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The Tracial Rokhlin Property for Countable Discrete Amenable Group Actions on Nuclear Tracially Approximately Divisible C*-Algebras

机译:核可弯曲的近似可分C *-代数上可数的离散可调整群行为的Tracetic Rokhlin属性

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摘要

In this dissertation we explore the question of existence of a property of group actions on C*-algebras known as the tracial Rokhlin property. We prove existence of the property in a very general setting as well as specialise the question to specific situations of interest.For every countable discrete elementary amenable group G, we show that there always exists a G-action ω with the tracial Rokhlin property on any unital simple nuclear tracially approximately divisible C*-algebra A. For the ω we construct, we show that if A is unital simple and Z-stable with rational tracial rank at most one and G belongs to the class of countable discrete groups generated by finite and abelian groups under increasing unions and subgroups, then the crossed product A ω G is also unital simple and Z-stable with rational tracial rank at most one.We also specialise the question to UHF algebras. We show that for any countable discrete maximally almost periodic group G and any UHF algebra A, there exists a strongly outer product type action α of G on A. We also show the existence of countable discrete almost abelian group actions with the "pointwise" Rokhlin property on the universal UHF algebra. Consequently we get many examples of unital separable simple nuclear C*-algebras with tracial rank zero and a unique tracial state appearing as crossed products.
机译:在本文中,我们探讨了在C *代数上的集体行动性质存在的问题,该性质被称为种族Rokhlin性质。我们证明了该属性在一个非常笼统的情况下存在,并将问题专门针对特定的关注情况。对于每个可数离散的基本可服从组G,我们表明在任何情况下,始终存在具有罗克林性质的G动作ω单位简单核tracially近似可分C *-代数A。对于ω,我们证明,如果A是单位简单且Z稳定且具有合理的tracial等级,且G属于有限元生成的可数离散群的类别以及在不断增加的并集和子组下的阿贝尔群,则交叉积AωG也是单位简单且Z稳定的,具有合理的阶数。我们也将问题专门针对UHF代数。我们表明,对于任何可数离散的最大近似周期群G和任何UHF代数A,G在A上都存在一个强烈的G的外积类型作用α。我们还显示了存在可数离散的近似阿贝尔群运动的“点向”罗克林通用UHF代数的性质。因此,我们得到了许多示例,它们的单位为零,且唯一的交易态以交叉乘积的形式出现,它是单位可分的简单核C *代数。

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    Sun Michael;

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