Nonunital Operator Systems and Noncommutative Convexity
Bibliographic record
Abstract
Abstract We establish the dual equivalence of the category of generalized (i.e., potentially nonunital) operator systems and the category of pointed compact noncommutative (nc) convex sets, extending a result of Davidson and the 1st author. We then apply this dual equivalence to establish a number of results about generalized operator systems, some of which are new even in the unital setting. For example, we show that the maximal and minimal C*-covers of a generalized operator system can be realized in terms of theC*-algebra of continuous nc functions on its nc quasistate space, clarifying recent results of Connes and van Suijlekom. We also characterize “C*-simple” generalized operator systems, that is, generalized operator systems with a simple minimal C*-cover, in terms of their nc quasistate spaces. We develop a theory of quotients of generalized operator systems that extends the theory of quotients of unital operator systems. In addition, we extend results of the 1st author and Shamovich relating to nc Choquet simplices. We show that a generalized operator system is a C*-algebra if and only if its nc quasistate space is an nc Bauer simplex with zero as an extreme point, and we show that a second countable locally compact group has Kazhdan’s property (T) if and only if for every action of the group on a C*-algebra, the set of invariant quasistates is the quasistate space of a C*-algebra.
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.000 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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 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".