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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Completely positive module maps and completely positive extreme maps
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by Sze-kai Tsui PDF
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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Additional Information
  • Sze-kai Tsui
  • Affiliation: Department of Mathematical Sciences, Oakland University, Rochester, Michigan 48309-4401
  • Email: tsui@vela.acs.oakland.edu
  • Received by editor(s): July 15, 1994
  • Communicated by: Palle E. T. Jorgensen
  • © Copyright 1996 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 124 (1996), 437-445
  • MSC (1991): Primary 46L05, 46L40
  • DOI: https://doi.org/10.1090/S0002-9939-96-03161-9
  • MathSciNet review: 1301050