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The structure of $C^{*}$-extreme points in spaces of completely positive linear maps on $C^{*}$-algebras


Authors: Douglas R. Farenick and Hongding Zhou
Journal: Proc. Amer. Math. Soc. 126 (1998), 1467-1477
MSC (1991): Primary 46L05; Secondary 46L30
DOI: https://doi.org/10.1090/S0002-9939-98-04282-8
MathSciNet review: 1443384
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Abstract: If $A$ is a unital $C^{*}$-algebra and if $H$ is a complex Hilbert space, then the set $S_{H}(A)$ of all unital completely positive linear maps from $A$ to the algebra $B(H)$ of continuous linear operators on $H$ is an operator-valued, or generalised, state space of $A$. The usual state space of $A$ occurs with the one-dimensional Hilbert space ${\mathbb{C}}$. The structure of the extreme points of generalised state spaces was determined several years ago by Arveson [Acta Math. 123(1969), 141-224]. Recently, Farenick and Morenz [Trans. Amer. Math. Soc. 349(1997), 1725-1748] studied generalised state spaces from the perspective of noncommutative convexity, and they obtained a number of results on the structure of $C^{*}$-extreme points. This work is continued in the present paper, and the main result is a precise description of the structure of the $C^{*}$-extreme points of the generalised state spaces of $A$ for all finite-dimensional Hilbert spaces $H$.


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Additional Information

Douglas R. Farenick
Affiliation: Department of Mathematics and Statistics, University of Regina, Regina, Saskatch- ewan, Canada S4S 0A2
Email: farenick@math.uregina.ca

Hongding Zhou
Affiliation: Department of Mathematics and Statistics, University of Regina, Regina, Saskatch- ewan, Canada S4S 0A2
Email: zhouho@math.uregina.ca

DOI: https://doi.org/10.1090/S0002-9939-98-04282-8
Keywords: Generalised state, completely positive map, $C^{*}$-convexity, $C^{*}$-extreme point
Received by editor(s): October 25, 1996
Additional Notes: This work is supported in part by the Natural Sciences and Engineering Research Council of Canada and by the Faculty of Graduate Studies and Research, University of Regina.
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1998 American Mathematical Society

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