Bounded Archimedean ℓ-algebras and Gelfand-Neumark-Stone duality
Bibliographic record
Abstract
By Gelfand-Neumark duality, the category C * Alg of commutative C *algebras is dually equivalent to the category of compact Hausdorff spaces, which by Stone duality, is also dually equivalent to the category uba of uniformly complete bounded Archimedean -algebras.Consequently, C * Alg is equivalent to uba , and this equivalence can be described through complexification.In this article we study uba within the larger category ba of bounded Archimedean -algebras.We show that uba is the smallest nontrivial reflective subcategory of ba , and that uba consists of exactly those objects in ba that are epicomplete, a fact that includes a categorical formulation of the Stone-Weierstrass theorem for ba .It follows that uba is the unique nontrivial reflective epicomplete subcategory of ba .We also show that each nontrivial reflective subcategory of ba is both monoreflective and epireflective, and exhibit two other interesting reflective subcategories of ba involving Gelfand rings and square closed rings.Dually, we show that Specker R-algebras are precisely the co-epicomplete objects in ba .We prove that the category spec of Specker R-algebras is a mono-coreflective subcategory of ba that is co-epireflective in a mono-coreflective subcategory of ba consisting of what we term -clean rings, a version of clean rings adapted to the ordertheoretic setting of ba .We conclude the article by discussing the import of our results in the setting of complex * -algebras through complexification.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".