Let A and B denote complex
Banach ∗-algebras and L(A,B) the space of continuous linear operators from
A into B. Let P ⊂ L(A,B) be the convex set of positive linear operators
of norm ≦ 1. If A has an identity, and if B is semi-simple and symmetric,
the multiplicative operators of P are shown to be extreme points of P. If,
on the other hand, it is assumed that, ∥T∥ = ∥Te∥ for τ ∈ P, then any
extreme point T of P satisfies TeTab = TaTb for all a,b ∈ A. With A as above
and B a B∗-algebra, the extreme points of P are multiplicative. Thus we
characterize the extreme points of P ⊂ L(A,C(X)) as the multiplicative operators.
The results are extended to include the case when A has an approximate
identity.
