Turbulence, orbit equivalence, and the classification of nuclear C*-algebras
Turbulence, orbit equivalence, and the classification of nuclear C*-algebras
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湍流、轨道等效和核 C* 代数的分类
DOI：
10.1515/crelle-2012-0053
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
Andrew S. Toms
中科院分区：
文献类型：
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作者：
I. Farah;Asger Tornquist;Andrew S. Toms
我们将核单可分C-代数同构关系的Borel基数限定为：它是动荡的，但Borel可约为Cuntz代数O2的自同构群在其闭子集上的作用。对可度量化的Choquet单形的仿射同胚也得到了同样的界。作为一个副产品，我们恢复的结果Kechris和Solecki，即，同胚的Rupta在希尔伯特立方体是Borel约化的波兰群作用。这些结果与核单C-代数的K-理论和迹的分类理论密切相关。为了进一步研究C-代数的Borel复杂性，证明了许多标准的C-代数结构和关系都是Borel的，并证明了基希贝格的O ~ 2-稳定性和嵌入定理的Borel版本.我们还找到了Kσ硬等价关系的一个C σ-代数证明.作者将这篇文章献给Greg Hjorth。
We bound the Borel cardinality of the isomorphism relation for nuclear simple separable C∗-algebras: It is turbulent, yet Borel reducible to the action of the automorphism group of the Cuntz algebra O2 on its closed subsets. The same bounds are obtained for affine homeomorphism of metrizable Choquet simplexes. As a by-product we recover a result of Kechris and Solecki, namely, that homeomorphism of compacta in the Hilbert cube is Borel reducible to a Polish group action. These results depend intimately on the classification theory of nuclear simple C∗-algebras by K-theory and traces. Both of necessity and in order to lay the groundwork for further study on the Borel complexity of C∗-algebras, we prove that many standard C∗-algebra constructions and relations are Borel, and we prove Borel versions of Kirchberg’s O2-stability and embedding theorems. We also find a C∗-algebraic witness for a Kσ hard equivalence relation. The authors dedicate this article to the memory of Greg Hjorth.