Extreme points in sets of positive linear maps on B(H)

AbstractThree main results are obtained: (1) If D is an atomic maximal Abelian subalgebra of B(H), P is the projection of B(H) onto D and h is a complex homomorphism on D, then h ∘ P is a pure state on B(H). (2) If {Pn} is a sequence of mutually orthogonal projections with rank(Pn) = n and ∑ Pn = I, P is the projection of B(H) onto {Pn}″ given by P(T)=∑tracen(T)Pn and h is a homomorphism on {Pn}″ such that h(Pn) = 0 for all n then h ∘ P induces a type II∞ factor representation of the Calkin algebra. (3) If M is a nonatomic maximal Abelian subalgebra of B(H) then there is an atomic maximal Abelian subalgebra D of B(H) and a large family {Φα} of ∗-homomorphisms from D onto M such that for each α, Φα ∘ P is an extreme point in the set of projections from B(H) onto M. (Here P denotes the projection of B(H) onto D.