C*-algebras of Essentially Tracially Rank One

Abstract

We introduce the notion of essential tracial rank at most one for simple $C^*$-algebras. We show that separable simple $C^*$-algebras with essential tracial rank at most one have several regularity properties, including stable rank one, strict comparison, and tracial approximate divisibility. A finite amenable simple $C^*$-algebras with this property is $\mathcal Z$-stable. We also prove the converse. Thus this property gives a new characterization of separable simple finite amenable $\mathcal Z$-stable $C^*$-algebras and shows that these algebras admit a one-dimensional tracial approximation. In the unital real-rank-zero case, this approximation can be strengthened to show that the algebra has tracial rank zero. This completes a long-standing project. As a byproduct, we prove that every tracial state on a unital separable simple finite amenable $C^*$-algebra is quasidiagonal and that the algebra itself is quasidiagonal. Consequently, every separable simple finite amenable $C^*$-algebra of finite nuclear dimension has finite decomposition rank. None of these results requires the UCT.