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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



$\mathit {C}{}^{*}$-extreme points in the generalised
state spaces of a $\mathit {C}{}^{*}$-algebra

Authors: Douglas R. Farenick and Phillip B. Morenz
Journal: Trans. Amer. Math. Soc. 349 (1997), 1725-1748
MSC (1991): Primary 46L05; Secondary 46L30
MathSciNet review: 1407488
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Abstract: In this paper we study the space $S_{H}(A)$ of unital completely positive linear maps from a $C^{*}$-algebra $A$ to the algebra $B(H)$ of continuous linear operators on a complex Hilbert space $H$. The state space of $A$, in this notation, is $S_{\mathbb {C}}(A)$. The main focus of our study concerns noncommutative convexity. Specifically, we examine the $C^{*}$-extreme points of the $C^{*}$-convex space $S_{H}(A)$. General properties of $C^{*}$-extreme points are discussed and a complete description of the set of $C^{*}$-extreme points is given in each of the following cases: (i) the cases $S_{{\mathbb {C}}^{2}}(A)$, where $A$ is arbitrary ; (ii) the cases $S_{{\mathbb {C}}^{r}}(A)$, where $A$ is commutative; (iii) the cases $S_{{\mathbb {C}}^{r}}(M_{n})$, where $M_{n}$ is the $C^{*}$-algebra of $n\times n$ complex matrices. An analogue of the Krein-Milman theorem will also be established.

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Additional Information

Douglas R. Farenick
Affiliation: Department of Mathematics and Statistics, University of Regina, Regina, Saskatchewan S4S 0A2, Canada

Phillip B. Morenz
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
Address at time of publication: Citadel Investment Group, 225 West Washington, Chicago, Illinois 60606

Keywords: Generalised state, $C^{*}$-convexity, quantum convexity, $C^{*}$-extreme point
Received by editor(s): November 17, 1994
Additional Notes: This work is supported in part by The Natural Sciences and Engineering Research Council of Canada through a research grant (Farenick) and a postdoctoral fellowship (Morenz).
Article copyright: © Copyright 1997 American Mathematical Society