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

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Wójtowicz M., Wiśniewska H.

Commentationes Mathematicae

2016

tom 56

nr 1

Let \(E\) be a fixed real function \(F\)-space, i.e., \(E\) is an order ideal in \(L_0(S,\Sigma,\mu)\) endowed with a monotone \(F\)-norm \(\|\|\) under which \(E\) is topologically complete. We prove that \(E\) contains an isomorphic (topological) copy of \(\omega\), the space of all sequences, if and only if \(E\) contains a lattice-topological copy \(W\) of \(\omega\). 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 \(\omega\). This solves partially the open problem set recently by W. Wnuk. The property of containing a copy of \(\omega\) by a Musielak−Orlicz space is characterized as follows. (1) A sequence space \(\ell_{\Phi}\), where \(\Phi = (\varphi_n)\), contains a copy of \(\omega\) iff \(\inf_{n \in \mathbb{N}} \varphi_n (\infty) = 0\), where \(\varphi_n (\infty) = \lim_{t \to \infty} \varphi_n (t)\). (2) If the measure \(\mu\) is atomless, then \(\omega\) embeds isomorphically into \(L_{\mathcal{M}} (\mu)\) iff the function \(\mathcal{M}_{\infty}\) is positive and bounded on some set \(A\in \Sigma\) of positive and finite measure, where \(\mathcal{M}_{\infty} (s) = \lim_{n \to \infty} \mathcal{M} (n, s)\), \(s\in S\). In particular, (1)' \(\ell_\varphi\) does not contain any copy of \(\omega\), and (2)' \(L_{\varphi} (\mu)\), with \(\mu\) atomless, contains a~copy \(W\) of \(\omega\) iff \(\varphi\) is bounded, and every such copy \(W\) is uncomplemented in \(L_{\varphi} (\mu)\).

Ograniczanie wyników

1 Commentationes Mathematicae

1 Wiśniewska H.

1 Wójtowicz M.

1 2016

Wyniki wyszukiwania

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