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 Γ from the Cuntz semigroup of A to the corresponding 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 Γ 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.

In this paper, we show that every separable simple tracially approximately divisible C *algebra has strict comparison, and, it is either purely infinite or has stable rank one. As a consequence, we show that every (non-unital) finite simple Z-stable C * -algebra has stable rank one.

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 Z-stable C *-algebras.

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