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Operator algebras with hyperarithmetic theory
Journal of Logic and Computation ( IF 0.7 ) Pub Date : 2020-09-23 , DOI: 10.1093/logcom/exaa059
Isaac Goldbring 1 , Bradd Hart 2
Affiliation  

We show that the following operator algebras have hyperarithmetic theory: the hyperfinite II|$_1$| factor |$\mathcal R$|⁠, |$L(\varGamma )$| for |$\varGamma $| a finitely generated group with solvable word problem, |$C^*(\varGamma )$| for |$\varGamma $| a finitely presented group, |$C^*_\lambda (\varGamma )$| for |$\varGamma $| a finitely generated group with solvable word problem, |$C(2^\omega )$| and |$C(\mathbb P)$| (where |$\mathbb P$| is the pseudoarc). We also show that the Cuntz algebra |$\mathcal O_2$| has a hyperarithmetic theory provided that the Kirchberg embedding problems have affirmative answers. Finally, we prove that if there is an existentially closed (e.c.) II|$_1$| factor (resp. |$\textrm{C}^*$|-algebra) that does not have hyperarithmetic theory, then there are continuum many theories of e.c. II|$_1$| factors (resp. e.c. |$\textrm{C}^*$|-algebras).

中文翻译:

超算术理论的算子代数

我们证明以下算子代数具有超算术理论:超有限II | $ _1 $ | 因子| $ \ mathcal R $ |⁠| $ L(\ varGamma)$ | | $ \ varGamma $ | | $ C ^ *(\ varGamma)$ |的一个带有可解单词问题的有限生成的组 为| $ \ varGamma $ | 一个有限表现组,| $ C ^ * _ \拉姆达(\ varGamma)$ | | $ \ varGamma $ | | $ C(2 ^ \ omega)$ |具有可解单词问题的有限生成的组 和| $ C(\ mathbb P)$ | (其中| $ \ mathbb P $ |是伪弧)。我们还显示了Cuntz代数| $ \ mathcal O_2 $ |有一个超算术理论,前提是Kirchberg嵌入问题具有肯定的答案。最后,我们证明如果存在存在闭式(ec)II | $ _1 $ | 不具有超算术理论的因数(分别为| $ \ textrm {C} ^ * $ |-代数),则存在ec II的许多连续论| $ _1 $ | 因子(分别为ec | $ \ textrm {C} ^ * $ |-代数)。
更新日期:2020-09-23
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