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 ϕπ associated with π such that ϕπ 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 ϕπ 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∈Ĝρ, then π is isolated in Ĝρ if and only if some nonzero positive‐definite function associated with π is a w*‐strongly exposed point of Pρ(G)0, where ρ is the left regular representation of G and Ĝρ is the reduced dual space of G. We prove that if Bρ(G) has the Radon–Nikodym property, then the set of isolated points of Ĝρ (so square‐integrable if G is unimodular) is dense in Ĝρ. 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 Ĝρ. In particular, for a countable discrete non‐amenable group G (for example the free group F2 on two generators), there is no closed point in its reduced dual space Ĝρ.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.000 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
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 source (direct Gemma or distilled Codex), 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".