Nonamenable simple -algebras with tracial approximation

Abstract

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 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.