Nathanial P. Brown
We give a detailed survey of the theory of quasidiagonal C*-algebras. The main structural results are presented and various functorial questions around quasidiagonality are discussed. In particular we look at what is currently known (and not known) about extensions, quotients, tensor products, ect. of quasidiagonal C*-algebras. We also point out how quasidiagonality is connected to some important open problems.
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We prove that the crossed product A x G of a separable, unital, quasidiagonal C*- algebra A by a discrete, countable, amenable, maximally almost periodic group G is quasidiagonal, provided that the action is almost periodic.
Spectral triples (of compact type) are constructed on arbitrary separable quasidiagonal C*-algebras. On the other hand an example of a spectral triple on a non-quasidiagonal algebra is presented.
I give an overview of recent developments in the structure and classification theory of separable, simple, nuclear C*-algebras. I will in particular focus on the role of quasidiagonality and amenability for classification, and on the regularity conjecture and its interplay with internal and external approximation properties.
We characterise quasidiagonality of the $C^*$-algebra of a cofinal $k$-graph in terms of an algebraic condition involving the coordinate matrices of the graph. This result covers all simple $k$-graph $C^*$-algebras. In the special case of cofinal $2$-graphs we further prove that AF-embeddability, quasidiagonality and stable finiteness of the $2$-graph algebra are all equivalent.
We study lifting properties for full product C*-algebras with amalgamation over ${\mathbb C}1$ and give new proofs for some results of Kirchberg and Pisier. We extend the result of Choi on the quasidiagonality of $C^*({\mathbb F}_n)$, proving that the free product with amalgamation over ${\mathbb C}1$ of a family of unital quasidiagonal C*-algebras is quasidiagonal.
In this paper, we show that a completely positive linear map is weakly nuclear if and only if its complexification is weakly nuclear. It is shown that a real $C^*$-algebra is exact if and only if its complexification is exact and similar case is provided for the quasidiaginality.
We prove that faithful traces on separable and nuclear C*-algebras in the UCT class are quasidiagonal. This has a number of consequences. Firstly, by results of many hands, the classification of unital, separable, simple and nuclear C*-algebras of finite nuclear dimension which satisfy the UCT is now complete. Secondly, our result links the finite to the general version of the Toms-Winter conjecture in the expected way and hence clarifies the relation between decomposition rank and nuclear dimension. Finally, we confirm the Rosenberg conjecture: discrete, amenable groups have quasidiagonal C*-…
In this paper we identify QD(A,B), the quasidiagonal classes in KK_1(A,B), in terms of K_*(A) and K_*(B), and we use these results in various applications. Here is our central result. Theorem: Suppose that A is in the category of separable nuclear C^*-algebras which satisfy the UCT and A is quasidiagonal relative to B. Then there is a natural isomorphism QD(A,B) = Pext (K_*(A), K_*(B))_0 . Thus quasidiagonality of KK-classes is indeed a topological invariant. We give several applications. Finally, we establish a converse to a theorem of Davidson, Herrero, and Salinas, giving conditions under w…
We introduce weakenings of two of the more prominent open problems in the classification of $\mathrm{C}^*$-algebras, namely the quasidiagonality problem and the UCT problem. We show that the a positive solution of the conjunction of the two weaker problems implies a positive solution of the original quasidiagonality problem as well as allows us to give a local, finitary criteria for the MF problem, which asks whether every stably finite $\mathrm{C}^*$-algebra is MF.
The subject of quasidiagonality is of much interest in many places - among other things, in the classification program for simple unital separable nuclear C*-algebras. In this note, we give two characterizations of nuclearity and quasidiagonality (for simple unital separable C*-algebras). Our first characterization is the nuclear analogue of Dadarlat's characterization of exact quasidiagonal C*-algebras. Our second characterizatiion is "dual" to the interesting (and important) Popa property first studied by Popa.
