two unital Kirchberg algebras with finitely generated K-groups, we introduce a property, called reciprocality, which is proved to be closely related to the homotopy theory of Kirchberg algebras. We show the Spanier--Whitehead duality for bundles of separable nuclear UCT C*-algebras with finitely generated K-groups and conclude that two reciprocal Kirchberg algebras share the same structure of their bundles.

The Rokhlin property vs. Rokhlin dimension 1 on unital Kirchberg algebras

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…

Reduced free products of unital AH algebras and Blackadar and Kirchberg's MF algebras

In the paper, we prove that reduced free products of unital AH algebras with respect to given faithful tracial states, in the sense of Voiculescu, are Blackadar and Kirhcberg's MF algebras. We also show that the reduced free products of unital AH algebras with respect to given faithful tracial states, under mild conditions, are not quasidiagonal. Therefore we conclude, for a large class of AH algebras, the Brown-Douglas-Fillmore extension semigroups of the reduced free products of these AH algebras with respect to given faithful tracial states are not groups. Our result is based on Haagerup an…

Continuous Fields of Kirchberg C*-algebras

In this paper we study the C*-algebras associated to continuous fields over locally compact metrisable zero dimensional spaces whose fibers are Kirchberg C*-algebras satisfying the UCT. We show that these algebras are inductive limits of finite direct sums of Kirchberg algebras and they are classified up to isomorphism by topological invariants.

Kirchberg代数的$G$-核

$G$-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。

Poly-$\mathbb{Z}$ group actions on Kirchberg algebras II

This is the second part of our serial work on the classification of poly-$\mathbb{Z}$ group actions on Kirchberg algebras. Based on technical results obtained in our previous work, we completely reduce the problem to the classification of continuous fields of Kirchberg algebras over the classifying spaces. As an application, we determine the number of cocycle conjugacy classes of outer $\mathbb{Z}^n$-actions on the Cuntz algebras.

动力 Kirchberg–Phillips 定理

The dynamical Kirchberg-Phillips theorem

本文建立任意局部紧群在 Kirchberg 代数上一类可和作用的余圈共轭分类，证明其由等变 Kasparov 理论决定，并解决及恢复相关猜想与结果。

Nonsplitting in Kirchberg's ideal-related KK-theory

A universal coefficient theorem in the setting of Kirchberg's ideal-related KK-theory was obtained in the fundamental case of a C*-algebra with one specified ideal by Bonkat and proved there to split, unnaturally, under certain conditions. Employing certain K-theoretical information derivable from the given operator algebras in a way introduced here, we shall demonstrate that Bonkat's UCT does not split in general. Related methods lead to information on the complexity of the K-theory which must be used to classify *-isomorphisms for purely infinite C*-algebras with one non-trivial ideal.

Non-amenable tight squeezes by Kirchberg 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.

One-parameter continuous fields of Kirchberg algebras with rational K-theory

We show that separable continuous fields over the unit interval whose fibers are stable Kirchberg algebras that satisfy the universal coefficient theorem in KK-theory and have rational K-theory groups are classified up to isomorphism by filtrated K-theory.

Razak-Jacelon代数的等变Kirchberg-Phillips型吸收

Equivariant Kirchberg-Phillips type absorption for the Razak-Jacelon algebra

本文证明简单可分核单调迹C*-代数上的强外顺从群作用在张量Razak-Jacelon代数后具有余循环共轭的唯一性，并用中心序列代数的不动点子代数给出刻画。

Kirchberg X-algebras with real rank zero and intermediate cancellation

A universal coefficient theorem is proved for C*-algebras over an arbitrary finite T_0-space X which have vanishing boundary maps. Under bootstrap assumptions, this leads to a complete classification of unital/stable real-rank-zero Kirchberg X-algebras with intermediate cancellation. Range results are obtained for (unital) purely infinite graph C*-algebras with intermediate cancellation and Cuntz-Krieger algebras with intermediate cancellation. Permanence results for extensions of these classes follow.

Non-cyclotomic Presentations of Modules and Prime-order Automorphisms of Kirchberg Algebras

We prove the following theorem: let $A$ be a UCT Kirchberg algebra, and let $α$ be a prime-order automorphism of $K_*(A)$, with $α([1_A])=[1_A]$ in case $A$ is unital. Then $α$ is induced from an automorphism of $A$ having the same order as $α$. This result is extended to certain instances of an equivariant inclusion of Kirchberg algebras. As a crucial ingredient we prove the following result in representation theory: every module over the integral group ring of a cyclic group of prime order has a natural presentation by generalized lattices with no cyclotomic summands.

