Bibliographic record
Abstract
Abstract. We show that every operator system (and hence every unital operator algebra) has sufficiently many boundary represen-tations to generate the C*-envelope. We solve a 45 year old problem of William Arveson that is central to his approach to non-commutative dilation theory. We show that every operator system and every unital operator algebra has sufficiently many boundary representations to completely norm it. Thus the C*-algebra generated by the image of the direct sum of these maps is the C*-envelope. This was a central problem left open in Arveson’s seminal work [2] on dilation theory for arbitrary operator algebras. In the intervening years, the existence of the C*-envelope was established, but a general argument producing boundary representations has not been available. Arveson [2, 3] reformulated the classical dilation theory of Sz. Nagy [14] so that it made sense for an arbitrary unital closed subalgebra A of a C*-algebra. A central theme was the use of completely pos-itive and completely bounded maps. He proposed the existence of a family of special representations of A, called boundary representations, which have unique completely positive extensions to C∗(A) that are irreducible ∗-representations. The set of boundary representations is a noncommutative analogue of the Choquet boundary of a function alge-bra, i.e. the set of points with unique representing measures. Arveson proposed that there should be sufficiently many boundary representa-tions, so that their direct sum recovers the norm on Mn(A) for all n ≥ 1. In this case, he showed that the C*-algebra generated by this direct sum enjoys an important universal property, and provides a re-alization of the C*-envelope of A.
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.005 | 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".