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 distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 teacher head, 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".