On the size of orbits in the duals of 𝐶*-algebras and convolution algebras
Bibliographic record
Abstract
We solve an open problem raised in [Trans. Amer. Math. Soc. 366 (2014), pp.4151–4171] concerning (infinite-dimensional) commutative semisimple Banach algebras A \mathcal {A} with a bounded approximate identity, namely, whether there always exists a functional f ∈ A ∗ f \in \mathcal {A}^* such that the orbit subspace A ∗ ∗ ◻ f \mathcal {A}^{**} \Box f of A ∗ \mathcal {A}^* is w ∗ w^* -closed and infinite-dimensional. Indeed, on the one hand, we show that no commutative C ∗ C^* -algebra shares this property. On the other hand, we prove that the answer is positive, in a strong sense, in the case of convolution algebras A = L 1 ( G ) \mathcal {A} = L_1(\mathcal {G}) , for large classes of locally compact groups G \mathcal {G} (commutativity is not needed): there exists f ∈ A ∗ f \in \mathcal {A}^* with maximal orbit, in fact, f f satisfies
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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 | 0.012 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.005 | 0.013 |
| Open science | 0.002 | 0.004 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.011 | 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".