A generalization of Gelfand-Naimark-Stone duality to completely regular spaces

Guram Bezhanishvili; Patrick J. Morandi; Bruce Olberding

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arXiv · 1812.07599

A generalization of Gelfand-Naimark-Stone duality to completely regular spaces

Guram Bezhanishvili·Patrick J. Morandi·Bruce Olberding

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Abstract

Gelfand-Naimark-Stone duality establishes a dual equivalence between the category ${\sf KHaus}$ of compact Hausdorff spaces and the category ${\boldsymbol{\mathit{uba}\ell}}$ of uniformly complete bounded archimedean $\ell$-algebras. We extend this duality to the category ${\sf CReg}$ of completely regular spaces. This we do by first introducing basic extensions of bounded archimedean $\ell$-algebras and generalizing Gelfand-Naimark-Stone duality to a dual equivalence between the category ${\boldsymbol{\mathit{ubasic}}}$ of uniformly complete basic extensions and the category ${\sf C}$ of compactifications of completely regular spaces. We then introduce maximal basic extensions and prove that the subcategory ${\boldsymbol{\mathit{mbasic}}}$ of ${\boldsymbol{\mathit{ubasic}}}$ consisting of maximal basic extensions is dually equivalent to the subcategory ${\sf SComp}$ of ${\sf Comp}$ consisting of Stone-Čech compactifications. This yields the desired dual equivalence for completely regular spaces since ${\sf CReg}$ is equivalent to ${\sf SComp}$.

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BibTeX RIS

Guram Bezhanishvili, Patrick J. Morandi, Bruce Olberding. 2020-02-16. A generalization of Gelfand-Naimark-Stone duality to completely regular spaces. https://arxiv.org/abs/1812.07599

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