A simple proof for a theorem of Luxemburg and Zaanen

In this paper a simple proof for the following theorem, due to Luxemburg and Zaanen is given: an Archimedean vector lattice A is Dedekind σ-complete if and only if A has the principal projection property and A is uniformly complete. As an application, we give a new and short proof for the following version of Freudenthal’s spectral theorem: let A be a uniformly complete vector lattice with the principal projection property and let 0 < u ∈ A. For any element w in A such that 0 w u there exists a sequence {sn: n = 1, 2, . . .} in A which satisfies 0 sn w (u), where each element sn is of the form k i=1 αipi , with real numbers α1, . . . , αk such that 0 αi 1 (i = 1, . . . , k) and mutually disjoint components p1, . . . , pk of u. © 2005 Elsevier Inc. All rights reserved