# On the Structure of the Completion of a Normed Lattice

On the Structure of the Completion of a Normed Lattice Let X be a normed lattice and Y be the norm completion of X with a natural embedding π : X → Y . By the Kawai- Luxemburg theorem, X is embedded as an order dense set and π preserves all suprema and infima iff X satisfies the condition (A o ) (i.e., the norm has pseudo σ-Lebesgue property). Let X o be the largest ideal in X having the condition (A o ); let Y (o) be the band in Y generated by π X o and Y (1) be the complementary band to Y (o) . The structure of Y and, in particular, of the bands Y (o) and Y (1) are studied. The conditions for Y (o) to be a projection band and π X o to be topologically dense in Y (o) are obtained. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# On the Structure of the Completion of a Normed Lattice

, Volume 11 (1) – Oct 13, 2006
19 pages

/lp/springer_journal/on-the-structure-of-the-completion-of-a-normed-lattice-wNEGE8E2bF
Publisher
Springer Journals
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-006-2020-3
Publisher site
See Article on Publisher Site

### Abstract

Let X be a normed lattice and Y be the norm completion of X with a natural embedding π : X → Y . By the Kawai- Luxemburg theorem, X is embedded as an order dense set and π preserves all suprema and infima iff X satisfies the condition (A o ) (i.e., the norm has pseudo σ-Lebesgue property). Let X o be the largest ideal in X having the condition (A o ); let Y (o) be the band in Y generated by π X o and Y (1) be the complementary band to Y (o) . The structure of Y and, in particular, of the bands Y (o) and Y (1) are studied. The conditions for Y (o) to be a projection band and π X o to be topologically dense in Y (o) are obtained.

### Journal

PositivitySpringer Journals

Published: Oct 13, 2006

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