  C. Akemann and G. Pedersen defined three concepts of semicontinuity for
self-adjoint elements of A**, the enveloping von Neumann algebra of a
C*-algebra A. We give the basic properties of the analogous concepts for
elements of pA**p, where p is a closed projection in A**. In other words, in
place of affine functionals on Q, the quasi-state space of A, we consider
functionals on F(p), the closed face of Q supported by p. We prove an
interpolation theorem: If h \geq k, where h is lower semicontinuous on F(p) and
k upper semicontinuous, then there is a continuous affine functional x on F(p)
such that x is between h and k. We also prove an interpolation-extension
theorem: Now h and k are given on Q, x is given on F(p) between h|F(p) and
k|F(p), and we seek to extend x to x on Q so that x is between h and k. We give
a characterization of p(M(A)_sa)p in terms of semicontinuity. And we give new
characterizations of operator convexity and strong operator convexity in terms
of semicontinuity.
