Nonamenable simple -algebras with tracial approximation

Abstract 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 same Elliott invariant as 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*} $K_{1}\left (A_z^{C_i}\right )=\{0\}$ ( $i=1,2$ ). show that ) is essentially tracially in class ${\mathscr Z}$ -stable -algebras nuclear dimension $1$ . stable rank one, strict comparison for positive elements no $2$ -quasitrace than state. also produce models nonexact (exact nuclear) which are -algebras, exhaust all possible weakly unperforated invariants. discuss some basic properties essential approximation.