The primitive ideal space of the C*-algebra of the affine semigroup of algebraic integers
Bibliographic record
Abstract
Abstract The purpose of this paper is to give a complete description of the primitive ideal space of the C*-algebra [R] associated to the ring of integers R in a number field K in the recent paper [5]. As explained in [5], [R] can be realized as the Toeplitz C*-algebra of the affine semigroup R ⋊ R× over R and as a full corner of a crossed product C0( ) ⋊ K ⋊ K*, where is a certain adelic space. Therefore Prim( [R]) is homeomorphic to the primitive ideal space of this crossed product. Using a recent result of Sierakowski together with the fact that every quasi-orbit for the action of K ⋊ K* on contains at least one point with trivial stabilizer we show that Prim( [R]) is homeomorphic to the quasi-orbit space for the action of K ⋊ K* on , which in turn may be identified with the power set of the set of prime ideals of R equipped with the power-cofinite topology.
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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) | 0.001 | 0.000 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.003 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.005 | 0.001 |
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".