Copies of the sequence space $\omega $ in $F$-lattices with applications to Musielak−Orlicz spaces

Marek Wójtowicz ; Halina Wiśniewska

Commentationes Mathematicae, Tome 56 (2016) no. 1.

Résumé

Let E be a fixed real function F -space, i.e., E is an order ideal in L 0 ( S , Σ , μ ) endowed with a monotone F -norm ∥ ∥ under which E is topologically complete. We prove that E contains an isomorphic (topological) copy of ω , the space of all sequences, if and only if E contains a lattice-topological copy W of ω . If E is additionally discrete, we obtain a much stronger result: W can be a projection band; in particular, E contains a complemented copy of ω . This solves partially the open problem set recently by W. Wnuk. The property of containing a copy of ω by a Musielak−Orlicz space is characterized as follows. (1) A sequence space ℓ Φ , where Φ = ( ϕ n ) , contains a copy of ω iff inf n ∈ ℕ ϕ n ( ∞ ) = 0 , where ϕ n ( ∞ ) = lim t → ∞ ϕ n ( t ) . (2) If the measure μ is atomless, then ω embeds isomorphically into L ℳ ( μ ) iff the function ℳ ∞ is positive and bounded on some set A ∈ Σ of positive and finite measure, where ℳ ∞ ( s ) = lim n → ∞ ℳ ( n , s ) , s ∈ S . In particular, (1)’ ℓ ϕ does not contain any copy of ω , and (2)’ L ϕ ( μ ) , with μ atomless, contains a copy W of ω iff ϕ is bounded, and every such copy W is uncomplemented in L ϕ ( μ ) .

Détail

Mots-clés : F-space, F-lattice, Musielak-Orlicz space, sequence space $\omega $,