Categories of abstract and noncommutative measurable spaces

Abstract: 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.

Equivalence and Dualities in Abstract Measurable Spaces and Operators

The paper explores the foundational aspects of abstract and noncommutative measurable spaces with an emphasis on categorical perspectives and dualities. It extends classical frameworks such as Gelfand and Loomis--Sikorski dualities to a broader context involving Boolean $\sigma$-algebras (Boos), $C^*$-algebras, and measurable spaces. The authors aim to provide a categorical and algebraic treatment of measure theory and its noncommutative counterparts, highlighting equivalences, adjunctions, and extensions of classical results.

Key Concepts and Frameworks

1. Boolean $\sigma$-algebras (Boos): These are abstract $\sigma$-completions of Boolean algebras and serve as a foundational tool to translate between algebraic and measurable structures. The paper builds on existing literature to develop a comprehensive theory of Boos, including notions like universal and regular $\sigma$-completions.
2. $C^*$-algebras and s: The paper introduces and discusses monotone $\sigma$-complete $C^*$-algebras (s), emphasizing their utility in extending measure theoretic properties to noncommutative spaces. A key focus is on the Pedersen--Baire envelope, which is a generalization of the Baire $\sigma$-algebra in the $C^*$-algebraic context.
3. Equivalences and Dualities: The study demonstrates an equivalence between commutative $C^*$-algebras and Boos using Gelfand and Stone dualities. This equivalence is fundamental for transferring concepts between the operator and algebraic representations of measurable spaces.
4. Measurable Gelfand Duality: Extending classical Gelfand duality, the paper establishes a measurable version that intertwines with the Loomis--Sikorski duality, thereby relating spectrums of $C^*$-algebras to measurable spaces.
5. Tensor Products: The paper extends the notion of tensor products in both commutative and noncommutative $C^*$-algebras, showing that various constructions are consistent across different categories, such as Pedersen--Baire envelopes.

Implications and Future Directions

The implications of these findings contribute to a deeper understanding of the algebraic structures underlying measure and functional analysis, especially in contexts where considerations of $\sigma$-completeness and noncommutativity are essential. The research hints at potential applications in quantum probability theory, where similar structures and dualities play essential roles in developing coherent mathematical models for quantum systems.

The exploration of tensor products and the establishment of equivalences provide foundational tools for future studies in operator algebras, potentially influencing how algebraic and categorical theories are applied in the analysis of measurable phenomena.

Conclusion

This study illustrates the power of categorical and algebraic methods in bridging classical measure theory and modern operator algebra frameworks. By offering a generalized view through the lens of dualities and equivalences, it extends the reach of established mathematical structures, allowing them to accommodate the complexities of noncommutative spaces. The insights on dualities and measurable spaces set the stage for further exploration of probabilistic and quantum systems in a cohesive mathematical frame.