The Isolated Points of Ĝ<sub>ρ</sub> and the <i>w</i><sup>*</sup>‐Strongly Exposed Points of <i>P</i><sub>ρ</sub>(<i>G</i>)<sub>0</sub>
Bibliographic record
Abstract
Let G be a separable locally compact group and let Ĝ be its dual space with Fell's topology. It is well known that the set P ( G ) of continuous positive‐definite functions on G can be identified with the set of positive linear functionals on the group C * ‐algebra C * ( G ). We show that if π is discrete in Ĝ, then there exists a nonzero positive‐definite function ϕ π associated with π such that ϕ π is a w * ‐strongly exposed point of P ( G ) 0 , where P ( G ) 0 ={ f ∈ P ( G ): f ( e )⩽ 1. Conversely, if some nonzero positive‐definite function ϕ π associated with π is a w * ‐strongly exposed point of P ( G ) 0 , then π is isolated in Ĝ. Consequently, G is compact if and only if, for every π∈Ĝ, there exists a nonzero positive‐definite function associated with π that is a w * ‐strongly exposed point of P ( G ) 0 . If, in addition, G is unimodular and π∈Ĝ ρ , then π is isolated in Ĝ ρ if and only if some nonzero positive‐definite function associated with π is a w * ‐strongly exposed point of P ρ ( G ) 0 , where ρ is the left regular representation of G and Ĝ ρ is the reduced dual space of G . We prove that if B ρ ( G ) has the Radon–Nikodym property, then the set of isolated points of Ĝ ρ (so square‐integrable if G is unimodular) is dense in Ĝ ρ . It is also proved that if G is a separable SIN‐group, then G is amenable if and only if there exists a closed point in Ĝ ρ . In particular, for a countable discrete non‐amenable group G (for example the free group F 2 on two generators), there is no closed point in its reduced dual space Ĝ ρ .
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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 | 0.006 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.001 | 0.003 |
| 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".