We study the connection between the UCT problem and Cartan subalgebras in C*-algebras. The UCT problem asks whether every separable nuclear C*-algebra satisfies the UCT, i.e., a noncommutative analogue of the classical universal coefficient theorem from algebraic topology. This UCT problem is one of the remaining major open questions in the structure and classification theory of simple nuclear C*-algebras. Since the class of separable nuclear C*-algebras is closed under crossed products by finite groups, it is a natural and important task to understand the behaviour of the UCT under such crossed products. We make a contribution towards a better understanding by showing that for certain approximately inner actions of finite cyclic groups on UCT Kirchberg algebras, the crossed products satisfy the UCT if and only if we can find Cartan subalgebras which are invariant under the actions of our finite cyclic groups. We also show that the class of actions we are able to treat is big enough to characterize the UCT problem, in the sense that every such action (even on a particular Kirchberg algebra, namely the Cuntz algebra $$\mathcal O_2$$ O 2 ) leads to a crossed product satisfying the UCT if and only if every separable nuclear C*-algebra satisfies the UCT. Our results rely on a new construction of Cartan subalgebras in certain inductive limit C*-algebras. This new tool turns out to be of independent interest. For instance, among other things, the second author has used it to construct Cartan subalgebras in all classifiable unital stably finite C*-algebras.

中文翻译：

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Cartan 子代数和 UCT 问题，II

我们研究了 UCT 问题与 C*-代数中的 Cartan 子代数之间的联系。UCT 问题询问是否每个可分离的核 C*-代数都满足 UCT，即代数拓扑中经典通用系数定理的非交换类似物。这个 UCT 问题是简单核 C*-代数的结构和分类理论中剩余的主要开放问题之一。由于可分离核 C*-代数类在有限群的交叉积下是封闭的，因此了解 UCT 在这种交叉积下的行为是一项自然而重要的任务。我们通过证明对于 UCT Kirchberg 代数上有限循环群的某些近似内部作用，为更好地理解做出了贡献，当且仅当我们能找到在有限循环群作用下不变的 Cartan 子代数时，交叉积才满足 UCT。我们还表明，我们能够处理的动作类别大到足以表征 UCT 问题，因为每个这样的动作（甚至在特定的 Kirchberg 代数上，即 Cuntz 代数 $$\mathcal O_2$$ O 2 ) 导致满足 UCT 的交叉乘积，当且仅当每个可分离核 C*-代数满足 UCT。我们的结果依赖于某些归纳极限 C*-代数中 Cartan 子代数的新构造。事实证明，这个新工具具有独立的利益。例如，除其他外，第二作者使用它在所有可分类的单位稳定有限 C*-代数中构造 Cartan 子代数。