Categories of abstract and noncommutative measurable spaces
Tobias Fritz, Antonio Lorenzin
TL;DR
The paper develops a comprehensive taxonomy of noncommutative and abstract measurable spaces by relating Boolean σ-algebras and monotone σ-complete C*-algebras through Loomis–Sikorski and Gelfand dualities, and by introducing Pedersen–Baire envelopes to bridge classical and quantum probability. It constructs measurable Gelfand duality, showing an equivalence between commutative σC*-algebras and Boolean σ-algebras, and extends these notions to probabilistic morphisms via Markov kernels, aligning with Markov-category perspectives. It further studies tensor products, establishing that for standard Borel spaces the universal and regular tensor products coincide, and proposes a maximal Pedersen–Baire tensor product to handle noncommutative cases, resulting in a symmetric monoidal structure suitable for quantum probability frameworks. The framework aims to provide well-behaved categorical structures for measure-theoretic probability beyond standard Borel spaces, offering concrete algebraic tools for analyzing noncommutative measurable spaces and their morphisms.
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 $σ$-complete $C^*$-algebras as well as categories of Boolean $σ$-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 $σ$-algebras and commutative monotone $σ$-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 $σ$-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.