Centrally pure C*-algebras

We show that a separable ⁎ C ⁎ -algebra A is Z -stable if and only if its uncorrected central sequence algebra A ′ ∩ A U is pure, if and only if Kirchberg's central sequence algebra F ( A ) is pure. More generally, we show that a ⁎ C ⁎ -algebra A is separably Z -stable if and only if the relative central sequence algebra B ′ ∩ A U is pure for every separable subalgebra B ⊆ A U .