The Toms-Winter conjecture: strict comparison implies Jiang-Su stability for simple nuclear C*-algebras

Theorem 1.1: strict comparison implies $A\cong A\otimes\mathcal Z$ for every separable simple nuclear non-elementary $A$, unital or not. More generally (Theorem 1.2) every separable nuclear algebra whose Cuntz semigroup is almost unperforated and fully almost divisible is $\mathcal Z$-stable, which answers the implication (iii) to (ii) of the nonsimple Toms-Winter problem (STW Problem LXXV) and the abstract Cuntz-semigroup question (STW Problem XXVI). Separability and nuclearity remain hypotheses; nonnuclear and non-separable algebras are not treated.

Theorem 1.1: strict comparison implies $A\cong A\otimes\mathcal Z$ for every separable simple nuclear non-elementary $A$, unital or not. More generally (Theorem 1.2) every separable nuclear algebra whose Cuntz semigroup is almost unperforated and fully almost divisible is $\mathcal Z$-stable, which answers the implication (iii) to (ii) of the nonsimple Toms-Winter problem (STW Problem LXXV) and the abstract Cuntz-semigroup question (STW Problem XXVI). Separability and nuclearity remain hypotheses; nonnuclear and non-separable algebras are not treated.