Crystallization of $$\hbox {C}^*$$-Algebras
Bibliographic record
Abstract
Abstract Given a $$\hbox {C}^*$$ C ∗ -algebra A with an almost periodic time evolution $$\sigma $$ σ , we define a new $$\hbox {C}^*$$ C ∗ -algebra $$A_c$$ A c , which we call the crystal of $$(A,\sigma )$$ ( A , σ ) , that represents the zero temperature limit of $$(A, \sigma )$$ ( A , σ ) . We prove that there is a one-to-one correspondence between the ground states of $$(A,\sigma )$$ ( A , σ ) and the states on $$A_c$$ A c , justifying the name. In order to investigate further the relation between low temperature equilibrium states on A and traces on $$A_c$$ A c , we define a Fock module $$\mathcal {F}$$ F over the crystal and construct a vacuum representation of A on $$\mathcal {F}$$ F . This allows us to show, under relatively mild assumptions, that for sufficiently large inverse temperatures $$\beta $$ β the $$\sigma $$ σ - $$\hbox {KMS}_\beta $$ KMS β -states on A are induced from traces on $$A_c$$ A c by means of the Fock module. In the second part, we compare the K-theoretic structures of A and $$A_c$$ A c . Previous work by various authors suggests that they have (rationally) isomorphic K-groups. We analyze this phenomenon in detail, confirming it under favorable conditions, but showing that, in general, there is apparently no easy way to relate these groups. As examples, we discuss in particular Exel’s results on semi-saturated circle actions, and recent results of Miller on the K-theory of inverse semigroup $$\hbox {C}^*$$ C ∗ -algebras. In relation to the latter, we introduce the notion of a scale N on an inverse semigroup I and define a new inverse semigroup $$I_c$$ I c , which we call the crystal of (I, N).
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.003 | 0.003 |
| Scholarly communication | 0.003 | 0.004 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.016 | 0.003 |
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".