Robert-Tikuisis Conjecture (C1): finite nuclear dimension implies Jiang-Su stability without elementary subquotients

Theorem 1.1: for separable nuclear $A$ with no nonzero elementary ideal subquotients, $\dim_{nuc}A<\infty$, $A\cong A\otimes\mathcal Z$ and $\dim_{nuc}A\le1$ are equivalent. Theorem 1.3: $\dim_{nuc}(A_0\otimes\mathcal Z)\le1$ for every separable nuclear $A_0$. Theorem 1.4 answers Kirchberg-Rordam's Question 6.2 (full orthogonal positive elements in a finite maximal tensor power of a unital algebra without characters). It does not address the strict-comparison part of the nonsimple problem, which the Cuntz-comparison companion treats.

Theorem 1.1: for separable nuclear $A$ with no nonzero elementary ideal subquotients, $\dim_{nuc}A<\infty$, $A\cong A\otimes\mathcal Z$ and $\dim_{nuc}A\le1$ are equivalent. Theorem 1.3: $\dim_{nuc}(A_0\otimes\mathcal Z)\le1$ for every separable nuclear $A_0$. Theorem 1.4 answers Kirchberg-Rordam's Question 6.2 (full orthogonal positive elements in a finite maximal tensor power of a unital algebra without characters). It does not address the strict-comparison part of the nonsimple problem, which the Cuntz-comparison companion treats.