ℓ1-contractive maps on noncommutative Lp-spaces
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Bibliographic record
Abstract
Let T:Lp(M)→Lp(N) be a bounded operator between two noncommutative Lp-spaces, 1⩽p<∞. We say that T is ℓ1-bounded (respectively ℓ1-contractive) if T⊗Iℓ1 extends to a bounded (respectively contractive) map from Lp(M;ℓ1) into Lp(N;ℓ1). We show that Yeadon's factorization theorem for Lp-isometries, 1⩽p≠2<∞, applies to an isometry T:L2(M)→L2(N) if and only if T is ℓ1-contractive. We also show that a contractive operator T:Lp(M)→Lp(N) is automatically ℓ1-contractive if it satisfies one of the following two conditions: either T is 2-positive; or T is separating, that is, for any disjoint a,b∈Lp(M) (i.e.\ a∗b=ab∗=0), the images T(a),T(b) are disjoint as well.
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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 | 0.002 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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 it