Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 46L05, 46L30

Retrieve articles in all journals with MSC (1991): 46L05, 46L30


Additional Information

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

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
Email: pmorenz@wfg.com

DOI: http://dx.doi.org/10.1090/S0002-9947-97-01877-1
PII: S 0002-9947(97)01877-1
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