Bibliographic record
Abstract
We consider a new class of potentially exotic group C*-algebras C ⁎ ( PF p ⁎ ( G ) ) for a locally compact group G , and its connection with the class of potentially exotic group C*-algebras C L p ⁎ ( G ) introduced by Brown and Guentner. Surprisingly, these two classes of C*-algebras are intimately related. By exploiting this connection, we show C L p ⁎ ( G ) = C ⁎ ( PF p ⁎ ( G ) ) for p ∈ ( 2 , ∞ ) , and the C*-algebras C L p ⁎ ( G ) are pairwise distinct for p ∈ ( 2 , ∞ ) when G belongs to a large class of nonamenable groups possessing the Haagerup property and either the rapid decay property or Kunze-Stein phenomenon by characterizing the positive definite functions that extend to positive linear functionals of C L p ⁎ ( G ) and C ⁎ ( PF p ⁎ ( G ) ) . This greatly generalizes earlier results of Okayasu (see [30] ) and the second author (see [40] ) on the pairwise distinctness of C L p ⁎ ( G ) for 2 < p < ∞ when G is either a noncommutative free group or the group SL ( 2 , R ) , respectively. As a byproduct of our techniques, we present two applications to the theory of unitary representations of a locally compact group G . Firstly, we give a short proof of the well-known Cowling-Haagerup-Howe Theorem, which presents sufficient condition implying the weak containment of a cyclic unitary representation of G in the left regular representation of G (see [14] ). Also we give a near solution to a 1978 conjecture of Cowling stated in [10] . This conjecture of Cowling states if G is a Kunze-Stein group and π is a unitary representation of G with cyclic vector ξ such that the map G ∋ s ↦ 〈 π ( s ) ξ , ξ 〉 belongs to L p ( G ) for some 2 < p < ∞ , then A π ⊆ L p ( G ) . We show B π ⊆ L p + ϵ ( G ) for every ϵ > 0 (recall A π ⊆ B π ).
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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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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 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".