On non-nuclear C*-algebras with the Local Lifting Property
Bibliographic record
Abstract
A C*-algebra has the lifting property (LP) if any ucp map to a quotient C*-algebra admits a ucp lift.It has the local lifting property (LLP) if every ucp map from a finite dimensional operator system admits a ucp lift.The latter, introduced by Kirchberg in 1993 [23], is dual to the weak expectation property introduced by Lance in 1973 [24].The Choi-Effros lifting theorem [10] implies that nuclear C*-algebras have the LLP, however non-nuclear examples are few and far between.Following a construction of Courtney [13], we present an exposition of results from various fields of C*-algebra theory which we will use to give another such example. ResuméUne C*-algèbre a la propriété de relèvement (local), si toute fonction unitale et complètement positive (ucp) à valeur dans une C*-algèbre quotient admet (localement) un relèvement ucp.Cette propriété de Kirchberg (1993) [23] est en dualité avec la propriété d'espérance faible de Lance (1973) [24].Grâce à un theorème de Choi et Effros [10], nous savons que les C*algèbres séparables et nucléaires ont la propriété de relèvement local.Pourtant, les exemples non-nucléaires ne sont pas nombreux.Suivant une construction de Courtney [13], nous donnons un exposé de divers résultats venant de la théorie des C*-algèbres et construisons un tel exemple en se basant sur ces résultats.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.002 | 0.006 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.010 | 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".