-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.

**1.**W.B. Arveson,*Subalgebras of -algebras*, Acta Math.**123**(1969), 141-224. MR**40:6274****2.**W.B. Arveson,*Subalgebras of -algebras,II*, Acta Math.**128**(1972), 271-308. MR**52:15035****3.**R. Bhat, V. Pati, and V.S. Sunder,*On some convex sets and their extreme points*, Math. Ann.**296**(1993), 637-648. MR**94f:46076****4.**J. Bunce and N. Salinas,*Completely positive maps on -algebras and the left matricial spectra of an operator*, Duke Math. J.**43**(1976), 747-777. MR**55:3798****5.**M.-D. Choi,*Completely positive linear maps on complex matrices*, Linear Algebra Appl.**10**(1975), 285-290. MR**51:12901****6.**E.G. Effros and S. Winkler,*Matrix convexity: operator analogues of the bipolar and Hahn-Banach theorems*, preprint, 1995.**7.**D.R. Farenick,*-convexity and matricial ranges*, Canad. J. Math**44**(1992), 280-297. MR**93j:46060****8.**D.R. Farenick and P.B. Morenz,*-extreme points of some compact -convex sets*, Proc. Amer. Math. Soc.**118**(1993), 765-775. MR**93i:46096****9.**I. Fujimoto,*CP-duality for - and -algebras*, J. Operator Theory**30**(1993), 201-216. MR**96b:46076****10.**A. Hopenwasser, R.L. Moore, and V.I. Paulsen,*-extreme points*, Trans. Amer. Math. Soc.**266**(1981), 291-307. MR**82f:46065****11.**L.J. Landau and R.F. Streater,*On Birkhoff's theorem for doubly stochastic completely positive maps of matrix algebras*, Linear Algebra Appl.**193**(1993), 107-127. MR**95c:47041****12.**R.I. Loebl,*A remark on unitary orbits*, Bull. Instit. Math. Acad. Sinica**7**(1979), 401-407. MR**80m:47035****13.**R.I. Loebl and V.I. Paulsen,*Some remarks on -convexity*, Linear Algebra Appl.**35**(1981), 63-78. MR**82b:46077****14.**P.B. Morenz,*The structure of -convex sets*, Canad. J. Math.**46**(1994), 1007-1026. MR**95k:46095****15.**R.R. Smith and J.D. Ward,*The geometric structure of generalized state spaces*, J. Funct. Anal.**40**(1981), 170-184. MR**82i:46094****16.**W.F. Stinespring,*Positive functions on -algebras*, Proc. Amer. Math. Soc.**6**(1955), 211-216. MR**16:1033b****17.**E. Størmer,*Positive linear maps of operator algebras*, Acta Math.**110**(1963), 233-278. MR**27:6145****18.**S.-K. Tsui,*Extreme -positive linear maps*, Proc. Edin. Math. Soc.**36**(1993), 123-131. MR**94b:46088****19.**S.-K. Tsui,*Completely positive module maps and completely positive extreme maps*, Proc. Amer. Math. Soc.**124**(1996), 437-445. MR**96d:46074**

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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