The Toms–Winter regularity equivalence
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For simple separable unital infinite-dimensional nuclear C*-algebras, do strict comparison, Jiang–Su absorption and finite nuclear dimension coincide? The remaining direction studied here is comparison implying absorption.
Why it matters and what counts as progress
Why it matters: Equivalence would unify three central regularity properties without restricted trace-boundary assumptions.
What counts as progress: The comparison-to-absorption implication throughout this unital class, or a counterexample; nonunital extensions require their own hypotheses.
How the problem developed
- 2010 · Conjecture [1]
Andrew S. Toms and Wilhelm Winter A contemporary account stated their conjecture that finite nuclear dimension, Jiang–Su stability, and strict comparison coincide for simple unital nuclear C*-algebras.
- 27 Feb 2015 · Partial resolution [2]
Yasuhiko Sato, Stuart White, and Wilhelm Winter Completed the equivalence of finite nuclear dimension and Jiang–Su stability in the monotracial case.
OpenAI Claimed the unital Toms–Winter equivalence in the simple separable unital infinite-dimensional nuclear setting, combining comparison, nuclear-dimension, and Jiang–Su absorption manuscripts.
AI contribution
Cuntz comparison and Jiang–Su absorption
For simple separable unital infinite-dimensional nuclear C*-algebras, the companion manuscripts claim strict comparison implies Jiang–Su absorption, completing the unital Toms–Winter equivalence with finite nuclear dimension. A separate extended-functional theorem also treats nonunital algebras under its own assumptions.
Sources
- Winter, Report on the classification of simple C*-algebras
- Sato, White, and Winter, Nuclear dimension and Z-stability
- OpenAI, Cuntz comparison and Jiang–Su absorption
- OpenAI, Nuclear dimension and Jiang–Su stability without elementary subquotients
- OpenAI, Tracial projection methods and uniform property Gamma
- Cuntz comparison and Jiang–Su absorption
- OpenAI mathematics release
- Tracial projection methods and uniform property Gamma
- Cuntz comparison and Jiang–Su absorption
- Uniform Gamma formalization scope
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