Type Decomposition and the Rectangular AFD Property for <i>W</i>*-TRO’s
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Bibliographic record
Abstract
Abstract We study the type decomposition and the rectangular AFD property for W *-TRO’s. Like von Neumann algebras, every W *-TRO can be uniquely decomposed into the direct sum of W *- TRO's of type I , type II , and type III . We may further consider W *-TRO's of type I m,n with cardinal numbers m and n , and consider W *-TRO's of type II λ,μ with λ, μ = 1 or ∞. It is shown that every separable stable W *-TRO (which includes type I ∞, ∞, type II ∞, ∞ and type III ) is TRO-isomorphic to a von Neumann algebra. We also introduce the rectangular version of the approximately finite dimensional property for W *-TRO’s. One of our major results is to show that a separable W *-TRO is injective if and only if it is rectangularly approximately finite dimensional. As a consequence of this result, we show that a dual operator space is injective if and only if its operator predual is a rigid rectangular space (equivalently, a rectangular space).
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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 | 0.001 | 0.002 |
| 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 | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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