Classification of certain continuous fields of Kirchberg algebras

We show that the K-theory cosheaf is a complete invariant for separable continuous fields with vanishing boundary maps over a finite-dimensional compact metrizable topological space whose fibers are stable Kirchberg algebras with rational K-theory groups satisfying the universal coefficient theorem. We provide a range result for fields in this class with finite-dimensional K-theory. There are versions of both results for unital continuous fields.

On pathological properties of fixed point algebras in Kirchberg algebras

We investigate how the fixed point algebra of a C*-dynamical system can differ from the underlying C*-algebra. For any exact group $Γ$ and any infinite group $Λ$, we construct an outer action of $Λ$ on the Cuntz algebra $\mathcal{O}_2$ whose fixed point algebra is almost equal to the reduced group C*-algebra ${\rm C}^\ast_{\rm r}(Γ)$. Moreover, we show that every infinite group admits outer actions on all Kirchberg algebras whose fixed point algebras fail the completely bounded approximation property.

Symmetries of Kirchberg algebras

Let A be a separable unital nuclear purely infinite simple C*-algebra satisfying the Universal Coefficient Theorem, and such that the K_0-class of the identity is zero. We prove that every automorphism of order two of the K-theory of A is implemented by an automorphism of A of order two. As a consequence, we prove that every countable Z/2Z-graded module over the representation ring of Z/2Z is isomorphic to the equivariant K-theory for some action of Z/2Z on a separable unital nuclear purely infinite simple C*-algebra. Along the way, we prove that every not necessarily finitely generated module…

单位 Kirchberg 代数的全扩张群

Total extension groups for unital Kirchberg algebras

为具有有限生成 K-群的单位 Kirchberg 代数引入分类层级及全扩张群不变量，其结合强、弱扩张群直和，为分类 Cuntz--Krieger 代数提供完全不变量与有效工具。

Representing Kirchberg algebras as inverse semigroup crossed products

In this note we show that a combinatorial model of Kirchberg algebras in the UCT, namely the Katsura algebras O_{AB}, can be expressed both as groupoid C*-algebras and as inverse semigroup crossed products. We use this picture to obtain results about simplicity, pure infiniteness and nuclearity of O_{AB} using different methods than those used by Katsura.

Purely infinite simple C*-algebras that are principal groupoid C*-algebras

From a suitable groupoid G, we show how to construct an amenable principal groupoid whose C*-algebra is a Kirchberg algebra which is KK-equivalent to C*(G). Using this construction, we show by example that many UCT Kirchberg algebras can be realised as the C*-algebras of amenable principal groupoids.

Continuous C*-algebras over topological spaces

We define continuous C*-algebras over a topological space X and establish some basic results. If X is a locally compact Hausdorff space, continuous C*-algebras over X are equivalent to ordinary continuous C_0(X)-algebras. The main purpose of our study is to prove that every continuous, full, separable, nuclear C*-algebra over X is KK(X)-equivalent to a stable Kirchberg algebra over X. (Here a Kirchberg algebra over X is a separable, nuclear, and strongly purely infinite C*-algebra over X with primitive ideal space homeomorphic to X.) In the case that X is a one-point space, this result is know…

A Non-Commutative Unitary Analogue of Kirchberg's Conjecture

The $C^{\ast}$-algebra $\mathcal{U}_{nc}(n)$ is the universal $C^{\ast}$-algebra generated by $n^2$ generators $u_{ij}$ that make up a unitary matrix. We prove that Kirchberg's formulation of Connes' embedding problem has a positive answer if and only if $\mathcal{U}_{nc}(2) \otimes_{\min} \mathcal{U}_{nc}(2)=\mathcal{U}_{nc}(2) \otimes_{\max} \mathcal{U}_{nc}(2)$. Our results follow from properties of the finite-dimensional operator system $\mathcal{V}_n$ spanned by $1$ and the generators of $\mathcal{U}_{nc}(n)$. We show that $\mathcal{V}_n$ is an operator system quotient of $M_{2n}$ and has…

分类简单核 $C^\ast$-代数的统一方法

A unified approach for classifying simple nuclear $C^\ast$-algebras

通过改编已有分类框架，本文为 Kirchberg--Phillips 定理提供了新证明，实现了对稳定有限与纯无限 $C^\ast$-代数的统一分类，且不依赖日内瓦定理。

张量范畴在Kirchberg代数上的作用

Actions of tensor categories on Kirchberg algebras

本文通过刻画非真C*-对应的Pimsner代数的单性，系统构造了酉张量范畴在Kirchberg代数上的外作用，并研究了融合环模作为K-群的可实现性，回答了Izumi关于3-上循环扭曲Cuntz代数作用的问题。

