|
The Krein-Milman theorem in operator convexity
Author(s):
Corran
Webster;
Soren
Winkler
Journal:
Trans. Amer. Math. Soc.
351
(1999),
307-322.
MSC (1991):
Primary 47D20;
Secondary 46A55, 46L89
Retrieve article in:
PDF DVI PostScript
This article is available free of charge
Abstract |
References |
Similar articles |
Additional information
Abstract:
We generalize the Krein-Milman theorem to the setting of matrix convex sets of Effros-Winkler, extending the work of Farenick-Morenz on compact C -convex sets of complex matrices and the matrix state spaces of C -algebras. An essential ingredient is to prove the non-commutative analogue of the fact that a compact convex set  may be thought of as the state space of the space of continuous affine functions on  .
References:
- 1.
- E. M. Alfsen, Compact Convex Sets and Boundary Integrals, Springer Verlag, New York, 1971. MR 56:3615
- 2.
- W. Arveson, Subalgebras of C
 -algebras, I, Acta Math., 123, 141-224, 1969. MR 40:6274 - 3.
- W. Arveson, Subalgebras of C
 -algebras, II, Acta Math., 128, 271-308, 1972. MR 52:15035 - 4.
- M-D Choi, Completely positive linear maps on complex matrices, Linear Algebra Appl., 10, 285-290, 1975. MR 51:12901
- 5.
- M-D Choi and E. G. Effros, Injectivity and operator spaces, J. Func. Anal., 24, 156-209, 1977. MR 55:3814
- 6.
- K. R. Davidson, A proof of the boundary theorem, Proc. Amer. Math. Soc., 82, 48-50, 1981. MR 82e:46075
- 7.
- D. Farenick, C
 -convexity and matricial ranges, Canad. J. Math., 44, 280-297, 1992. MR 93j:46060 - 8.
- D. Farenick and P. Morenz, C
 -extreme points of some compact C -convex sets, Proc. Amer. Math. Soc., 118, 765-775, 1993. MR 93i:46096 - 9.
- D. Farenick and P. Morenz, C
 -extreme points in the generalised state spaces of a C -algebra, Trans. Amer. Math. Soc., 349, 1725-1748, 1997. MR 97h:46100 - 10.
- D. Fujimoto, CP-duality for C
 - and W -algebras, J. Operator Theory, 30, 201-216, 1993. MR 96b:46076 - 11.
- E. Effros and S. Winkler, Matrix convexity: Operator analogues of the Bipolar and Hahn-Banach theorems, J. of Functional Analysis, 144, 117-152, 1997. MR 98e:46066
- 12.
- A. Hopenwasser, R. L. Moore and V. I. Paulsen, C
 -extreme points, Trans. Amer. Math. Soc., 266, 291-307, 1981. MR 82f:46065 - 13.
- R. V. Kadison, A representation theory for commutative topological algebra, Mem. Amer. Math. Soc., 7, 1951. MR 13:360b
- 14.
- R. Loebl, A remark on unitary orbits, Bull. Inst. Math. Acad. Sinica, 7, 401-407, 1979. MR 80m:47035
- 15.
- R. Loebl and V. I. Paulsen, Some remarks on C
 -convexity, Linear Algebra Appl., 78, 63-78, 1981. MR 82b:46077 - 16.
- C. Le Merdy, On the duality of operator spaces, Canad. Math. Bull., 38, 334-346, 1995. MR 97a:46022
- 17.
- P. Morenz, The structure of C
 -convex sets, Canad. J. Math., 46, 1007-1026, 1994. MR 95k:46095 - 18.
- Z.-J. Ruan, Subspaces of C
 -algebras, J. Funct. Anal., 29, 217-230, 1988. MR 89k:46082 - 19.
- N. Salinas, Extensions of C
 -algebras and  -normal operators, Bull. Amer. Math. Soc., 82, 143-146, 1976. MR 58:30297 - 20.
- R. R. Smith and J. D. Ward, Matrix ranges for Hilbert space operators, Amer. J. Math. 102, 1031-1081, 1980. MR 82a:46067
- 21.
- S. Winkler, Matrix convexity, Ph. D. thesis, UCLA, 1996.
- 22.
- G. Wittstock, On Matrix Order and Convexity, in Functional Analysis, Survey and Recent Results III, Math. Studies 90, 175-188, 1984. MR 86a:46077
Similar Articles:
Retrieve articles in Transactions of the American Mathematical Society
with MSC
(1991):
47D20,
46A55, 46L89
Retrieve articles in all Journals with MSC
(1991):
47D20,
46A55, 46L89
Additional Information:
Corran
Webster
Affiliation:
Department of Mathematics, University of California, Los Angeles, California 90095
Address at time of publication:
Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email:
Corran.Webster@math.tamu.edu
Soren
Winkler
Affiliation:
University of Wales, Swansea, Singleton Park, Swansea SA2 8PP, UK
Email:
SWI@simcorp.dk
DOI:
10.1090/S0002-9947-99-02364-8
PII:
S 0002-9947(99)02364-8
Keywords:
Krein-Milman,
non-commutative convexity
Received by editor(s):
January 22, 1997
Additional Notes:
The first author was supported by the NSF and the second author by the EPSRC and the EU
Copyright of article:
Copyright
1999,
American Mathematical Society
|
Comments: webmaster@ams.org