Hui Li
We pose a conjecture on the K-theory of the self-similar $k$-graph C*-algebra of a standard product of odometers. We generalize the C*-algebra $\mathcal{Q}_S$ to any subset of $\mathbb{N}^\times \setminus \{1\}$ and then realize it as the self-similar $k$-graph C*-algebra of a standard product of odometers.
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In this paper, we study the boundary quotient C*-algebras associated to products of odometers. One of our main results shows that the boundary quotient C*-algebra of the standard product of $k$ odometers over $n_i$-letter alphabets ($1\le i\le k$) is always nuclear, and that it is a UCT Kirchberg algebra if and only if $\{\ln n_i: 1\le i\le k\}$ is rationally independent, if and only if the associated single-vertex $k$-graph C*-algebra is simple. To achieve this, one of our main steps is to construct a topological $k$-graph such that its associated Cuntz-Pimsner C*-algebra is isomorphic to the…
Finite Blaschke Symbols and the $K$-Theory of Scalar Odometer $C^*$-Algebras
研究标量里程表\(C^*\) - 代数\(\mathcal A_\xi\)，证明其含紧算子，等距标量符号下\(W_\xi\)是Fredholm的条件，以及其在Calkin代数中像与里程表边界商同构等，得出不同次数有限Blaschke符号生成非同构\(C^*\) - 代数的结论。
We describe both the Bunce-Deddens C*-algebras and their Toeplitz versions, as crossed products of commutative C*-algebras by partial automorphisms. In the latter case, the commutative algebra has, as its spectrum, the union of the Cantor set and a copy the set of natural numbers N, fitted together in such a way that N is an open dense subset. The partial automorphism is induced by a map that acts like the odometer map on the Cantor set while being the translation by one on N. From this we deduce, by taking quotients, that the Bunce-Deddens C*-algebras are isomorphic to the (classical) crossed…
The external Kasparov product is used to construct odd and even spectral triples on crossed products of $C^*$-algebras by actions of discrete groups which are equicontinuous in a natural sense. When the group in question is $\Z$ this gives another viewpoint on the spectral triples introduced by Belissard, Marcolli and Reihani. We investigate the properties of this construction and apply it to produce spectral triples on the Bunce-Deddens algebra arising from the odometer action on the Cantor set and some other crossed products of AF-algebras.
Homology and twisted C*-algebras for self-similar actions and Zappa-Szép products
本文研究自相似作用对应的小范畴Zappa-Szép积的范畴同调，证明其可由双复形谱序列计算，并据此定义扭曲C*-代数并证明其基本结构性质。
Finitely Summable $K$-homology, the Index Pairing, and Cantor Minimal Systems
本文研究 Cantor 极小系统交叉积 $C^*$-代数的指标配对，利用 Putnam 轨道破缺 AF-子代数证明可通过 Connes 迹公式计算任意配对，并证明里程计代数具有一致有限可和 $K$-同调。
Let $M=M_1*M_2$ be a nontrivial tracial free product of finite von Neumann algebras. We prove that any amenable subalgebra of $M$ that has a diffuse intersection with $M_1$ is in fact contained in $M_1$. This has been proved by C. Houdayer under more general circumstances.
