Hyperbolic Group <i>C</i><sup>*</sup>-Algebras and Free-Product <i>C</i><sup>*</sup>-Algebras as Compact Quantum Metric Spaces
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Bibliographic record
Abstract
Abstract Let ℓ be a length function on a group G , and let M ℓ denote the operator of pointwise multiplication by ℓ on ℓ 2 ( G ). Following Connes, M ℓ can be used as a “Dirac” operator for C* r ( G ). It defines a Lipschitz seminorm on C* r ( G ), which defines a metric on the state space of C* r ( G ). We show that if G is a hyperbolic group and if ℓ is a word-length function on G , then the topology from this metric coincides with the weak-* topology (our definition of a “compact quantum metric space”). We show that a convenient framework is that of filtered C * -algebras which satisfy a suitable “Haagerup-type” condition. We also use this framework to prove an analogous fact for certain reduced free products of C * -algebras.
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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 | 0.008 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.000 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.002 | 0.000 |
| Research integrity | 0.000 | 0.002 |
| Insufficient payload (model declined to judge) | 0.001 | 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