Representations of Orlicz lattices. (English) Zbl 0566.46018

Wnuk, Witold

Diss. Math. 235, 62 p. (1984).

The paper is devoted to Orlicz lattices which are generalizations of abstract \(L_ p\)-spaces. An Orlicz lattice is defined as a vector lattice with an orthogonal additive modular (in the sense of Nakano). The main purpose of the paper is to present conditions for an Orlicz lattice to be mapped, with the preservation of order and topological properties, into some known function space.
The paper consists of four parts. The first is of an introductory nature and it contains several well known facts from theories of vector lattices and measures. A few theorems on order-topological properties of Musielak- Orlicz spaces are included in the second part. The third part contains the main representation theorem, giving a representation of Orlicz lattices by super order dense sublattices in Musielak-Orlicz spaces. There are also theorems on representations of Orlicz lattices, especially by \(L_ p\)-spaces \((0\leq p<\infty)\). The last part is devoted to ultraproducts of some Orlicz lattices. Bibliographical notes and historical comments complete the paper.

MSC:

46E30	Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46B42	Banach lattices
03C20	Ultraproducts and related constructions

Keywords:

Orlicz lattices; abstract \(L_ p\)-spaces; orthogonal additive modular; vector lattices and measures; order-topological properties of Musielak- Orlicz spaces; representation theorem; super order dense sublattices in Musielak-Orlicz spaces; ultraproducts