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Completely positive module maps and completely positive extreme maps. (English) Zbl 0846.46037
Summary: Let A, B be unital C * -algebras and P (A,B) be the set of all completely positive linear maps of A into B. In this article we characterize the extreme elements in P (A,B,p), p=Φ(1) for all ΦP (A,B,p), and pure elements in P (A,B) in terms of a self-dual Hilbert module structure induced by each Φ in P (A,B). Let P (B(H)) R be the subset of P (B(H),B(H)) consisting of R-module maps for a von Neumann algebra RB(). We characterize normal elements in P (B(H)) R to be extreme. Results here generalize various earlier results by Choi, Paschke and Lin.

46L05General theory of C * -algebras
46L40Automorphisms of C * -algebras
46H25Normed modules and Banach modules, topological modules