On Kirchberg's Embedding Problem

Kirchberg's Embedding Problem (KEP) asks whether every separable C$^*$ algebra embeds into an ultrapower of the Cuntz algebra $\mathcal{O}_2$. In this paper, we use model theory to show that this conjecture is equivalent to a local approximate nuclearity condition that we call the existence of good nuclear witnesses. In order to prove this result, we study general properties of existentially closed C$^*$ algebras. Along the way, we establish a connection between existentially closed C$^*$ algebras, the weak expectation property of Lance, and the local lifting property of Kirchberg. The paper c…

Weakly Exact von Neumann Algebras

The theory of exact C*-algebras was introduced by Kirchberg and has been influential in recent development of C*-algebras. A fundamental result on exact C*-algebras is a local characterization of exactness. The notion of weakly exact von Neumann algebras was also introduced by Kirchberg. In this paper, we give a local characterization of weak exactness. As a corollary, we prove that a discrete group is exact if and only if its group von Neumann algebra is weakly exact. The proof naturally involves the operator space duality.

Some Examples of Blackadar and Kirchberg's MF Algebras

In the paper, we provide some examples of MF algebras by considering minimal or maximal tensor products of MF algebras and crossed products of MF algebras by finite groups or an integer group. We also present some examples of C$^*$-algebras, whose BDF extension semigroup is not group. These examples include, for example, $ C_r^*(F_n)\otimes_{max} C^*(F_n) $, $ C_r^*(F_n)\otimes_{min} C^*(F_n) $, $C_r^*(H_1\ast H_2)$ with $2\le |H_1|<\infty$ and $3\le |H_2|<\infty$ where $|H_1|$, $|H_2|$ are the orders of thr groups $H1$, or $H_2$ respectively, and many others.

Cuntz代数中的C*-对角线具有康托尔谱

C*-diagonals with Cantor spectrum in Cuntz algebras

本文证明了Cuntz代数中存在具有康托尔谱的C*-对角线，并构建了相关代数的主以太群状模型。

超越部分作用、内可 amenable 性和 Kirchberg 代数模型的群oid

On groupoids beyond partial actions, inner amenability, and models for Kirchberg algebras

本文构造了首个非内可amenable且非部分作用变换群oid的例子，解决了Anantharaman--Delaroche和Exel的问题，并展示了Deaconu--Renault群oid不来源于离散群的部分作用，同时提供了变换群oid模型。

Kirchberg--Wassermann exactness vs exactness: reduction to the unimodular totally disconnected case

We show that in order to prove that every second countable locally compact groups with exact reduced group C*-algebra is exact in the dynamical sense (i.e. KW-exact) it suffices to show this for totally disconnected groups.

Deaconu-Renault 群胚中的轨道破缺及 UCT Kirchberg 代数的模型

Orbit-breaking in Deaconu-Renault groupoids and models for UCT Kirchberg algebras

本文将轨道破缺推广到满射局部同胚，构造出开子群胚，其C*-代数为UCT Kirchberg代数，并给出到O_2的嵌入，实现所有稳定UCT Kirchberg代数及K-群同态诱导的*-同态。

A quantum complexity approach to the Kirchberg Embedding Problem

The Kirchberg Embedding Problem (KEP) asks if every C*-algebra embeds into an ultrapower of the Cuntz algebra $\cal O_2$. Motivated by the recent refutation of the Connes Embedding Problem using the quantum complexity result MIP*=RE, we establish two quantum complexity consequences of a positive solution to KEP. Both results involve almost-commuting strategies to nonlocal games.

KK-theory of circle actions with the Rokhlin property

We investigate the structure of circle actions with the Rokhlin property, particularly in relation to equivariant $KK$-theory. Our main results are $\mathbb{T}$-equivariant versions of celebrated results of Kirchberg: any Rokhlin action on a separable, nuclear C*-algebra is $KK^\mathbb{T}$-equivalent to a Rokhlin action on a Kirchberg algebra; any Rokhlin action on an exact separable C*-algebra embeds equivariantly into $\mathcal{O}_2$ (with its unique Rokhlin action); and two circle actions with the Rokhlin property on a Kirchberg algebra are conjugate if and only if they are $KK^\mathbb{T}$-…

