Rokhlin dimension: duality, tracial properties, and crossed products
Rokhlin dimension: duality, tracial properties, and crossed products
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DOI：
10.1017/etds.2019.68
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发表时间：
2017-09
影响因子：
0.9
通讯作者：
Eusebio Gardella;Ilan Hirshberg;Luis Santiago
中科院分区：
文献类型：
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作者：
Eusebio Gardella;Ilan Hirshberg;Luis Santiago
我们研究了有限Rokhlin维数的紧群作用，特别是与交叉产品。例如，我们的特点，这种行动的，推广以前的部分结果的Rokhlin属性。作为应用，我们确定了它们的交叉积的理想结构。在交换塔的假设下，我们证明了通过这种作用取交叉积保持了一些相关的C ^{\ast}$-代数类，包括：$D $-吸收$C ^{\ast}$-代数，其中$D $是强自吸收$C ^{\ast}$-代数;稳定$C ^{\ast}$-代数;$C ^{\ast}$-具有有限核维度（或分解秩）的代数;$C ^{\ast}$-具有有限稳定秩（或真实的秩）的代数;以及$C ^{\ast}$-代数，其$K $-理论是平凡的、有理数的，或$n $-可整除$n\in\mathbb {N}$。核性和通用系数定理（UCT）的组合也被证明是保存这些行动。其中一些结果是新的，即使在充分研究的情况下的Rokhlin属性。此外，在一些技术假设下，我们证明了有限的Rokhlin维数与交换塔意味着（弱）tracial Rokhlin性质。在我们的论点的核心是一定的局部近似的交叉产品的连续$C（X）$-代数的纤维是稳定同构的基础代数。在某些感兴趣的情况下计算空间X，并利用它的描述在满足UCT的单位AF-代数和单位基希贝格代数上构造了一个$\mathbb {Z}_{2}$-作用，其有交换塔和无交换塔的Rokhlin维数是有限的但不一致.
We study compact group actions with finite Rokhlin dimension, particularly in relation to crossed products. For example, we characterize the duals of such actions, generalizing previous partial results for the Rokhlin property. As an application, we determine the ideal structure of their crossed products. Under the assumption of so-called commuting towers, we show that taking crossed products by such actions preserves a number of relevant classes of $C^{\ast }$ -algebras, including: $D$ -absorbing $C^{\ast }$ -algebras, where $D$ is a strongly self-absorbing $C^{\ast }$ -algebra; stable $C^{\ast }$ -algebras; $C^{\ast }$ -algebras with finite nuclear dimension (or decomposition rank); $C^{\ast }$ -algebras with finite stable rank (or real rank); and $C^{\ast }$ -algebras whose $K$ -theory is either trivial, rational, or $n$ -divisible for $n\in \mathbb{N}$ . The combination of nuclearity and the universal coefficient theorem (UCT) is also shown to be preserved by these actions. Some of these results are new even in the well-studied case of the Rokhlin property. Additionally, and under some technical assumptions, we show that finite Rokhlin dimension with commuting towers implies the (weak) tracial Rokhlin property. At the core of our arguments is a certain local approximation of the crossed product by a continuous $C(X)$ -algebra with fibers that are stably isomorphic to the underlying algebra. The space $X$ is computed in some cases of interest, and we use its description to construct a $\mathbb{Z}_{2}$ -action on a unital AF-algebra and on a unital Kirchberg algebra satisfying the UCT, whose Rokhlin dimensions with and without commuting towers are finite but do not agree.