We characterize the solvable Lie groups of the form ${\mathbb R}^m\rtimes {\mathbb R}$, whose $C^*$-algebras are quasidiagonal. Using this result, we determine the connected simply connected solvable Lie groups of type~I whose $C^*$-algebras are strongly quasidiagonal. As a by-product, we give also examples of amenable Lie groups with non-quasidiagonal $C^*$-algebras.
Quasidiagonality and the finite section method
本文研究希尔伯特空间上的拟对角算子，利用有限截面方法（伽辽金方法）分析其收敛性，并给出收敛速度估计，对带状算子显式给出有限截面与收敛速度。
We introduce the decomposition rank, a notion of covering dimension for nuclear C^*-algebras. The decomposition rank generalizes ordinary covering dimension and has nice permanence properties; in particular, it behaves well with respect to direct sums, quotients, inductive limits, unitization and quasidiagonal extensions. Moreover, it passes to hereditary subalgebras and is invariant under stabilization. It turns out that the decomposition rank can be finite only for strongly quasidiagonal C^*-algebras and that it is closely related to the classification program.
We examine the question of quasidiagonality for C*-algebras of discrete amenable groups from a variety of angles. We give a quantitative version of Rosenberg's theorem via paradoxical decompositions and a characterization of quasidiagonality for group C*-algebras in terms of embeddability of the groups. We consider several notable examples of groups, such as topological full groups associated with Cantor minimal systems and Abels' celebrated example of a finitely presented solvable group that is not residually finite, and show that they have quasidiagonal C*-algebras. Finally, we study strong …
This is a survey of recent progress in the structure and classification theory of nuclear C*-algebras. In particular, I outline how the Universal Coefficient Theorem ensures a positive answer to the quasidiagonality question in the presence of faithful traces. This has strong consequences for the regularity conjecture and the classification problem for separable, simple, nuclear C*-algebras. Moreover, it entails a positive solution to Rosenberg's conjecture on quasidiagonality of reduced C*-algebras of discrete amenable groups. This note is largely based on a joint paper with Aaron Tikuisis an…
We study quasidiagonality and local reflexivity for $C^{*}$-algebras which are $C^*$-module over another $C^*$-algebra with compatible actions. We introduce and study a notion of amenability for vector valued traces.
Suppose B is an ultraproduct of finite dimensional C^\ast-algebras. We consider mapping and injectability properties for separable C^\ast-algebras into B. In the case of approximately finite C^\ast-algebras, we obtain a classification of these mappings up to inner conjugacy. Using a Theorem of Voiculescu, we show that for nuclear C^\ast-algebras injectability into an ultraproduct of finite dimensional C^\ast-algebras is equivalent to quasidiagonality.
In this note we address a question of Don Hadwin: "Which groups have strongly quasidiagonal C*-algebras?" In recent work we showed that all finitely generated virtually nilpotent groups have strongly quasidiagonal C*-algebras, while together with Carrión and Dadarlat we showed that most wreath products fail to have strongly quasidiagonal C*-algebras. These two results raised the question of whether or not strong quasidiagonality could characterize virtual nilpotence among finitely generated groups. The purpose of this note is to provide examples of finitely generated groups (in fact of the for…
I show that quasidiagonality and AF embeddability are equivalent properties for traceless $\mathrm C^\ast$-algebras and are characterised in terms of the primitive ideal space. For nuclear $\mathrm C^\ast$-algebras the same characterisation determines when Connes and Higson's $E$-theory can be unsuspended.
Deformations of Wreath Products
本文给出了可分精确 C*-代数连通性的新刻画，证明了增广理想具连通性的离散可数顺从群类在广义圈积下封闭，并探讨了非交换伯努利作用约化交叉积 C*-代数的拟对角性。
In the current article, we prove the cross product $C^*$-algebra by a Rokhlin action of finite group on a strongly quasidiagonal $C^*$-algbra is strongly quasidiagonal again. We also show that a just-infinite $C^*$-algebra is quasidiagonal if and only if it is inner quasidiagonal. Finally, we compute the topological free entropy dimension in just-infinite $C^*$-algebras.
