Completely distributive CSL algebras

with no complements in

Author:
J. A. Erdos

Journal:
Proc. Amer. Math. Soc. **124** (1996), 1127-1131

MSC (1991):
Primary 47D25; Secondary 47B10

DOI:
https://doi.org/10.1090/S0002-9939-96-03134-6

MathSciNet review:
1301023

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Abstract | References | Similar Articles | Additional Information

Abstract: Anoussis and Katsoulis have obtained a criterion for the space to have a closed complement in , where is a completely distributive commutative subspace lattice. They show that, for a given , the set of for which this complement exists forms an interval whose endpoints are harmonic conjugates. Also, they establish the existence of a lattice for which has no complement for any . However, they give no specific example. In this note an elementary demonstration of a simple example of this phenomenon is given. From this it follows that for a wide range of lattices , fails to have a complement for any .

**[AN]**M. Anoussis and E. G. Katsoulis,*Complemented subspaces of spaces and CSL algebras*, J. London Math. Soc. (2)**45**(1992), 301--313. MR**93i:47064****[DP]**K. R. Davidson and S. C. Power,*Failure of the distance formula*, J. London Math. Soc. (2)**32**(1984), 157--165. MR**87e:47056****[F]**J. Froelich,*Compact operators, invariant subspaces and spectral synthesis*, J. Funct. Anal.**81**(1988), 1--37. MR**90b:47078****[OS]**V. Olevskii and M. Solomyak,*An estimate for Schur multipliers in classes*, Linear Algebra Appl. MR**95f:47050**

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Additional Information

**J. A. Erdos**

Email:
J.ERDOS@uk.ac.kcl

DOI:
https://doi.org/10.1090/S0002-9939-96-03134-6

Keywords:
Commutative subspace lattice,
complemented subspace,
von Neumann-Schatten class

Received by editor(s):
October 3, 1994

Communicated by:
Palle E. T. Jorgensen

Article copyright:
© Copyright 1996
American Mathematical Society