Convolution of Trace Class Operators over Locally Compact Quantum Groups
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Bibliographic record
Abstract
Abstract We study locally compact quantum groups 𝔾 through the convolution algebras L 1 (𝔾) and (T(L 2 (𝔾)); ⊳). We prove that the reduced quantum groupC*-algebraC 0 (𝔾) can be recovered fromthe convolution ⊳ by showing that the right T(L 2 (𝔾))-module 〈K(L 2 (𝔾)) ⊳ T(L 2 (𝔾))〉 is equal to C 0 (𝔾). On the other hand, we show that the left T(L 2 (𝔾)(-module 〈T(L 2 (𝔾)) ⊳ K(L 2 (𝔾))〉 is isomorphic to the reduced crossed product C 0 (Ĝ ) r⋉C 0 (𝔾), and hence is a much larger C*-subalgebra of B(L 2 (𝔾)). We establish a natural isomorphism between the completely bounded right multiplier algebras of L 1 (𝔾) and (T(L 2 (𝔾)); ⊳), and settle two invariance problems associated with the representation theorem of Junge–Neufang–Ruan (2009). We characterize regularity and discreteness of the quantum group 𝔾 in terms of continuity properties of the convolution . on T(L 2 (𝔾))⊳and settle two invariance problems associated with the representation theorem of Junge–Neufang–Ruan (2009). We characterize regularity and discreteness of the quantum group 𝔾 in terms of continuity properties of the convolution ⊳ on T(L 2 (𝔾)). We prove that if 𝔾 is semiregular, then the space 〈T(L 2 (𝔾)) ⊳ B(L 2 (𝔾))〉 of right 𝔾-continuous operators on L 2 (𝔾), which was introduced by Bekka (1990) for L 1 (𝔾), is a unital C*-subalgebra of B(L 2 (𝔾)). In the representation framework formulated by Neufang–Ruan–Spronk (2008) and Junge–Neufang–Ruan, we show that the dual properties of compactness and discreteness can be characterized simultaneously via automatic normality of quantum group bimodule maps on B(L 2 (𝔾)). We also characterize some commutation relations of completely bounded multipliers of (T(L 2 (𝔾)); ⊳) over B(L 2 (𝔾)).
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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 | 0.002 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
Machine scores (provisional)
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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 it