Hannes Thiel
We show that every separable C*-algebra of real rank zero that tensorially absorbs the Jiang-Su algebra contains a dense set of generators. It follows that in every classifiable, simple, nuclear C*-algebra, a generic element is a generator.
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We compute the generator rank of a subhomgeneous C*-algebra in terms of the covering dimension of the pieces of its primitive ideal space corresponding to irreducible representations of a fixed dimension. We deduce that every Z-stable ASH-algebra has generator rank one, which means that a generic element in such an algebra is a generator. This leads to a strong solution of the generator problem for classifiable, simple, nuclear C*-algebras: a generic element in each such algebra is a generator. Examples of Villadsen show that this is not the case for all separable, simple, nuclear C*-algebras.
Generators of II_1 Factors
在2005年，Shen引入了一个新的不变量$\mathcal G(N)$，它是一个具有固定忠实迹的扩散冯诺依曼代数$N$的不变量，他使用这个不变量给出了一个统一的方法来证明大类的${\mathrm{II}}_1$因子$M$是单生成的。本文主要研究这种不变量的性质。我们将$\mathcal G(M)$与${\mathrm{II}}_1$因子$M$的自伴生成元的数目相关联：如果$\mathcal G(M)<n/2$，则$M$由$n+1$自伴算子生成，而如果$M$由$n+1$自伴算子生成，则$\mathcal G(M)\leq n/2$。不变的$\mathcal G(\cdot)$在扩增下表现良好，满足所有$t>0$的$\mathcal G(M_t)=t^{-2}\mathcal G(M)$。特别地，如果对于任何特定的$r>1$有$\mathcal G(\mathcal L\mathbb F_r)>0$，则自由群因子是成对非同构的，并且对于足够大的$r$值不是单独生成的。估计给出形成自由产品和传递到有限指数子因子和基本建设。我们还研究了一个版本的不变$\mathcal G_{\text{sa}}(M)$定义只使用自伴算子，这是证明满足$\mathcal G_{\text{sa}}(M)=2\mathcal G(M)$。最后，我们给出不等式有关的一个数量参与的计算$\mathcal G(M…
We show that every AF-algebra is generated by a single operator. This was previously unclear, since the invariant that assigns to a C*-algebra its minimal number of generators lacks natural permanence properties. In particular, it may increase when passing to ideals or inductive limits. To obtain a better behaved theory, we not only ask if a C*-algebra is generated by $n$ elements, but also if generating $n$-tuples are dense. This defines the generator rank, which we show has many natural permanence properties: it does not increase when passing to ideals, quotients or inductive limits.
In this article, we show that, if $S\in \mathcal{B}(H)$ is irreducible and essential unitary, then $\{S,SS^*\}$ is a hyperrigid generator for the unital $C^*$-algebra $\mathcal{T}$ generated by $\{S,SS^*\}$. We prove that, if $T$ is an operator in $\mathcal{B}(H)$ that generates an unital $C^*$-algebra $\mathcal{A}$ then $\{T,T^*T,TT^*\}$ is a hyperrigid generator for $\mathcal{A}$. As a corollary it follows that, if $T\in \mathcal{B}(H)$ is normal then $\{T,TT^*\}$ is hyperrigid generator for the unital $C^*$-algebra generated by $T$ and if $T\in \mathcal{B}(H)$ is unitary then $\{T\}$ is hyp…
Additive mappings in algebras of unbounded operators preserving operators of rank one
设$ \mathcal D$是希尔伯特空间$\mathcal H$中的稠密线性流形，$L^+(\mathcal D)$是所有线性算子$A$使得$A \mathcal D \subset \mathcal D, A^* \mathcal D \subset \mathcal D$的 *-代数.用$F(\mathcal D)$表示由所有有限秩算子组成的$L^+(\mathcal D)$的 *-理想。给出了保秩为1的$F(\mathcal D)$算子或保秩为1的投影的可加映射的结构的一个刻划。P. uSemrl给出了巴拿赫空间上有界算子代数的相应结果。
Symmetry in the Cuntz Algebra on two generators
我们研究了交换两个正则等距的$\mathcal{O}_{2}$的自同构的结构。我们的主要观察结果是，该作用的不动点C*-子代数同构于$\mathcal{O}_{2}$，并详细讨论了交叉积与不动点子代数之间的关系.
