Extensions of pure states and projections of norm one II

It is shown that if a C*-algebra A contains a semi-scattered C*-algebra B s uch that the pure states of B and the zero functional extend uniquely to A, and the canonical mapping (B) over cap --> (A) over cap is injective, then there exists a (unique) projection of norm one R : A --> B. In certain cir cumstances, the conditional expectation R can be effected by a unitary aver aging process using unitary elements in the centre of the multiplier algebr a M(B).