Completely positive module maps and completely positive extreme maps

Tsui, Sze-kai

doi:10.1090/S0002-9939-96-03161-9

Completely positive module maps and completely positive extreme maps
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Proc. Amer. Math. Soc. 124 (1996), 437-445 Request permission

Abstract:

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

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