Centrally pure C*-algebras

We show that a separable C*-algebra $A$ is $\mathcal{Z}$-stable if and only if its uncorrected central sequence algebra $A^{\mathrm{\prime }}\cap A_{\mathcal{U}}$ 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 $\mathcal{Z}$-stable if and only if the relative central sequence algebra $B^{\mathrm{\prime }}\cap A_{\mathcal{U}}$ is pure for every separable subalgebra $B\subseteq A_{\mathcal{U}}$.