Abstract. In this paper the discussion of normal spectral approximation in a unital $C^*$-algebra is continued. For the approximation by positive or self-adjoint elements and by positive contractions in the $C^*$-norm as well as in a second norm the extreme points of the convex set of all approximants are studied. The main purpose of this paper is to develop sufficient conditions for an approximant to be such an extreme point. As an application many extreme points for the approximation of normal elements are constructed. Moreover for the approximation in the $C^*$-algebra of all bounded linear operators on a complex Hilbert space the number of extreme points is completely determined.

AMS Subject Classification (1991): 47A58

Received August 30, 1993. (Registered under 5578/2009.)