Extensions of Pure States and Projections of Norm One

We show that the extension property for pure states of a C*-subalgebra B of a C*-algebra A leads to the existence of a projection of norm one R: A→B in the case where B is liminal with Hausdorff primitive ideal space. Furthermore, R is given by a “Dixmier process” in which the averaging is effected by a group of unitary elements in the centre of the multiplier algebra M(B). These results generalize earlier work of J. Anderson and the author for the case when B is a masa of A. Various applications are given in the context of inductive limit algebras such as AF algebras and, more generally, Kumjianʹs ultraliminary C*-algebras.