A non-commutative Fej\\'{e}r theorem for crossed products, the\n approximation property, and applications
Bibliographic record
Abstract
We prove that a locally compact group has the approximation property (AP),\nintroduced by Haagerup-Kraus, if and only if a non-commutative Fej\\'{e}r\ntheorem holds for the associated $C^*$- or von Neumann crossed products. As\napplications, we answer three open problems in the literature. Specifically, we\nshow that any locally compact group with the AP is exact. This generalizes a\nresult by Haagerup-Kraus, and answers a problem raised by Li. We also answer a\nquestion of B\\'{e}dos-Conti on the Fej\\'{e}r property of discrete\n$C^*$-dynamical systems, as well as a question by Anoussis-Katavolos-Todorov\nfor all locally compact groups with the AP. In our approach, which relies on\noperator space techniques, we develop a notion of Fubini crossed product for\nlocally compact groups, and a dynamical version of the AP for actions\nassociated with $C^*$- or $W^*$-dynamical systems.\n
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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 | 0.001 | 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 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| 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".