Lp{L_{p}}-representations of discrete quantum groups
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Bibliographic record
Abstract
Abstract Given a locally compact quantum group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝔾</m:mi> </m:math> {\mathbb{G}} , we define and study representations and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>C</m:mi> <m:mo>∗</m:mo> </m:msup> </m:math> {\mathrm{C}^{\ast}} -completions of the convolution algebra <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>L</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>𝔾</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {L_{1}(\mathbb{G})} associated with various linear subspaces of the multiplier algebra <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>C</m:mi> <m:mi>b</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>𝔾</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {C_{b}(\mathbb{G})} . For discrete quantum groups <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝔾</m:mi> </m:math> {\mathbb{G}} , we investigate the left regular representation, amenability and the Haagerup property in this framework. When <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝔾</m:mi> </m:math> {\mathbb{G}} is unimodular and discrete, we study in detail the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>C</m:mi> <m:mo>∗</m:mo> </m:msup> </m:math> {\mathrm{C}^{\ast}} -completions of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>L</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>𝔾</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {L_{1}(\mathbb{G})} associated with the non-commutative <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>L</m:mi> <m:mi>p</m:mi> </m:msub> </m:math> {L_{p}} -spaces <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>L</m:mi> <m:mi>p</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>𝔾</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {L_{p}(\mathbb{G})} . As an application of this theory, we characterize (for each <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo>[</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>∞</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {p\in[1,\infty)} ) the positive definite functions on unimodular orthogonal and unitary free quantum groups <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝔾</m:mi> </m:math>
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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 | 0.003 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.002 |
| 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 it