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On finite elements in sublattices of Banach lattices

✍ Scribed by Z. L. Chen; M. R. Weber

Publisher
John Wiley and Sons
Year
2007
Tongue
English
Weight
162 KB
Volume
280
Category
Article
ISSN
0025-584X

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✦ Synopsis

Abstract

Let E be a Banach lattice. Let H stand for a sublattice, an ideal or a band in E, and denote by Φ~1~(E) and Φ~1~(H) the ideals of finite elements in the vector lattices E and H, respectively. In this paper we first present some sufficient conditions and some counterexamples for the inclusions Φ~1~(H) ⊂ Φ~1~(E) and Φ~1~(E) ∩ H ⊂ Φ~1~(H) to hold or not. For closed ideals (and therefore bands) H there always holds Φ~1~(H) ⊂ Φ~1~(E). If H is a projection band then even P~H~ Φ~1~(E) = Φ~1~(E) ∩ H = Φ~1~(H). It is proved that every finite element of E is also finite both in its Dedekind completion Ê and in its bidual space E ″. Some results concerning the finite elements in direct sums of Banach lattices are also included. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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