Decomposition rank of UHF-absorbing C∗-algebras

Abstract

Let $A$ be a unital separable simple ${\mathrm{C}}^{\ast }$-algebra with a unique tracial state. We prove that if $A$ is nuclear and quasidiagonal, then $A$ tensored with the universal uniformly hyperfinite (UHF) algebra has decomposition rank at most one. We then prove that $A$ is nuclear, quasidiagonal, and has strict comparison if and only if $A$ has finite decomposition rank. For such $A$, we also give a direct proof that $A$ tensored with a UHF algebra has tracial rank zero. Using this result, we obtain a counterexample to the Powers–Sakai conjecture.