We use product systems of $C^*$-correspondences to introduce twisted $C^*$-algebras of topological higher-rank graphs. We define the notion of a continuous $\mathbb{T}$-valued $2$-cocycle on a topological higher-rank graph, and present examples of such cocycles on large classes of topological higher-rank graphs. To every proper, source-free topological higher-rank graph $Λ$, and continuous $\mathbb{T}$-valued $2$-cocycle $c$ on $Λ$, we associate a product system $X$ of $C_0(Λ^0)$-correspondences built from finite paths in $Λ$. We define the twisted Cuntz--Krieger algebra $C^*(Λ,c)$ to be the C…
On measuring unboundedness of the $H^\infty$-calculus for generators of analytic semigroups
通过估计映射$H^{\infty}\rightarrow \mathcal{B}(X)$：$f\mapsto f(A)T(\varepsilon)$对于$\varepsilon$在零附近的有界$b(\varepsilon)$，研究了$H^\infty$ -演算的有界性.这里，$-A$生成解析半群$T$，而$H^{\infty}$是严格包含$A$的谱的域上的有界解析函数的空间。我们表明，$b(\varepsilon)=\mathcal{O}(|\log\varepsilon|)$在一般情况下，而$b(\varepsilon)=\mathcal{O}(1)$有界结石。这推广了Vitse的一个结果，并补充了Haase和Rozendaal关于非解析半群的工作。我们讨论了我们的边界的清晰度，并表明，单平方函数估计产量$b(\varepsilon)=\mathcal{O}(\sqrt{|\log\varepsilon|})$。
Rank of a co-doubly commuting submodule is 2
证明了非平凡余双交换子模的秩是$2$ .更准确地说，假设$φ, ψ\in H^\infty(\mathbb{D})$是两个内部函数。如果$\mathcal{Q}_φ = H^2(\mathbb{D})/ φH^2(\mathbb{D})$和$\mathcal{Q}_ψ = H^2(\mathbb{D})/ ψH^2(\mathbb{D})$，则\[ \mbox{rank~}(\mathcal{Q}_φ \otimes \mathcal{Q}_ψ)^\perp = 2. \]一个直接的结果如下：设$\mathcal{S}$是$H^2(\mathbb{D}^2)$的余双交换子模。则$\mbox{rank~} \mathcal{S} = 1$当且仅当$\mathcal{S} = ΦH^2(\mathbb{D}^2)$对于某个单变量内部函数$Φ\in H^\infty(\mathbb{D}^2)$。这回答了R. G. Douglas和R.杨
The similarity of irreducible operators in factors
本文研究了可分因子中与最大阿贝尔自共轭子代数相关的生成元，探讨了可分因子中与不可约算子相似的正规算子的特征。
On the generators of quantum dynamical semigroups with invariant subalgebras
本文研究具有不变子代数的 GKLS 生成元与 CP 映射刻画问题，证明二者在弱闭 *-代数情形下本质等价，并给出原子代数情形的范式。
$K$-theory and homotopies of 2-cocycles on higher-rank graphs
本文继续研究局部紧Hausdorff广群$\mathcal{G}$上的2-上圈的同伦$ω= \{ω_t\}_{t \in [0,1]}$何时引起扭广群$C^*$ -代数的$K$ -理论群的同构问题：$K_*(C^*(\mathcal{G}, ω_0)) \cong K_*(C^*(\mathcal{G}, ω_1)).$特别是，我们建立在Kumjian，Pask，证明了如果$\mathcal{G} = \mathcal{G}_Λ$是与无源的行有限高秩图$Λ$相关联的无限路广群，并且$\{c_t\}_{t \in [0,1]}$是$Λ$上的2-上循环的同伦，则是$K_*(C^*(\mathcal{G}_Λ, σ_{c_0})) \cong K_*(C^*(\mathcal{G}_Λ, σ_{c_1})),$，其中$σ_{c_t}$表示与$Λ$上的2-上循环$c_t$相关联的$\mathcal{G}_Λ$上的2-上循环。我们还证明了一个技术性结果（定理3.3），即局部紧Hausdorff广群$\mathcal{G}$上的2-上圈同伦产生上半连续$C^*$ -丛.
In a companion paper [ABS1] we introduced the stable $\infty$-category of noncommutative CW-spectra, which we denoted $\mathtt{NSp}$. Let $\mathcal{M}$ denote the full spectrally enriched subcategory of $\mathtt{NSp}$ whose objects are the non-commutative suspension spectra of matrix algebras. In [ABS1] we proved that $\mathtt{NSp}$ is equivalent to the $\infty$-category of spectral presheaves on $\mathcal{M}$. In this paper we investigate the structure of $\mathcal{M}$, and derive some consequences regarding the structure of $\mathtt{NSp}$. To begin with, we introduce a rank filtration of $\m…
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