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：

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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）$ | 为| $ \ 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）$ | 为| $ \ 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）。