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

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- by Douglas R. Farenick and Hongding Zhou PDF
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**126**(1998), 1467-1477 Request permission

## 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
- 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
- © Copyright 1998 American Mathematical Society
- 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