Completely positive module maps and completely positive extreme maps

Abstract

Let A , B A,B be unital C ∗ C^* -algebras and P ∞ ( A , B ) P_\infty (A,B) be the set of all completely positive linear maps of A A into B B . In this article we characterize the extreme elements in P ∞ ( A , B , p ) P_\infty (A,B,p) , p = Φ ( 1 ) p=\Phi (1) for all Φ ∈ P ∞ ( A , B , p ) \Phi \in P_\infty (A,B,p) , and pure elements in P ∞ ( A , B ) P_\infty (A,B) in terms of a self-dual Hilbert module structure induced by each Φ \Phi in P ∞ ( A , B ) P_\infty (A,B) . Let P ∞ ( B ( H ) ) R P_\infty (B(H))_R be the subset of P ∞ ( B ( H ) , B ( H ) ) P_\infty (B(H), B(H)) consisting of R R -module maps for a von Neumann algebra R ⊆ B ( H ) R\subseteq B(\mathbb {H}) . We characterize normal elements in P ∞ ( B ( H ) ) R P_\infty (B(H))_R to be extreme. Results here generalize various earlier results by Choi, Paschke and Lin.