具有Rokhlin性质的有限群的异常作用的分类

Classification of anomalous actions of finite groups with the Rokhlin property

该研究将Rokhlin G作用的分类推广到异常G作用情形，明确Kirchberg代数上带Rokhlin性质的G核的分类依据，还探究了相关K理论存在的障碍。

Automorphisms of Cuntz-Krieger algebras

We prove that the natural homomorphism from Kirchberg's ideal-related KK-theory, KKE(e, e'), with one specified ideal, into Hom_Λ (\underline{K}_{E} (e), \underline{K}_{E} (e')) is an isomorphism for all extensions e and e' of separable, nuclear C*-algebras in the bootstrap category N with the K-groups of the associated cyclic six term exact sequence being finitely generated, having zero exponential map and with the K_{1}-groups of the quotients being free abelian groups. This class includes all Cuntz-Krieger algebras with exactly one non-trivial ideal. Combining our results with the results o…

Comparison properties of asymptotically tracially approximation C*-algebras

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).

Classification of $\mathcal O_\infty$-stable $C^\ast$-algebras

I present a proof of Kirchberg's classification theorem: two separable, nuclear, $\mathcal O_\infty$-stable $C^\ast$-algebras are stably isomorphic if and only if they are ideal-related $KK$-equivalent. In particular, this provides a more elementary proof of the Kirchberg--Phillips theorem which is isolated in the paper to increase readability of this important special case.

A few remarks on exact C(X)-algebras

We extend in this paper several results of E. Kirchberg, S. Wassermann and the author dealing with continuous fields of C*--algebras to the semi-continuous case. We provide a new characterisation of separable lower semi-continuity C*--bundles and we present exactness criteria for C(X)-algebras and unital bisimplifiable Hopf \cst-algebras.

Finite group actions on certain stably projectionless C*-algebras with the Rohlin property

We introduce the Rohlin property and the approximate representability for finite group actions on stably projectionless C*-algebras and study their basic properties. We give some examples of finite group actions on the Razak-Jacelon algebra and show some classification results of these actions. This study is based on the work of Izumi, Robert's classification theorem and Kirchberg's central sequence C*-algebras.

A class of $C^*$-algebras generalizing both graph algebras and homeomorphism $C^*$-algebras IV, pure infiniteness

This is the final one in the series of papers where we introduce and study the $C^*$-algebras associated with topological graphs. In this paper, we get a sufficient condition on topological graphs so that the associated $C^*$-algebras are simple and purely infinite. Using this result, we give one method to construct all Kirchberg algebras as $C^*$-algebras associated with topological graphs.

互反Cuntz--Krieger代数

Reciprocal Cuntz--Krieger algebras

本文通过K-理论对偶构造Kirchberg代数的互反对偶代数，具体实现Cuntz--Krieger代数的互反代数，并证明其自同构群基本群同构且保持规范作用。

保卡坦 *-自同构群：紧致阿贝尔群的实现与障碍

Cartan-preserving *-automorphism groups: realization and obstructions for compact abelian groups

研究 C*-代数中保卡坦子代数的自同构群，通过扭曲 étale 群胚描述，得到相关定理类似物。研究卡坦固定自同构群，证明 UCT 基尔希贝格代数的相关性质，及扩张有效群胚产生的 C*-代数的性质，还给出不等价卡坦子代数存在性。

Tsirelson's problem and Kirchberg's conjecture

Tsirelson's problem asks whether the set of nonlocal quantum correlations with a tensor product structure for the Hilbert space coincides with the one where only commutativity between observables located at different sites is assumed. Here it is shown that Kirchberg's QWEP conjecture on tensor products of C*-algebras would imply a positive answer to this question for all bipartite scenarios. This remains true also if one considers not only spatial correlations, but also spatiotemporal correlations, where each party is allowed to apply their measurements in temporal succession; we provide an ex…

Weak Exactness for C*-algebras and Application to Condition (AO)

We generalize Kirchberg's weak exactness to inclusions of C*-algebras in von Neumann algebras and study some characterizations and permanence properties which are similar to those of exact groups. We then consider a similar condition to Ozawa's condition (AO) with our weak exactness and generalize Ozawa's theorem for bi-exact groups. As a corollary, we give new examples of prime factors.

$C^*$-algebras with finite complexity

Complexity rank for $C^*$-algebras was introduced by the second author and Yu for applications towards the UCT: very roughly, this rank is at most $n$ if you can repeatedly cut the $C^*$-algebra in half at most $n$ times, and end up with something finite dimensional. In this paper, we study complexity rank, and also a weak complexity rank that we introduce; having weak complexity rank at most one can be thought of as `two-colored local finite-dimensionality'. We first show that for separable, unital, and simple $C^*$-algebras, weak complexity rank one is equivalent to the conjunction of nuclea…