[Paper Review] Toward a non-commutative Gelfand duality: Boolean locally separated toposes and Monoidal monotone complete $C^{*}$-categories

Simon Henry|arXiv (Cornell University)|Jan 28, 2015

Advanced Operator Algebra Research7 references3 citations

TL;DR

This paper establishes a reconstruction theorem for boolean locally separated toposes using their associated symmetric monoidal monotone complete $C^*$-categories of Hilbert bundles and square-integrable Hilbert bundles. It shows that such toposes are classifying toposes for non-degenerate normal symmetric monoidal $*$-representations of these categories, suggesting a non-commutative generalization of Gelfand duality.

ABSTRACT

** Draft Version ** To any boolean topos one can associate its category of internal Hilbert spaces, and if the topos is locally separated one can consider a full subcategory of square integrable Hilbert spaces. In both case it is a symmetric monoidal monotone complete $C^{*}$-category. We will prove that any boolean locally separated topos can be reconstructed as the classifying topos of "non-degenerate" monoidal normal $*$-representations of both its category of internal Hilbert spaces and its category of square integrable Hilbert spaces. This suggest a possible extension of the usual Gelfand duality between a class of toposes (or more generally localic stacks or localic groupoids) and a class of symmetric monoidal $C^{*}$-categories yet to be discovered.

Motivation & Objective

• To extend Gelfand duality beyond commutative $C^*$-algebras to non-commutative settings via topos-theoretic and operator algebraic structures.
• To establish a reconstruction theorem for boolean locally separated toposes using their internal categories of Hilbert bundles and square-integrable Hilbert bundles.
• To explore whether symmetric monoidal monotone complete $C^*$-categories can serve as non-commutative analogues of topological spaces or groupoids.
• To identify the necessary axioms and structures on $C^*$-categories that would allow such a duality to hold in general.
• To investigate how completeness, weak convergence, and duality in $C^*$-categories relate to geometric reconstruction in topos theory.

Proposed method

• Associate to any boolean topos $\mathcal{T}$ the symmetric monoidal monotone complete $C^*$-category $\mathcal{H}(\mathcal{T})$ of internal Hilbert bundles.
• For locally separated toposes, define the full subcategory $\mathcal{H}^{\text{red}}(\mathcal{T})$ of square-integrable Hilbert bundles.
• Construct a geometric morphism from the classifying topos of non-degenerate normal symmetric monoidal $*$-representations to the original topos $\mathcal{T}$.
• Use internal logic and Kripke-Joyal semantics to analyze the structure of separating objects and their representations.
• Prove that the category of points of $\mathcal{T}$ is equivalent to the category of such representations, via a construction relying on nets of operators $V_U$.
• Leverage booleanness and local separation to ensure monotone completeness and the existence of asymptotic duals, enabling reconstruction.

Experimental results

Research questions

• RQ1 Can boolean locally separated toposes be reconstructed from their symmetric monoidal $C^*$-categories of Hilbert bundles?
• RQ2 What conditions on a symmetric monoidal $C^*$-category ensure it arises as $\mathcal{H}^{\text{red}}(\mathcal{T})$ for some boolean locally separated topos $\mathcal{T}$?
• RQ3 How can the notion of 'good' or 'non-degenerate normal' representations be axiomatized purely in terms of $C^*$-categorical structures?
• RQ4 What additional structures or properties (e.g., weak convergence, weighted limits, duality) are essential for generalizing this duality beyond the boolean locally separated case?
• RQ5 Can this framework be extended to a full non-commutative Gelfand duality between localic groupoids/stacks and symmetric monoidal $C^*$-categories?

Key findings

• Any boolean locally separated topos $\mathcal{T}$ is the classifying topos for non-degenerate normal symmetric monoidal $*$-representations of $\mathcal{H}^{\text{red}}(\mathcal{T})$.
• The reconstruction of $\mathcal{T}$ from $\mathcal{H}^{\text{red}}(\mathcal{T})$ relies on a net of operators $V_U$ that asymptotically realize duality, crucially enabled by the booleanness and local separation of $\mathcal{T}$.
• The category $\mathcal{H}^{\text{red}}(\mathcal{T})$ is a symmetric monoidal monotone complete $C^*$-category without a unit object, yet still admits a full reconstruction.
• The unreduced category $\mathcal{H}(\mathcal{T})$ also classifies $\mathcal{T}$, and the reconstruction from $\mathcal{H}^{\text{red}}(\mathcal{T})$ implies the reconstruction from $\mathcal{H}(\mathcal{T})$.
• The results suggest that monotone complete $C^*$-categories with appropriate structures (e.g., weak convergence, duality) may serve as non-commutative analogues of topological spaces.
• The framework provides a potential unification of Gelfand duality, $W^*$-algebra duality, and Doplicher-Roberts reconstruction, suggesting a broader non-commutative duality.