[论文解读] Nuclear dimension and Z-stability of pure C*-algebras

Wilhelm Winter|arXiv (Cornell University)|Jun 14, 2010

Advanced Operator Algebra Research参考文献 23被引用 10

一句话总结

该论文证明了：对于满足迹纯性条件（即强迹m-比较与迹m̄-几乎可除性）的分离的、简单的、非初等的、单位元C*-代数，若其局部有限核维数，则其Z-稳定。关键结果表明，此类代数在张量积意义下吸收Jiang–Su代数Z，从而解决了Toms–Winter猜想的一个主要情形，并通过K-理论不变量完成了AH代数及近似子同态代数的分类结果。

ABSTRACT

In this article I study a number of topological and algebraic dimension type properties of simple C*-algebras and their interplay. In particular, a simple C*-algebra is defined to be (tracially) (m,\bar{m})-pure, if it has (strong tracial) m-comparison and is (tracially) \bar{m}-almost divisible. These notions are related to each other, and to nuclear dimension. The main result says that if a separable, simple, nonelementary, unital C*-algebra A with locally finite nuclear dimension is (m,\bar{m})-pure, then it absorbs the Jiang-Su algebra Z tensorially. It follows that A is Z-stable if and only if it has the Cuntz semigroup of a Z-stable C*-algebra. The result may be regarded as a version of Kirchberg's celebrated theorem that separable, simple, nuclear, purely infinite C*-algebras absorb the Cuntz algebra O_\infty tensorially. As a corollary we obtain that finite nuclear dimension implies Z-stability for separable, simple, nonelementary, unital C*-algebras; this settles an important case of a conjecture by Toms and the author. The main result also has a number of consequences for Elliott's program to classify nuclear C*-algebras by their K-theory data. In particular, it completes the classification of simple, unital, approximately homogeneous algebras with slow dimension growth by their Elliott invariants, a question left open in the Elliott-Gong-Li classification of simple AH algebras. Another consequence is that for simple, unital, approximately subhomogeneous algebras, slow dimension growth and Z-stability are equivalent. In the case where projections separate traces, this completes the classification of simple, unital, approximately subhomogeneous algebras with slow dimension growth by their ordered K-groups.

研究动机与目标

• 建立简单、单位元C*-代数中核维数与Z-稳定性之间的联系。
• 定义并分析迹(m, m̄)-纯性作为结合迹m-比较与迹m̄-几乎可除性的正则性条件。
• 通过证明有限核维数蕴含Z-稳定性，解决Toms–Winter猜想的一个主要情形，适用于分离的、简单的、非初等的、单位元C*-代数。
• 通过其有序K-理论完成对简单、单位元、近似子同态C*-代数的分类。

提出的方法

• 引入（迹性地）(m, m̄)-纯C*-代数的概念，结合强迹m-比较与迹m̄-几乎可除性。
• 利用Robert关于核维数蕴含m-比较的结果，并将其扩展至迹m-比较与m̄-几乎可除性。
• 应用Kirchberg的覆盖数技术，将核维数与可除性性质联系起来。
• 利用完全正定零阶映射与可分解的完全正逼近，构造所需的*-同态与酉元。
• 借助Cuntz半群及其同构不变量刻画Z-稳定性。
• 综合运用上述工具，证明在有限核维数下，迹(m, m̄)-纯性蕴含Z-稳定性。

实验结果

研究问题

• RQ1 对于分离的、简单的、非初等的、单位元C*-代数，有限核维数是否蕴含Z-稳定性？
• RQ2 迹(m, m̄)-纯性是否可用于刻画核维数局部有限的C*-代数的Z-稳定性？
• RQ3 对于近似子同态C*-代数，Z-稳定性是否等价于缓慢维数增长？
• RQ4 当核维数有限时，C*-代数的Cuntz半群是否决定其Z-稳定性？
• RQ5 简单、单位元、近似齐次C*-代数在缓慢维数增长下是否完全由其Elliott不变量分类？

主要发现

• 若一个分离的、简单的、非初等的、单位元C*-代数具有局部有限核维数，并且对某个m, m̄ ∈ ℕ为迹(m, m̄)-纯，则其Z-稳定。
• 有限核维数蕴含迹(m, m̄)-纯性，而后者又蕴含Z-稳定性。
• 对于此类代数，Z-稳定性等价于其Cuntz半群同构于A ⊗ ℤ的Cuntz半群。
• 对于近似子同态C*-代数，Z-稳定性与缓慢维数增长等价。
• 具有缓慢维数增长且投影分离迹的简单、单位元、近似子同态C*-代数类，由其有序K-理论完全分类。
• 对于近似齐次C*-代数，缓慢维数增长、Z-稳定性和有限分解秩三者等价，从而通过Elliott不变量完成其分类。