It is shown that a separable C*-algebra is inner quasidiagonal if and only if it has a separating family of quasidiagonal irreducible representations. As a consequence, a separable C*-algebra is a strong NF algebra if and only if it is nuclear and has a separating family of quasidiagonal irreducible representations. We also obtain some permanence properties of the class of inner quasidiagonal C*-algebras.
Recently, it was proved by Tikuisis, White and Winter that any faithful trace on a separable, nuclear $C^\ast$-algebras in the UCT class is quasidiagonal. Building on their work, we generalise the result, and show that any faithful, amenable trace on a separable, exact $C^\ast$-algebras in the UCT class is quasidiagonal. We also prove that any amenable trace on a separable, exact, quasidiagonal $C^\ast$-algebra in the UCT class is quasidiagonal.
In the paper, we give two new characterizations of separable inner quasidiagonal C*-algebras. Base on these characterizations, we show that a unital full free product of two inner quasidiagonal C*-algebras is inner quasidiagonal again. As an application, we show that a unital full free product of two inner quasidiagoanl C*-algebras with amalgmation over a full matrix algebra is inner quasidiagonal. Meanwhile, we conclude that a unital full free product of two AF algebras with amalgamation over a finite-dimensional C*-algebra is inner quasidiagonal if there are faithful tracial states on each o…
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 study the validity of the Blackadar-Kirchberg conjecture for extensions of separable, nuclear, quasidiagonal $C^*$-algebras that satisfy the UCT. More specifically, we show that the conjecture for the extension has an affirmative answer if the ideal lies in a class of $C^*$-algebras that is closed under local approximations and contains all separable ASH algebras, as well as certain classes of simple, unital $C^*$-algebras and crossed products of unital $C^*$-algebras with the integers.
Let T be a quasidiagonal operator on a separable Hilbert space. Then T is the norm limit of operators, each of which generate a finite dimensional C*-algebra, if and only if the C*-algebra generated by T is exact.
We study the extension problem for quasidiagonal (QD) C*-algebras (i.e. when is an extension of QD C*-algebras again QD?). The main positive result states that in many instances an extension will remain QD provided that a certain boundary arising from the K-theory of the extension vanishes. A K-theoretic Hahn-Banach type property is also introduced for QD C*-algebras. We show that every nuclear QD C*-algebra has this Hahn-Banach property if and only if the extension problem for many QD C*-algebras is completely determined by a boundary map on K-theory.
Let $A$ be an amenable separable \CA and $B$ be a non-unital but $σ$-unital simple \CA with continuous scale. We show that two essential extensions $τ_1$ and $τ_2$ of $A$ by $B$ are approximately unitarily equivalent if and only if $$ [τ_1]=[τ_2] {\rm in} KL(A, M(B)/B). $$ If $A$ is assumed to satisfy the Universal Coefficient Theorem, there is a bijection from approximate unitary equivalence classes of the above mentioned extensions to $KL(A, M(B)/B).$ Using $KL(A, M(B)/B),$ we compute exactly when an essential extension is quasidiagonal. We show that quasidiagonal extensions may not be appro…
In the paper, we consider the question whether a unital full amalgamated free product of quasidiagonal C*-algebras is quasidiagonal again. We give a sufficient condition such that a unital full amalgamated free product of quasidiagonal C*-algebras with amalgamation over a finite dimensional C*- algebra is quasidiagonal. Applying this result, we conclude that a unital full free product of two AF algebras with amalgamation over a finite-dimensional C*-algebra is AF if there are faithful tracial states on each of these two AF algebras such that the restrictions on the common subalgebra agree.
