Bibliographic record
Abstract
For a Banach space X denote by \mathcal{L}(X) the algebra of bounded linear operators on X , by \mathcal{K}(X) the compact operator ideal on X , and by \mathit{Cal}(X) = \mathcal{L}(X)/\mathcal{K}(X) the Calkin algebra of X . We prove that \mathit{Cal}(X) can be an infinite-dimensional reflexive Banach space, even isomorphic to a Hilbert space. More precisely, for every Banach space U with a normalized unconditional basis (u_{s})_{s=1}^{\infty} not having a c_{0} asymptotic version we construct a Banach space \mathfrak{X}_{U} and a sequence of mutually annihilating projections (I_{s})_{s=1}^{\infty} on \mathfrak{X}_{U} , i.e., I_{s}I_{t} = 0 for s\neq t , such that \mathcal{L}(\mathfrak{X}_{U}) = \mathcal{K}(\mathfrak{X}_{U})\oplus[(I_{s})_{s=1}^{\infty}]\oplus\mathbb{C}I and (I_{s})_{s=1}^{\infty} is equivalent to (u_{s})_{s=1}^{\infty} . In particular, \mathit{Cal}(\mathfrak{X}_{U}) is isomorphic, as a Banach algebra, to the unitization of U with coordinatewise multiplication. Banach spaces U meeting these criteria include \ell_{p} and (\bigoplus_{n}\ell_{\infty}^{n})_{p} , 1\leq p<\infty , with their unit vector bases, L_{p} , 1 <p<\infty , with the Haar system, the asymptotic- \ell_{1} Tsirelson space and Schlumprecht space with their usual bases, and many others.
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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 | 0.002 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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".