标题： 抽象和非交换可测空间的范畴 显示英文标题

标题： Categories of abstract and noncommutative measurable spaces

摘要： Gelfand 对偶性是一个基本结果，它证明了将一般的单位$C^*$- 代数视为紧致Hausdorff空间的非交换版本是合理的。 受这种观点的启发，我们探讨了非交换可测空间应该是什么样的。 这引导我们考虑单调的$\sigma$- 完备$C^*$- 代数范畴以及布尔$\sigma$- 代数范畴，这些可以被视为抽象的可测空间。 受到寻找良好非交换可测空间概念的驱动，我们提供了这些范畴的一个统一概述，包括可测空间范畴，并通过函子、伴随和等价形式化它们之间的关系。 这包括布尔$\sigma$- 代数与交换单调$\sigma$- 完备$C^*$- 代数之间的等价，以及后一范畴与可测空间范畴之间的 Gelfand 型对偶伴随。 这种对偶限制为两个等价：一个涉及广泛用于概率论的标准波莱尔空间，另一个涉及更一般的贝尔可测空间。 此外，这一结果可以有一个概率版本，其中态射分别是$\sigma$-正规CPU映射和马尔可夫核。我们希望这些进展也能有助于目前寻找一个表现良好的马尔可夫范畴以处理测度论概率的标准波莱尔设定之外的问题——这是当前最前沿的一个开放问题。 显示英文摘要

摘要： Gelfand duality is a fundamental result that justifies thinking of general unital $C^*$-algebras as noncommutative versions of compact Hausdorff spaces. Inspired by this perspective, we investigate what noncommutative measurable spaces should be. This leads us to consider categories of monotone $\sigma$-complete $C^*$-algebras as well as categories of Boolean $\sigma$-algebras, which can be thought of as abstract measurable spaces. Motivated by the search for a good notion of noncommutative measurable space, we provide a unified overview of these categories, alongside those of measurable spaces, and formalize their relationships through functors, adjunctions and equivalences. This includes an equivalence between Boolean $\sigma$-algebras and commutative monotone $\sigma$-complete $C^*$-algebras, as well as a Gelfand-type duality adjunction between the latter category and the category of measurable spaces. This duality restricts to two equivalences: one involving standard Borel spaces, which are widely used in probability theory, and another involving the more general Baire measurable spaces. Moreover, this result admits a probabilistic version, where the morphisms are $\sigma$-normal cpu maps and Markov kernels, respectively. We hope that these developments can also contribute to the ongoing search for a well-behaved Markov category for measure-theoretic probability beyond the standard Borel setting - an open problem in the current state of the art.