On finite elements in sublattices of Banach lattices

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)