Andrew S. Toms
We observe almost divisibility for the original Cuntz semigroup of a simple AH algebra with strict comparison. As a consequence, the properties of strict comparison, finite nuclear dimension, and Z-stability are equivalent for such algebras, confirming partially a conjecture of Winter and the author.
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We construct Cartan subalgebras in all classifiable stably finite C*-algebras. Together with known constructions of Cartan subalgebras in all UCT Kirchberg algebras, this shows that every classifiable simple C*-algebra has a Cartan subalgebra.
It is shown that the coloured isomorphism class of a unital, simple, $\mathcal{Z}$-stable, separable amenable C$^*$-algebra satisfying the Universal Coefficient Theorem (UCT) is determined by its tracial simplex.
We show that certain extensions of classifiable C*-algebra are strongly classified by the associated six-term exact sequence in K-theory together with the positive cone of K_{0}-groups of the ideal and quotient. We apply our result to give a complete classification of graph C*-algebras with exactly one ideal.
In this paper, we prove that unital homomorphisms from continuous functions on a compact metric space to matrices over a C*-algebra with tracial rank at most one are approximately diagonalizable. We also consider some generalizations of this result.
Given a simple, acyclic dimension group $G_{0}$ and countable, torsion-free, abelian group $G_{1}$, we construct a minimal, amenable, étale equivalence relation $R$ on a Cantor set whose associated groupoid $C^{*}$-algebra, $C^{*}(R)$, is tracially AF, and hence classifiable in the Elliott classification scheme for simple, amenable, separable $C^{*}$-algebras, and with $K_{*}(C^{*}(R)) \cong(G_{0}, G_{1})$.
We classify extensions of certain classifiable C*-algebras using the six term exact sequence in K-theory together with the positive cone of the K_0-groups of the distinguished ideal and quotient. We then apply our results to a class of C*-algebras arising from substitutional shift spaces.
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.
Exotic circle actions on classifiable C*-algebras
本文构造了一个单核可分单位稳定有限Z-稳定C*-代数上的圆作用，其交叉积简单但非Z-稳定，展示了奇异圆作用的例子。
We study the $H^3$ invariant of a group homomorphism $ϕ:G \rightarrow \mathrm{Out}(A)$, where $A$ is a classifiable C$^*$-algebra. We show the existence of an obstruction to possible $H^3$ invariants arising from considering the unitary algebraic $K_1$ group. In particular, we prove that when $A$ is the Jiang--Su algebra $\mathcal{Z}$ this invariant must vanish. We deduce that the unitary fusion categories $\mathrm{Hilb}(G, ω)$ for non-trivial $ω\in H^3(G, \mathbb{T})$ cannot act on $\mathcal{Z}$.
The admissible KMS bundles on classifiable C$^*$-algebras
本文证明了在可分类 C$^*$-代数和 UCT Kirchberg 代数上，对于满足特定拓扑条件的单纯形丛，均存在实现该单纯形的流。
Irreducible representations of nilpotent groups generate classifiable C*-algebras
本文证明有限生成幂零群不可约表示生成的 C*-代数满足 UCT，并结合既有结果确立其按 Elliott 不变量的可分类性，同时刻画为同伦平凡余循环扭曲群 C*-代数的中心截断。
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）。
Classifiable C*-algebras have quantum symmetries
本文证明可分类简单C*-代数（如Jiang-Su代数）具有离散量子群外作用，并由此构造新的超有限子因子及SU_q(2)的极小作用。
We prove that the automorphisms of any separable C*-algebra that does not have continuous trace are not classifiable by countable structures up to unitary equivalence. This implies a dichotomy for the Borel complexity of the relation of unitary equivalence of automorphisms of a separable unital C*-algebra: Such relation is either smooth or not even classifiable by countable structures.
