Invariant means on weakly almost periodic functionals with application to quantum groups
Bibliographic record
Abstract
Abstract Let ${\mathcal A}$ be a Banach algebra, and let $\varphi $ be a nonzero character on ${\mathcal A}$ . For a closed ideal I of ${\mathcal A}$ with $I\not \subseteq \ker \varphi $ such that I has a bounded approximate identity, we show that $\operatorname {WAP}(\mathcal {A})$ , the space of weakly almost periodic functionals on ${\mathcal A}$ , admits a right (left) invariant $\varphi $ -mean if and only if $\operatorname {WAP}(I)$ admits a right (left) invariant $\varphi |_I$ -mean. This generalizes a result due to Neufang for the group algebra $L^1(G)$ as an ideal in the measure algebra $M(G)$ , for a locally compact group G. Then we apply this result to the quantum group algebra $L^1({\mathbb G})$ of a locally compact quantum group ${\mathbb G}$ . Finally, we study the existence of left and right invariant $1$ -means on $ \operatorname {WAP}(\mathcal {T}_{\triangleright }({\mathbb G}))$ .
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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) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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".