-extreme points in the generalised

state spaces of a -algebra

Authors:
Douglas R. Farenick and Phillip B. Morenz

Journal:
Trans. Amer. Math. Soc. **349** (1997), 1725-1748

MSC (1991):
Primary 46L05; Secondary 46L30

DOI:
https://doi.org/10.1090/S0002-9947-97-01877-1

MathSciNet review:
1407488

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we study the space of unital completely positive linear maps from a -algebra to the algebra of continuous linear operators on a complex Hilbert space . The state space of , in this notation, is . The main focus of our study concerns noncommutative convexity. Specifically, we examine the -extreme points of the -convex space . General properties of -extreme points are discussed and a complete description of the set of -extreme points is given in each of the following cases: (i) the cases , where is arbitrary ; (ii) the cases , where is commutative; (iii) the cases , where is the -algebra of 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

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:
https://doi.org/10.1090/S0002-9947-97-01877-1

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