Homotopical Stable Ranks for Certain $C^{\ast}$-algebras Associated to Groups

纯度和稳定秩一对于某些群扩张的减少扭曲群C*代数

Pureness and stable rank one for reduced twisted group $\mathrm{C}^\ast$-algebras of certain group extensions

本文证明了具有PHP性质的离散群生成的减少扭曲群C*代数是完全自洽的，并展示了某些有限由G扩张的群的减少扭曲C*代数具有稳定秩一且为纯，从而严格比较。

$C^\ast$ -无自由子群的单群

$C^\ast$-simple groups without free subgroups

构造了第一个不含非循环自由子群的非平凡群的例子，证明了其约化$C^\ast$ -代数是单的，且迹唯一。这就回答了德·拉·哈普的一个问题。提供了扭转和无挠的例子。特别地，我们证明了秩为$m\ge 2$且奇指数$n$足够大的自由Burnside群$B(m,n)$的约化$C^\ast$ -代数是简单的，且迹唯一.

Maps preserving the local spectral subspace of skew-product of operators

Let $B(H)$ be the algebra of all bounded linear operators on an infinite-dimensional complex Hilbert space $H$. For $T \in B(H)$ and $λ\in \mathbb{C}$, let $H_{T}(\{λ\})$ denotes the local spectral subspace of $T$ associated with $\{λ\}$. We prove that if $φ:B(H)\rightarrow B(H)$ be an additive map such that its range contains all operators of rank at most two and satisfies $$H_{φ(T)φ(S)^{\ast}}(\{λ\})= H_{TS^{\ast}}(\{λ\})$$ for all $T, S \in B(H)$ and $λ\in \mathbb{C}$, then there exist a unitary operator $V$ in $B(H)$ and a nonzero scalar $μ$ such that $φ(T) = μTV^{\ast}$ for all $T \in B(H…

无自利的约化自由积与图积C*-代数

Selfless reduced free products and graph products of $\mathrm{C}^\ast$-algebras

本文在温和假设下证明无需快速衰减性质即可得C*-代数的完全无自利约化自由积与图积，进而构造出多类具有特定性质的新C*-代数例子。

C$^{\ast}$-代数中可分离的单位邻域

Separable neighbourhood of identity in C$^{\ast}$-algebras

研究双射C$^{\ast}$-代数中可分离元素的结构，探讨单位元素周围可分离邻域的存在性和大小，通过估计合同有界范数来表征代数结构特性，解决Musat和Rørdam的最新猜想。

Games on AF-algebras

We analyze $\mathrm{C}^\ast$-algebras, particularly AF-algebras, and their $K_0$-groups in the context of the infinitary logic $\mathcal{L}_{ω_1 ω}$. Given two separable unital AF-algebras $A$ and $B$, and considering their $K_0$-groups as ordered unital groups, we prove that $K_0(A) \equiv_{ω\cdot α} K_0(B)$ implies $A \equiv_αB$, where $M \equiv_βN$ means that $M$ and $N$ agree on all sentences of quantifier rank at most $β$. This implication is proved using techniques from Elliott's classification of separable AF-algebras, together with an adaptation of the Ehrenfeucht-Fraïssé game to the m…

Invariant ergodic measures and the classification of crossed product $C^\ast$-algebras

Let $α: G\curvearrowright X$ be a minimal free continuous action of an infinite countable amenable group on an infinite compact metrizable space. In this paper, under the hypothesis that the invariant ergodic probability Borel measure space $E_G(X)$ is compact and zero-dimensional, we show that the action $α$ has the small boundary property. This partially answers an open problem in dynamical systems that asks whether a minimal free action of an amenable group has the small boundary property if its space $M_G(X)$ of invariant Borel probability measures forms a Bauer simplex. In addition, under…

来自Anzai流的$C^\ast$ -代数及其$K$ -群

$C^\ast$-algebras from Anzai flows and their $K$-groups

研究$n$ -维环面$\mathbb{T}^n$上的Anzai流生成的$C^{*}$ -代数$\mathcal{A}_{n,θ}$.证明了该代数是$(n+2)$ -维连通单连通幂零李群$F_n$的格子群$\mathfrak{D}_n$的群$C^{*}$ -代数的单商，而$C^{*}$-代数对应的李代数是一般丝状李代数$\mathfrak{f}_{n}$ . $C^{*}(\mathfrak{D}_n)$的其他单无限维代数也被刻画和表示为低维环面上的单仿射Furstenberg变换群$C^*$ -代数上的矩阵代数。研究了$\mathcal{A}_{n,θ}$的$K$ -群和$C^{*}(\mathfrak{D}_n)$的其它单等价群，Pimsner-Voiculescu六项正合列是一个有用的工具.尽可能明确地研究了$\mathcal{A}_{n,θ}$的$K$ -群的秩，并证明了它与$\mathbb{T}^n$的更一般的变换群$C^*$ -代数的秩相同，包括Furstenberg变换群\text{$C^*$-algebras} $A_{F_{f,θ}}$ .一个错误（关于这些$K$ -组）在文献中得到解决。