A note on when amenable traces are quasidiagonal
本文证明了带有忠实顺从迹的可分精确$C^*$-代数，其所有顺从迹均为拟对角的性质在同伦下保持不变。
The hyperfinite II_1 factor is not quasidiagonal
通过构造一个 MF C*-代数 A，证明其与超有限 II_1 因子的张量积含真等距，从而表明稳定有限性不保持张量积，且超有限 II_1 因子非拟对角。
A quasidiagonal $C^*$-algebra with a nonquasidiagonal maximal tensor square
构造一个可分单幺剩余有限维 $C^*$-代数 $A$，其极大张量平方含真等距，从而 $A$ 拟对角线而 $A\ max A$ 非拟对角线，通过性质 $(T)$ 子群与膨胀共轭实现。
Quasidiagonal traces need not form a face
本文构造了一个可分单位剩余有限维C*-代数，其拟对角迹不构成迹态空间的面，通过加权表示阶梯连接两个副本并证明拟对角迹对两副本等权。
We continue the study of $\mathcal{OL}_\infty$ structure of nuclear $C^*$-algebras initiated by Junge, Ozawa and Ruan. In particular, we prove if $\mathcal{OL}_\infty(A)<1.005,$ then $A$ has a separating family of irreducible, stably finite representations. As an application we give examples of nuclear, quasidiagonal $C^*$-algebras $A$ with $\mathcal{OL}_\infty(A)>1.$
We show that the group C*-algebra of any elementary amenable group is quasidiagonal. This is an offspring of recent progress in the classification theory of nuclear C*-algebras.
We give characterizations of quasitriangular operator algebras along the line of Voiculescu's characterization of quasidiagonal $C^*$-algebras.
Let A be a unital separable simple C*-algebra with a unique tracial state. We prove that if A is nuclear and quasidiagonal, then A tensored with the universal UHF-algebra has decomposition rank at most one. Then it is proved that A is nuclear, quasidiagonal and has strict comparison if and only if A has finite decomposition rank. For such A, we also give a direct proof that A tensored with a UHF-algebra has tracial rank zero. Applying this characterization, we obtain a counter-example to the Powers-Sakai conjecture.
A trace on a C*-algebra is amenable (resp. quasidiagonal) if it admits a net of completely positive, contractive maps into matrix algebras which approximately preserve the trace and are approximately multiplicative in the 2-norm (resp. operator norm). Using that the double commutant of a nuclear C*-algebras is hyperfinite, it is easy to see that traces on nuclear C*-algebras are amenable. A recent result of Tikuisis, White, and Winter shows that faithful traces on separable, nuclear C*-algebras in the UCT class are quasidiagonal. We give a new proof of this result using the extension theory of…
Finite dimensional approximations for Hilbert space operators and applications in Quantum Mechanics
本研究建立希尔伯特空间线性算子拟对角与Følner型逼近的统一框架，将其推广至无界稠定可闭算子并拓展经典结论，引入稀疏Følner序列并明确其与拟对角逼近的关联，通过加权移位示例验证且应用于含Weyl代数的量子力学模型。
We analyze the dichotomy amenable/paradoxical in the context of (discrete, countable, unital) semigroups and corresponding semigroup rings. We consider also Følner's type characterizations of amenability and give an example of a semigroup whose semigroup ring is algebraically amenable but has no Følner sequence. In the context of inverse semigroups $S$ we give a characterization of invariant measures on $S$ (in the sense of Day) in terms of two notions: $domain$ $measurability$ and $localization$. Given a unital representation of $S$ in terms of partial bijections on some set $X$ we define a n…
The present article is a review of recent developments concerning the notion of Følner sequences both in operator theory and operator algebras. We also give a new direct proof that any essentially normal operator has an increasing Følner sequence $\{P_n\}$ of non-zero finite rank projections that strongly converges to 1. The proof is based on Brown-Douglas-Fillmore theory. We use Følner sequences to analyze the class of finite operators introduced by Williams in 1970. In the second part of this article we examine a procedure of approximating any amenable trace on a unital and separable C*-alge…
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