Nuclear dimension and 𝒵-stability of non-simple C*-algebras
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Bibliographic record
Abstract
We investigate the interplay of the following regularity properties for non-simple <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper C Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">C</mml:mi> </mml:mrow> <mml:mo> ∗ </mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">\mathrm C^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -algebras: finite nuclear dimension, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper Z"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">Z</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal Z</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -stability, and algebraic regularity in the Cuntz semigroup. We show that finite nuclear dimension implies algebraic regularity in the Cuntz semigroup, provided that known type I obstructions are avoided. We demonstrate how finite nuclear dimension can be used to study the structure of the central sequence algebra, by factorizing the identity map on the central sequence algebra, in a manner resembling the factorization arising in the definition of nuclear dimension. Results about the central sequence algebra are used to attack the conjecture that finite nuclear dimension implies <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper Z"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">Z</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {Z}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -stability, for sufficiently non-type I, separable <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper C Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">C</mml:mi> </mml:mrow> <mml:mo> ∗ </mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">\mathrm C^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -algebras. We prove this conjecture in the following cases: (i) the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper C Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">C</mml:mi> </mml:mrow> <mml:mo> ∗ </mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">\mathrm C^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -algebra has no simple purely infinite ideals of quotients and its primitive ideal space has a basis of compact open sets, (ii) the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper C Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">C</mml:mi> </mml:mrow> <mml:mo> ∗ </mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">\mathrm C^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -algebra has no purely infinite quotients and its primitive ideal space is Hausdorff. In particular, this covers <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper C Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">C</mml:mi> </mml:mrow> <mml:mo> ∗ </mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">\mathrm C^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -algebras with finite decomposition rank and real rank zero. Our results hold more generally for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper C Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">C</mml:mi> </mml:mrow> <mml:mo> ∗ </mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">\mathrm C^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -algebras with locally finite nuclear dimension which are <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper M comma upper N right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>,</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(M,N)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -pure (a regularity condition of the Cuntz semigroup).
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
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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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