Turbulence, orbit equivalence, and the classification of nuclear <i>C<sup>*</sup> </i>-algebras
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Bibliographic record
Abstract
Abstract. We bound the Borel cardinality of the isomorphism relation for nuclear simple separable C * -algebras: It is turbulent, yet Borel reducible to the action of the automorphism group of the Cuntz algebra <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>𝒪</m:mi> <m:mn>2</m:mn> </m:msub> </m:math> $\mathcal {O}_2$ on its closed subsets. The same bounds are obtained for affine homeomorphism of metrizable Choquet simplexes. As a by-product we recover a result of Kechris and Solecki, namely, that homeomorphism of compacta in the Hilbert cube is Borel reducible to a Polish group action. These results depend intimately on the classification theory of nuclear simple C * -algebras by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">K</m:mi> </m:math> $\mathrm {K}$ -theory and traces. Both of necessity and in order to lay the groundwork for further study on the Borel complexity of C * -algebras, we prove that many standard C * -algebra constructions and relations are Borel, and we prove Borel versions of Kirchberg's <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>𝒪</m:mi> <m:mn>2</m:mn> </m:msub> </m:math> $\mathcal {O}_2$ -stability and embedding theorems. We also find a C * -algebraic witness for a <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>K</m:mi> <m:mi>σ</m:mi> </m:msub> </m:math> $K_\sigma $ hard equivalence relation.
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.006 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.002 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
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