Completely bounded representations of convolution algebras of locally compact quantum groups
Bibliographic record
Abstract
Given a locally compact quantum group G, we study the structure of completely bounded homomorphisms π:L1(G)→B(H), and the question of when they are similar to ∗-homomorphisms. By analogy with the cocommutative case (representations of the Fourier algebra A(G)), we are led to consider the associated map π∗:L1♯(G)→B(H) given by π∗(ω)=π(ω♯)∗. We show that the corepresentation Vπ of L∞(G) associated to π is invertible if and only if both π and π∗ are completely bounded. Moreover, we show that the co-efficient operators of such representations give rise to completely bounded multipliers of the dual convolution algebra $L^1(\\hat \\mathbb G)$. An application of these results is that any (co)isometric corepresentation is automatically unitary. An averaging argument then shows that when G is amenable, π is similar to a *-homomorphism if and only if π∗ is completely bounded. For compact Kac algebras, and for certain cases of A(G), we show that any completely bounded homomorphism π is similar to a *-homomorphism, without further assumption on π∗. Using free product techniques, we construct new examples of compact quantum groups G such that L1(G) admits bounded, but not completely bounded, representations.
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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) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.002 |
| 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".