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

Wójtowicz M., Wiśniewska H.

Commentationes Mathematicae

2016

Vol 56, No. 1

103--117

Let E be a fixed real function F-space, i.e., E is an order ideal in L0(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 ℓΦ, where Φ=(φn), contains a copy of ω iff infn∈Nφn(∞)=0, where φn(∞)=limt→∞φn(t). (2) If the measure μ is atomless, then ω embeds isomorphically into LM(μ) iff the function M∞ is positive and bounded on some set A∈Σ of positive and finite measure, where M∞ (s)=limn→∞M(n,s), s∈S. In particular, (1)' ℓφ does not contain any copy of ω, and (2)' Lφ(μ), with μ atomless, contains a~copy W of ω iff φ is bounded, and every such copy W is uncomplemented in Lφ(μ).