We analyze the algebraic structure of the Connes fusion tensor product (CFTP) in the case of bi-finite Hilbert modules over a von Neumann algebra M. It turns out that all complications in its definition disappear if one uses the closely related bi-modules of bounded vectors. We construct an equivalence of monoidal categories with duality between a category of Hilbert bi-modules over M with CFTP and some natural category of bi-modules over M with the usual relative algebraic tensor product. The results are surely well-known to the experts. We hope that the exposition and presentation is useful …
Let $(G, Λ)$ be a self-similar $k$-graph with a possibly infinite vertex set $Λ^0$. We associate a universal C*-algebra $\mathcal{O}_{G,Λ}$ to $(G,Λ)$. The main purpose of this paper is to investigate the ideal structures of $\mathcal{O}_{G,Λ}$. We prove that there exists a one-to-one correspondence between the set of all $G$-hereditary and $G$-saturated subsets of $Λ^0$ and the set of all gauge-invariant and diagonal-invariant ideals of $\mathcal{O}_{G,Λ}$. Under some conditions, we characterize all primitive ideas of $\mathcal{O}_{G,Λ}$. Moreover, we describe the Jacobson topology of some co…
Some remarks on Reduced $C^*$-algebras of semigroup dynamical systems and product systems
研究半群动力系统的简化的交叉积和乘积系统的简化的$C^*$-代数的精确性，证明在合理假设下，简化的交叉积精确当且仅当原代数精确。
Using an inequality due to Ricard and Xu, we give a different proof of Paul Skoufranis's recent result showing that the strong convergence of possibly non-commutative random variables $X^{(k)}\to X$ is stable under reduced free product with a fixed non-commutative random variable $Y$. In fact we obtain a more general fact: assuming that the families $X^{(k)}=\{X_i^{(k)}\}$ and $Y^{(k)}=\{Y_j^{(k)}\}$ are $*$-free as well as their limits (in moments) $X =\{X_i \}$ and $Y =\{Y_j\}$, the strong convergences $X^{(k)}\to X$ and $Y^{(k)}\to Y$ imply that of $\{X^{(k)},Y^{(k)} \}$ to $\{X ,Y\}$. Phra…
It is shown that projectionless C*-algebras that tensorially absorb the Jiang-Su algebra have the property that every element is a limit of products of two nilpotents. This is then used to classify the approximate unitary equivalence classes of positive elements in such C*-algebras using traces.
Well known results related to the compactness of Hankel operators of one complex variable are extended to little Hankel operators of two complex variables. Critical to these considerations is the result of Ferguson and Lacey characterizing the boundedness of the little Hankel operators in terms of the product BMO of S.-Y. Chang and R. Fefferman.
Semigroups of self-similar actions and higher rank Baumslag-Solitar semigroups
本文开创高阶Baumslag-Solitar半群及其C*-代数研究，刻画两类极端情形的von Neumann代数因子性与Cartan子代数，并利用自相似高阶图工具推广秩1情形。
Remarks on Villadsen algebras, II: A generalized construction and the comparison radius function
本文扩展了Villadsen代数的分类方法，探讨了Bratteli AF代数的广义构造，并研究了不同种子空间对代数区分的影响。
In this informal expository note, we present a universal, formulaic construction of the free product of rigid $C^*$-2-categories. This construction allows for a straightforward, purely categorical formulation of the free composition of subfactors and planar algebras considered by Bisch and Jones. As an application, we explain the results of Tarrago and Wahl on free wreath products of compact quantum groups in this categorical language.
Some remarks on decay in countable groups and amalgamated free products
本文研究可数群关于长度函数的次指数衰减概念，推广快速衰减性质，给出具有SD但不具有RD的群例，并证明融合自由积中衰减永存定理的最优性。
We show that the mathematical structures in a recent work of Bultinck-Mariena-Williamson-Sahinoglu-Haegemana-Verstraete are the same as those of flat symmetric bi-unitary connections and the tube algebra in subfactor theory. More specifically, a system of flat symmetric bi-unitary connections arising from a subfactor with finite index and finite depth satisfies all their requirements for tensors and the tube algebra for such a subfactor and the anyon algebra for such tensors are isomorphic up to the normalization constants. Furthermore, the matrix product operator algebras arising from tensors…
A remark on non-commutative $L^p$-spaces
本文显式刻画了张量积 von Neumann 代数对应的 Haagerup 与 Kosaki 非交换 $L^p$ 空间，并说明其在算子代数量子信息理论中的用途。
We study non-commutative algebraic geometry of Artin, Serre and Tate in terms of the operator algebras. Namely, we define the Serre $C^*$-algebra $\mathcal{A}_X$ of a projective variety $X$ as the norm-closure of a representation of the twisted homogeneous coordinate ring of $X$ by the linear operators on a Hilbert space $\mathcal{H}$. It is proved that $X$ is homeomorphic to the space of all irreducible representations of the crossed product of $\mathcal{A}_X$ by an automorphism of $\mathcal{A}_X$. The case of rational elliptic curves $X$ is considered in detail.
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