The structure of -extreme points in spaces of completely positive linear maps on -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 | References | Similar Articles | Additional Information

Abstract: If is a unital -algebra and if is a complex Hilbert space, then the set of all unital completely positive linear maps from to the algebra of continuous linear operators on is an operator-valued, or generalised, state space of . The usual state space of occurs with the one-dimensional Hilbert space . 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 -extreme points. This work is continued in the present paper, and the main result is a precise description of the structure of the -extreme points of the generalised state spaces of for all finite-dimensional Hilbert spaces .

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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