标题： 非交换 Cantor-Bendixson 导数和散射$C^*$-代数 显示英文标题

标题： Noncommutative Cantor-Bendixson derivatives and scattered $C^*$-algebras

摘要： 我们分析通过连续应用康托尔-本迪克斯森导数得到的序列，针对非交换散射$C^*$-代数$\mathcal A$，使用由$\mathcal A$的最小投影生成的理想$\mathcal I^{At}(\mathcal A)$。借助它，我们以类似于散射紧致或局部紧致豪斯多夫空间和超原子布尔代数的交换情况的方式，提出了一些关于散射$C^*$-代数的基本结果。它还使我们能够提出一些在经典拓扑学中“基数序列”计划所激发的问题，在非交换背景下进行表述。这导致了一些新的非交换散射$C^*$-代数的构造和新的开放问题。 特别是，我们构造了一个类型$I$$C^*$-代数，它是稳定理想$\mathcal A_\alpha$的归纳极限，沿着不可数的极限序数$\lambda$，使得$\mathcal A_{\alpha+1}/\mathcal A_\alpha$与可分希尔伯特空间上所有紧算子的代数$*$同构，而$\mathcal A_{\alpha+1}$对每个$\alpha<\lambda$都是$\sigma$-单位且稳定的，但$\mathcal A$不是稳定的，并且$\mathcal A$的所有理想都是$\mathcal A_\alpha$的形式。 特别地，$\mathcal A$是一个不可分的$C^*$-代数，没有理想在稳定的理想中是极大的。 这回答了 M. R\ordam 在不可分情况下的一个问题。 以上所有$C^*$-代数$A_\alpha$和$A$都满足 AF 代数定义的以下版本：任何有限子集都可以从一个有限维子代数中近似。 基于本文中发展的语言，另外两种复杂的构造在单独的论文中进行了介绍。 显示英文摘要

摘要： We analyze the sequence obtained by consecutive applications of the Cantor-Bendixson derivative for a noncommutative scattered $C^*$-algebra $\mathcal A$, using the ideal $\mathcal I^{At}(\mathcal A)$ generated by the minimal projections of $\mathcal A$. With its help, we present some fundamental results concerning scattered $C^*$-algebras, in a manner parallel to the commutative case of scattered compact or locally compact Hausdorff spaces and superatomic Boolean algebras. It also allows us to formulate problems which have motivated the "cardinal sequences" programme in the classical topology, in the noncommutative context. This leads to some new constructions of noncommutative scattered $C^*$-algebras and new open problems. In particular, we construct a type $I$ $C^*$-algebra which is the inductive limit of stable ideals $\mathcal A_\alpha$, along an uncountable limit ordinal $\lambda$, such that $\mathcal A_{\alpha+1}/\mathcal A_\alpha$ is $*$-isomorphic to the algebra of all compact operators on a separable Hilbert space and $\mathcal A_{\alpha+1}$ is $\sigma$-unital and stable for each $\alpha<\lambda$, but $\mathcal A$ is not stable and where all ideals of $\mathcal A$ are of the form $\mathcal A_\alpha$. In particular, $\mathcal A$ is a nonseparable $C^*$-algebra with no ideal which is maximal among the stable ideals. This answers a question of M. R\ordam in the nonseparable case. All the above $C^*$-algebras $A_\alpha$s and $A$ satisfy the following version of the definition of an AF algebra: any finite subset can be approximated from a finite-dimensional subalgebra. Two more complex constructions based on the language developed in this paper are presented in separate papers.