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## Commentationes Mathematicae

2016 | 56 | 1 |
Tytuł artykułu

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

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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 $$L_{\varphi} (\mu)$$.
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2016
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2016-10-13
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