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 ubaℓ of uniformly complete bounded Archimedean ℓ-algebras.Consequently, C * Alg is equivalent to ubaℓ, and this equivalence can be described through complexification.In this article we study ubaℓ within the larger category baℓ of bounded Archimedean ℓ-algebras.We show that ubaℓ is the smallest nontrivial reflective subcategory of baℓ, and that ubaℓ consists of exactly those objects in baℓ that are epicomplete, a fact that includes a categorical formulation of the Stone-Weierstrass theorem for baℓ.It follows that ubaℓ is the unique nontrivial reflective epicomplete subcategory of baℓ.We also show that each nontrivial reflective subcategory of baℓ is both monoreflective and epireflective, and exhibit two other interesting reflective subcategories of baℓ involving Gelfand rings and square closed rings.Dually, we show that Specker R-algebras are precisely the co-epicomplete objects in baℓ.We prove that the category spec of Specker R-algebras is a mono-coreflective subcategory of baℓ that is co-epireflective in a mono-coreflective subcategory of baℓ consisting of what we term ℓ-clean rings, a version of clean rings adapted to the ordertheoretic setting of baℓ.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 machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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 source (direct Gemma or distilled Codex), 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".