Interpolation by projections in C*-algebras

If x is a self-adjoint element in a unital C*-algebra A, and if p(delta) and q(delta) denote the spectral projections of x corresponding to the intervals ]delta, infinity[ and ] -infinity, -delta[, we show that there is a projection p in A such that p(delta) <= p <= 1 - q(delta), provided that delta > dist {x, A(sa)(-1). This result extends to unbounded operators affiliated with a C*-algebra, and has applications to certain other distance functions.