Let A be a separable (not necessarily unital) simple $C^*$-algebra with strict comparison. We show that if A has tracial approximate oscillation zero, then A has stable rank one and the canonical map $\Gamma $ from the Cuntz semigroup of A to the corresponding lower-semicontinuous affine function space is surjective. The converse also holds. As a by-product, we find that a separable simple $C^*$-algebra which has almost stable rank one must have stable rank one, provided it has strict comparison and the canonical map $\Gamma $ is surjective.

We construct two types of unital separable simple $C^*$-algebras: $A_z^{C_1}$ and $A_z^{C_2}$, one exact but not amenable, the other nonexact. Both have the same Elliott invariant as the Jiang–Su algebra – namely, $A_z^{C_i}$ has a unique tracial state,

$$ \begin{align*} \left(K_0\left(A_z^{C_i}\right), K_0\left(A_z^{C_i}\right)_+, \left[1_{A_z^{C_i}} \right]\right)=(\mathbb{Z}, \mathbb{Z}_+,1), \end{align*} $$

and $K_{1}\left (A_z^{C_i}\right )=\{0\}$ ($i=1,2$). We show that $A_z^{C_i}$ ($i=1,2$) is essentially tracially in the class of separable ${\mathscr Z}$-stable $C^*$-algebras of nuclear dimension $1$. $A_z^{C_i}$ has stable rank one, strict comparison for positive elements and no $2$-quasitrace other than the unique tracial state. We also produce models of unital separable simple nonexact (exact but not nuclear) $C^*$-algebras which are essentially tracially in the class of simple separable nuclear ${\mathscr Z}$-stable $C^*$-algebras, and the models exhaust all possible weakly unperforated Elliott invariants. We also discuss some basic properties of essential tracial approximation.

We revisit the notion of tracial approximation for unital simple $C^*$-algebras. We show that a unital simple separable infinite dimensional $C^*$-algebra A is asymptotically tracially in the class of $C^*$-algebras with finite nuclear dimension if and only if A is asymptotically tracially in the class of nuclear $\mathcal {Z}$-stable $C^*$-algebras.