A weak expectation property for operator modules, injectivity and amenable actions
Bibliographic record
Abstract
We introduce an equivariant version of the weak expectation property (WEP) at the level of operator modules over completely contractive Banach algebras [Formula: see text]. We prove a number of general results — for example, a characterization of the [Formula: see text]-WEP in terms of an appropriate [Formula: see text]-injective envelope, and also a characterization of those [Formula: see text] for which [Formula: see text]-WEP implies WEP. In the case of [Formula: see text], we recover the [Formula: see text]-WEP for [Formula: see text]-[Formula: see text]-algebras in recent work of Buss–Echterhoff–Willett [A. Buss, S. Echterhoff and R. Willett, The maximal injective crossed product, to appear in Ergodic Theory Dynam. Systems, https://doi.org/10.1017/etds.2019.25 ]. When [Formula: see text], we obtain a dual notion for operator modules over the Fourier algebra. These dual notions are related in the setting of dynamical systems, where we show that a [Formula: see text]-dynamical system [Formula: see text] with [Formula: see text] injective is amenable if and only if [Formula: see text] is [Formula: see text]-injective if and only if the crossed product [Formula: see text] is [Formula: see text]-injective. Analogously, we show that a [Formula: see text]-dynamical system [Formula: see text] with [Formula: see text] nuclear and [Formula: see text] exact is amenable if and only if [Formula: see text] has the [Formula: see text]-WEP if and only if the reduced crossed product [Formula: see text] has the [Formula: see text]-WEP.
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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 | 0.002 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.005 |
| Scholarly communication | 0.003 | 0.007 |
| Open science | 0.001 | 0.004 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.006 | 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".