Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Completely distributive CSL algebras
with no complements in $\mathcal C_p$


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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Anoussis and Katsoulis have obtained a criterion for the space $\operatorname{Alg}\,\mathcal L\cap\mathcal C_p$ to have a closed complement in $\mathcal C_p$, where $\mathcal L$ is a completely distributive commutative subspace lattice. They show that, for a given $\mathcal L$, the set of $p$ for which this complement exists forms an interval whose endpoints are harmonic conjugates. Also, they establish the existence of a lattice $\mathcal L$ for which $\operatorname{Alg}\,\mathcal L\cap\mathcal C_p$ has no complement for any $p\not=2$. 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 $\mathcal L$, $\operatorname{Alg}\,\mathcal L\cap\mathcal C_p$ fails to have a complement for any $p\not=2$.


References [Enhancements On Off] (What's this?)

  • [AN] M. Anoussis and E. G. Katsoulis, Complemented subspaces of $\mathcal C_p$ 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 $\mathcal S_p$ classes, Linear Algebra Appl. MR 95f:47050

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 47D25, 47B10

Retrieve articles in all journals with MSC (1991): 47D25, 47B10


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

American Mathematical Society