Marius Dadarlat, Cornel Pasnicu
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.
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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.
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.
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.
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.
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。
We prove that every action of a finite group all of whose Sylow subgroups are cyclic on the K-theory of a Kirchberg algebra can be lifted to an action on the Kirchberg algebra. The proof uses a construction of Kirchberg algebras generalizing the one of Cuntz-Krieger algebras, and a result on modules over finite groups. As a corollary, every automorphism of the K-theory of a Kirchberg algebra can be lifted to an automorphism of the Kirchberg algebra with same order.
We give a construction of Kirchberg algebras from graphs. By using product graphs in the construction we are able to provide models for general (UCT) Kirchberg algebras while maintaining the explicit generators and relations of the underlying graphs.
We classify a large class of Z^2-actions on the Kirchberg algebras employing the Kasparov group KK^1 as the space of classification invariants.
We show that a Kirchberg algebra is semiprojective if and only if it is KK-semiprojective. In particular, this shows that a Kirchberg algebra in the UCT-class is semiprojective if and only if its K-theory is finitely generated, thereby giving a positive solution to a longstanding conjecture by Blackadar.
In this note we show that every Kirchberg algebra in the UCT class is the $C^{\ast}$-algebra of a Hausdorff, ample, amenable and locally contracting groupoid. The non-unital case follows from Spielberg's graph-based models for Kirchberg algebras. Our contribution is the unital case and our proof builds on Spielberg's construction.
We give a partial answer to the question for the precise value of the nuclear dimension of UCT-Kirchberg algebras raised by W. Winter and J. Zacharias. It is shown that every Kirchberg algebra in the UCT-class with torsion free $K_1$-group has nuclear dimension 1.
Toward the complete classification of poly-$\mathbb{Z}$ group actions on Kirchberg algebras, we prove several fundamental theorems that are used in the classification. In addition, as an application of them, we classify outer actions of poly-$\mathbb{Z}$ groups of Hirsch length not greater than three on unital Kirchberg algebras up to $KK$-trivial cocycle conjugacy.
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.
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。
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。
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。
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。
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.
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.
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.
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。
$C^*$-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。
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。
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。
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.
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。
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.
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.
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…
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…
$G$-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 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…
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.
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.
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…
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.
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。
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…
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.
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…
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.
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…
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…
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。
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。
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.
A locally compact group $G$ has the factorization property if the map $$C^*(G)\odot C^*(G)\ni a\otimes b\mapsto λ(a)ρ(b)\in\mathcal B(L^2(G))$$ is continuous with respect to the minimal C*-norm. This paper seeks to initiate a rigorous study of this property in the case of locally compact groups which, in contrast to the discrete case, has been relatively untouched. A partial solution to the question of when the factorization property passes to continuous embeddings is given -- a question which traces back to Kirchberg's seminal work on the topic and is known to be false in general. It is also …
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。
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.
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。
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 $KK^\mathbb{T}$-…
In this paper, we study continuous Rokhlin property of $\mathrm{C}^*$-dynamical systems using techniques of equivariant $\mathrm{KK}$-theory and quantum group theory. In particular, we determine the $\mathrm{KK}$-equivalence class and give a classification of Kirchberg $G$-algebras when the $G$ is a compact Lie group with Hodgkin condition.
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…
We introduce and study the continuous Rokhlin property for actions of compact groups on C*-algebras. An important technical result is a characterization of the continuous Rokhlin property in terms of asymptotic retracts. As a consequence, we derive strong KK-theoretical obstructions to the continuous Rokhlin property. Using these, we show that the UCT is preserved under formation of crossed products and passage to fixed point algebras by such actions, even in the absence of nuclearity. Our analysis of the KK-theory of the crossed product allows us to prove a $\mathbb{T}$-equivariant version of…
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…
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.
We answer a question of E. Kirchberg (personal communication): does the relative commutant of a separable C*-algebra in its ultrapower depend on the choice of the ultrafilter?
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.
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。
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。
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