Boundary quotient C*‐algebras of semigroups
Bibliographic record
Abstract
We study two classes of operator algebras associated with a unital subsemigroup P $P$ of a discrete group G $G$ : one related to universal structures and one related to co-universal structures. First we provide connections between universal C*-algebras that arise variously from isometric representations of P $P$ that reflect the space J $\mathcal {J}$ of constructible right ideals, from associated Fell bundles, and from induced partial actions. This includes connections of appropriate quotients with the strong covariance relations in the sense of Sehnem. We then pass to the reduced representation C λ ∗ ( P ) $\mathrm{C}^*_\lambda (P)$ , and we consider the boundary quotient ∂ C λ ∗ ( P ) $\partial \mathrm{C}^*_\lambda (P)$ related to the minimal boundary space. We show that ∂ C λ ∗ ( P ) $\partial \mathrm{C}^*_\lambda (P)$ is co-universal in two different classes: (a) with respect to the equivariant constructible isometric representations of P $P$ ; and (b) with respect to the equivariant C*-covers of the reduced non-selfadjoint semigroup algebra A ( P ) $\mathcal {A}(P)$ . If P $P$ is an Ore semigroup, or if G $G$ acts topologically freely on the minimal boundary space, then ∂ C λ ∗ ( P ) $\partial \mathrm{C}^*_\lambda (P)$ coincides with the usual C*-envelope C env ∗ ( A ( P ) ) $\mathrm{C}^*_{\text{env}}(\mathcal {A}(P))$ in the sense of Arveson. This covers total orders, finite type and right-angled Artin monoids, the Thompson monoid, multiplicative semigroups of non-zero algebraic integers, and the a x + b $ax+b$ -semigroups over integral domains that are not a field. In particular, we show that P $P$ is an Ore semigroup if and only if there exists a canonical ∗ $*$ -isomorphism from ∂ C λ ∗ ( P ) $\partial \mathrm{C}^*_\lambda (P)$ , or from C env ∗ ( A ( P ) ) $\mathrm{C}^*_{\text{env}}(\mathcal {A}(P))$ , onto C λ ∗ ( G ) $\mathrm{C}^*_\lambda (G)$ . If any of the above holds, then A ( P ) $\mathcal {A}(P)$ is shown to be hyperrigid.
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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 | 0.004 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.002 | 0.003 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.003 | 0.004 |
| Research integrity | 0.000 | 0.004 |
| 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 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".