Extreme points of weakly closed -modules

Authors:
Dong Zhe and Lu Shijie

Journal:
Proc. Amer. Math. Soc. **130** (2002), 461-469

MSC (2000):
Primary 47L75

DOI:
https://doi.org/10.1090/S0002-9939-01-06075-0

Published electronically:
July 25, 2001

MathSciNet review:
1862126

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we first characterize the rank one operators in the preannihilator of a weakly closed -module . Using this characterization for the rank one operators in , a complete description of the extreme points of the unit ball is given. Finally, we show how to apply the techniques of the present paper to other operator systems and characterize their extreme points.

**[1]**M.Anoussis and E.G.Katsoulis,*A non-selfadjoint Russo-Dye theorem*, Math. Ann. 304 (1996), 685-699. MR**97f:47042****[2]**M.Anoussis and E.G.Katsoulis,*Compact operators and the geometric structure of C-algebras*, Proc. Amer. Math. Soc. 124 (1996), 2115-2122. MR**96i:46068****[3]**J.Arazy and B.Solel,*Isometries of non-selfadjoint operator algebras*, J. Funct. Anal. 90 (1990), 284-305. MR**91c:47085****[4]**K.R.Davidson,*The Russo-Dye theorem in nest algebras*, Proc. Amer. Math. Soc.(10) 126 (1998), 3055-3059. MR**2000f:47109****[5]**J.A.Erdos and S.C.Power,*Weakly closed ideals of nest algebras*, J. Operator Theory 7 (1982), 219-235. MR**84a:47056****[6]**T.D.Hudson, E.G.Katsoulis and D.R.Larson,*Extreme points in triangular UHF algebras*, Trans. Amer. Math. Soc. 349 (1997), 3391-3400. MR**97m:47058****[7]**R.V.Kadison,*Isometries of operator algebras*, Ann. of Math.(2) 54 (1951), 325-338. MR**13:256a****[8]**Lu Fanyan and Lu Shijie,*Finite rank operators in some ideals of nest algebras*, Acta. Math. Sinica (Chinese) (1) 41 (1998), 113-118.**[9]**R.L.Moore and T.T.Trent,*Extreme points of certain operator algebras*, Indiana U. Math. J. 36 (1987), 645-650. MR**89d:47103****[10]**R.L.Moore and T.T.Trent,*Isometries of nest algebras*, J. Funct. Anal. 86 (1989), 180-209. MR**90k:47096**

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Additional Information

**Dong Zhe**

Affiliation:
Department of Mathematics, Zhejiang University, Hangzhou 310027, People’s Republic of China

Address at time of publication:
Institute of Mathematics, Fudan University, Shanghai 200433, People’s Republic of China

Email:
dzhe8@china.com

**Lu Shijie**

Affiliation:
Department of Mathematics, Zhejiang University, Hangzhou 310027, People’s Republic of China

DOI:
https://doi.org/10.1090/S0002-9939-01-06075-0

Keywords:
Weakly closed $\mathcal{ T(N)}$--module,
preannihilator,
extreme point,
contractive perturbation

Received by editor(s):
November 15, 1999

Received by editor(s) in revised form:
June 26, 2000

Published electronically:
July 25, 2001

Additional Notes:
This work was supported by the National Natural Science Foundation of China.

Communicated by:
David R. Larson

Article copyright:
© Copyright 2001
American Mathematical Society