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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Completely positive module maps and completely positive extreme maps

Author: Sze-kai Tsui
Journal: Proc. Amer. Math. Soc. 124 (1996), 437-445
MSC (1991): Primary 46L05, 46L40
MathSciNet review: 1301050
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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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Sze-kai Tsui
Affiliation: Department of Mathematical Sciences, Oakland University, Rochester, Michigan 48309-4401

Keywords: Pure completely positive linear maps, extreme completely linear maps, module maps, strongly independent, Hilbert module representations
Received by editor(s): July 15, 1994
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1996 American Mathematical Society