We construct C*-diagonals with connected spectra in all classifiable stably finite C*-algebras which are unital or stably projectionless with continuous scale. For classifiable stably finite C*-algebras with torsion-free $K_0$ and trivial $K_1$, we further determine the spectra of the C*-diagonals up to homeomorphism. In the unital case, the underlying space turns out to be the Menger curve. In the stably projectionless case, the space is obtained by removing a non-locally-separating copy of the Cantor space from the Menger curve. We show that each of our classifiable C*-algebras has continuum…
We study the range of a classifiable class ${\cal A}$ of unital separable simple amenable $C^*$-algebras which satisfy the Universal Coefficient Theorem. The class ${\cal A}$ contains all unital simple AH-algebras. We show that all unital simple inductive limits of dimension drop circle $C^*$-algebras are also in the class. This unifies some of the previous known classification results for unital simple amenable $C^*$-algebras. We also show that there are many other $C^*$-algebras in the class. We prove that, for any partially ordered, simple weakly unperforated rationally Riesz group $G_0$ wi…
Free actions of polynomial growth Lie groups and classifiable C*-algebras
本文证明了多项式增长的连通Lie群在有限维局部紧空间上的自由作用具有有限管维数，进而得出相关交叉积C*-代数具有有限核维数，并证明了与某些非周期点集相关的C*-代数是可分类的。
Let ${\cal A}_1$ be the class of all unital separable simple $C^*$-algebras $A$ such that $A\otimes U$ has tracial rank at most one for all UHF-algebras of infinite type. It has been shown that amenable ${\cal Z}$-stable $C^*$-algebras in ${\cal A}_1$ which satisfy the Universal Coefficient Theorem can be classified up to isomorphism by the Elliott invariant. We show that $A\in {\cal A}_1$ if and only if $A\otimes B$ has tracial rank at most one for one of unital simple infinite dimensional AF-algebra $B.$ In fact, we show that $A\in {\cal A}_1$ if and only if $A\otimes B\in {\cal A}_1$ for so…
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。
We study the complexity of the classification problem for Cartan subalgebras in von Neumann algebras. We construct a large family of II$_1$ factors whose Cartan subalgebras up to unitary conjugacy are not classifiable by countable structures, providing the first such examples. Additionally, we construct examples of II$_1$ factors whose Cartan subalgebras up to conjugacy by an automorphism are not classifiable by countable structures. Finally, we show directly that the Cartan subalgebras of the hyperfinite II$_1$ factor up to unitary conjugacy are not classifiable by countable structures, and d…
We prove that a simple, separable, nuclear, purely infinite classifiable $C^*$-algebra is weakly semiprojective if and only if its $K$-groups are direct sums of cyclic groups.
It is proved that classifiable simple separable nuclear purely infinite C*-algebras having finitely generated K-theory and torsion-free K_1 are semiprojective. This is accomplished by exhibiting these algebras as C*-algebras of infinite directed graphs.
In this paper we consider the question of what abelian groups can arise as the $K$-theory of $\mathrm{C}^*$-algebras arising from minimal dynamical systems. We completely characterize the $K$-theory of the crossed product of a space $X$ with finitely generated $K$-theory by an action of the integers and show that crossed products by a minimal homeomorphisms exhaust the range of these possible $K$-theories. Moreover, we may arrange that the minimal systems involved are uniquely ergodic, so that their $\mathrm{C}^*$-algebras are classified by their Elliott invariants. We also investigate the $K$…
We prove that simple, separable, monotracial UHF $L^{p}$-operator algebras are not classifiable up to (complete) isomorphism using countable structures, such as K-theoretic data, as invariants. The same assertion holds even if one only considers UHF $L^{p}$-operator algebras of tensor product type obtained from a diagonal system of similarities. For $p=2$, it follows that separable nonselfadjoint UHF operator algebras are not classifiable by countable structures up to (complete) isomorphism. Our results, which answer a question of N. Christopher Phillips, rely on Borel complexity theory, and p…
Metrics on trace spaces
本文针对实秩零、稳定秩一的可分类$\text{C}^*$-代数，用最优输运方法给出了$^*$-同态酉轨道距离的迹数据计算方式，还利用迹Lipschitz结构表征了自同态的统计特征。
Quantum metric Choquet simplices
本文提出具备Riesz插值性质的量子度量Choquet单形概念，给出其C*代数构造、两类实例及连续场相关连续性结论，可用于稳定有限可分类C*代数范畴的距离度量。
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。
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.
Applications of $\mathrm{C}^*$-classification
本文探讨了可分类C*代数的背景及其在分类中的应用，重点在于核维度和通用系数定理的代数结构。
Cuntz--Pimsner algebras of partial automorphisms twisted by vector bundles II: Nuclear dimension
本文证明最小作用且底空间紧、无限、有限覆盖维数时，向量丛扭曲的部分自同构的 Cuntz--Pimsner 代数在 Elliott 程序下可分类，并建立迹与共形测度的双射，推广了部分交叉积及破轨子代数的相关结果。
Let A be a unital separable C*-algebra. We observe that A is type I if and only if the CNT-entropy of every inner automorphism of A is zero.
