A non-commutative unitary analogue of Kirchberg's conjecture
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Bibliographic record
Abstract
The $C^{\ast}$-algebra $\mathcal{U}_{nc}(n)$ is the universal $C^{\ast}$-algebra generated by $n^2$ generators $u_{ij}$ that make up a unitary matrix. We prove that Kirchberg's formulation of Connes' embedding problem has a positive answer if and only if $\mathcal{U}_{nc}(2) \otimes_{\min} \mathcal{U}_{nc}(2)=\mathcal{U}_{nc}(2) \otimes_{\max} \mathcal{U}_{nc}(2)$. Our results follow from properties of the finite-dimensional operator system $\mathcal{V}_n$ spanned by $1$ and the generators of $\mathcal{U}_{nc}(n)$. We show that $\mathcal{V}_n$ is an operator system quotient of $M_{2n}$ and has the OSLLP. We obtain necessary and sufficient conditions on $\mathcal{V}_n$ for there to be a positive answer to Kirchberg's problem. Finally, in analogy with recent results of Ozawa, we show that a form of Tsirelson's problem related to $\mathcal{V}_n$ is equivalent to Connes' Embedding problem.
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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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.004 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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