Nonamenable simple $C^*$-algebras with tracial approximation

Xuanlong Fu ; Huaxin Lin

Forum of Mathematics, Sigma, Tome 10 (2022)

Résumé

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.

Détail

DOI : 10.1017/fms.2021.79