Consider a minimal free topological dynamical system $(X, T, \mathbb{Z}^d)$. It is shown that the comparison radius of the crossed product C*-algebra $\mathrm{C}(X) \rtimes \mathbb{Z}^d$ is at most the half of the mean topological dimension of $(X, T, \mathbb{Z}^d)$. As a consequence, the C*-algebra $\mathrm{C}(X) \rtimes \mathbb{Z}^d$ is classifiable if $(X, T, \mathbb{Z}^d)$ has zero mean dimension.
We prove that if two nonzero homomorphisms from the Cuntz algebra O_infinity to a purely infinite simple C*-algebra have the same class in KK-theory, and if either both are unital or both are nonunital, then they are approximately unitarily equivalent. It follows that O_infinity is classifiable in the sense of Rordam. In particular, Rordam's classification theorem for direct limits of matrix algebras over even Cuntz algebras extends to direct limits involving both matrix algebras over even Cuntz algebras and corners of O_infinity for which the K_0 group can be an arbitrary countable abelian gr…
We establish $\mathcal{Z}$-stability for crossed products of outer actions of amenable groups on $\mathcal{Z}$-stable $C^*$-algebras under a mild technical assumption which we call McDuff property with respect to invariant traces. We obtain such result using a weak form of dynamical comparison, which we verify in great generality. We complement our results by proving that McDuffness with respect to invariant traces is automatic in many cases of interest. This is the case, for instance, for every action of an amenable group $G$ on a classifiable $C^*$-algebra $A$ whose trace space $T(A)$ is a B…
Let $\mathcal{A}$ be a $*$-algebra and $\mathcal{M}$ be a $*$-$\mathcal A$-bimodule, we study the local properties of $*$-derivations and $*$-Jordan derivations from $\mathcal{A}$ into $\mathcal{M}$ under the following orthogonality conditions on elements in $\mathcal A$: $ab^*=0$, $ab^*+b^*a=0$ and $ab^*=b^*a=0$. We characterize the mappings on zero product determined algebras and zero Jordan product determined algebras. Moreover, we give some applications on $C^*$-algebras, group algebra, matrix algebras, algebras of locally measurable operators and von Neumann algebras.
Generic classification of the quasi-free flows on the Cuntz algebra $\mathcal{O}_2$
本文研究 Cuntz 代数上拟自由流的一般分类，证明 $\mathcal{O}_2$ 情形可由唯一 KMS 态的逆温度分类，并建立 $\mathcal{O}_n$ 上一类流的一般等变 $\mathcal{Z}$-稳定性及对偶作用的 Rokhlin 性质。
We address the classification problem for graph $C^*$-algebras of finite graphs (finitely many edges and vertices), containing the class of Cuntz-Krieger algebras as a prominent special case. Contrasting earlier work, we do not assume that the graphs satisfy the standard condition (K), so that the graph $C^*$-algebras may come with uncountable ideal structures. We find that in this generality, stable isomorphism of graph $C^*$-algebras does not coincide with the geometric notion of Cuntz move equivalence. However, adding a modest condition on the graphs, the two notions are proved to be mutual…
We prove that the C*-algebra of a directed graph $E$ is liminal iff the graph satisfies the finiteness condition: if $p$ is an infinite path or a path ending with a sink or an infinite emitter, and if $v$ is any vertex, then there are only finitely many paths starting with $v$ and ending with a vertex in $p$. Moreover, C*(E) is Type I precisely when the circuits of $E$ are either terminal or transitory, i.e., $E$ has no vertex which is on multiple circuits, and $E$ satisfies the weaker condition: for any infinite path $λ$, there are only finitely many vertices of $λ$ that get back to $λ$ in an…
We give complete descriptions of the tracial states on both the universal and reduced crossed products of a C*-dynamical system consisting of a unital C*-algebra and a discrete group. In particular, we also answer the question of when the tracial states are in canonical bijection with the invariant tracial states on the original C*-algebra. This generalizes the unique trace property for discrete groups. The analysis simplifies greatly in various cases, for example when the conjugacy classes of the original group are all finite, and in other cases gives previously known results, for example whe…
Every separable nondegenerate C*-correspondence over a commutative C*-algebra with discrete spectrum is isomorphic to a graph correspondence.
