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Extensions of Pure States and Projections of Norm One

Abstract 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, Kumjian's ultraliminary C *-algebras.