Decomposition rank of UHF-absorbing \(C^\ast\)-algebras. (English) Zbl 1317.46041

Matui, Hiroki; Sato, Yasuhiko

Duke Math. J. 163, No. 14, 2687-2708 (2014).

Summary: Let \(A\) be a unital separable simple \(C^\ast\)-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.

MSC:

46L35	Classifications of \(C^*\)-algebras
46L06	Tensor products of \(C^*\)-algebras
46L55	Noncommutative dynamical systems

Keywords:

\(C^*\)-algebras; Jiang-Su algebra; decomposition rank; tracial rank; Powers-Sakai conjecture