The Rank Ratio Function and the Radius of Comparison
该研究针对Cuntz半群引入秩比函数与比较半径函数，前者可控制元素秩的振荡并用于计算相对比较半径，后者可构造具有不同相对比较半径的不可分类C*-代数。
We study closedness of the range, adjointability and generalized invertibility of modular operators between Hilbert modules over locally C*-algebras of coefficients. Our investigations and the recent results of M. Frank [Characterizing C*-algebras of compact operators by generic categorical properties of Hilbert C*-modules, {\it J. K-Theory} {\bf 2} (2008), 453-462] reveal a number of equivalence properties of the category of Hilbert modules over locally C*-algebras which characterize precisely the inverse limit of C*-algebras of the C*-algebra of compact operators.
Given a not-necessarily Hausdorff, topologically free, twisted étale groupoid $(G, L)$, we consider its "essential groupoid C*-algebra", denoted $C^*_{ess}(G, L)$, obtained by completing $C_c(G, L)$ with the smallest among all C*-seminorms coinciding with the uniform norm on $C_c(G^0)$. The inclusion of C*-algebras $(C_0(G^0), C^*_{ess}(G, L))$ is then proven to satisfy a list of properties characterizing it as what we call a "weak Cartan inclusion". We then prove that every weak Cartan inclusion $(A, B)$, with $B$ separable, is modeled by a topologically free, twisted étale groupoid, as above…
Let ${\mathcal N}$ be a nest on a complex Banach space $X$ and let $\mbox{ Alg}{\mathcal N}$ be the associated nest algebra. We say that an operator $Z\in \mbox{ Alg}{\mathcal N}$ is an all-derivable point of $\mbox{ Alg}{\mathcal N}$ if every linear map $δ$ from $\mbox{ Alg}{\mathcal N}$ into itself derivable at $Z$ (i.e. $δ$ satisfies $δ(A)B+Aδ(B)=δ(Z)$ for any $A,B \in \mbox{ Alg}{\mathcal N}$ with $AB=Z$) is a derivation. In this paper, it is shown that every injective operator and every operator with dense range in $\mbox{Alg}{\mathcal N}$ are all-derivable points of $\mbox{Alg}{\mathcal …
Characterizing C*-algebras of compact operators by generic categorical properties of Hilbert C*-modules
本文通过发现 Hilbert C*-模范畴中新的泛范畴性质，精确刻画了紧算子的 C*-代数，推广了 Magajna 和 Schweizer 的相关结果。
We study to what extent group $C^\ast$-algebras are characterized by their unitary groups. A complete characterization of which Abelian group $C^\ast$-algebras have isomorphic unitary groups is obtained. We compare these results with other unitary-related invariants of $C^\ast(Γ)$, such as the $K$-theoretic $K_1(C^\ast(Γ))$ and find that $C^\ast$-algebras of nonisomorphic torsion-free Abelian groups may have isomorphic $K_1$-groups, in sharp contrast with the well-known fact that $C^\ast(Γ)$ (even $Γ$) is characterized by the topological group structure of its unitary group when $Γ$ is torsion…
Characterizing convex trace ranges in finite atomic von Neumann algebras
本文研究了有限原子 von Neumann 代数中凸迹范围的刻画，通过非递增正序列的性质给出了迹范围为凸集的充分必要条件，并描述了相关集合的极值点。
In a previous paper we showed how the main theorems characterizing operator algebras and operator modules, fit neatly into the framework of the `noncommutative Shilov boundary', and more particularly via the left multiplier operator algebra of an operator space. As well as giving new characterization theorems, the approach of that paper allowed many of the hypotheses of the earlier theorems to be eliminated. Recent progress of the author with Effros and Zarikian now enables weak*-versions of these characterization theorems. For example, we prove a result analogous to Sakai's famous characteriz…
On Rieffel's conjecture characterizing a deformed algebra as Heisenberg smooth operators
本研究刻画了海森堡群典范作用下的光滑向量，证明其与左乘算子代数重合，解决了Rieffel关于形变代数的相关猜想，涉及C$^*$-代数、希尔伯特模与形变量子化。
We transfer a large part of the circle of theorems characterizing the generalization of classical $H^\infty$ known as `weak* Dirichlet algebras', to Arveson's noncommutative setting of subalgebras of finite von Neumann algebras.
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