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

Marek Wójtowicz, and Halina Wiśniewska. "Copies of the sequence space $\omega $ in $F$-lattices with applications to Musielak−Orlicz spaces." Commentationes Mathematicae 56.1 (2016): null. <http://eudml.org/doc/292477>.

@article{MarekWójtowicz2016,
abstract = {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 $\Vert \Vert $ 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 $\varphi _n (\infty ) = \lim _\{t \rightarrow \infty \} \varphi _n (t)$. (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 $\mathcal \{M\}_\{\infty \} (s) = \lim _\{n \rightarrow \infty \} \mathcal \{M\} (n, s)$, $s\in S$. 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 $L_\{\varphi \} (\mu )$.},
author = {Marek Wójtowicz, Halina Wiśniewska},
journal = {Commentationes Mathematicae},
keywords = {F-space; F-lattice; Musielak-Orlicz space; sequence space $\omega $; },
language = {eng},
number = {1},
pages = {null},
title = {Copies of the sequence space $\omega $ in $F$-lattices with applications to Musielak−Orlicz spaces},
url = {http://eudml.org/doc/292477},
volume = {56},
year = {2016},
}

TY - JOUR
AU - Marek Wójtowicz
AU - Halina Wiśniewska
TI - Copies of the sequence space $\omega $ in $F$-lattices with applications to Musielak−Orlicz spaces
JO - Commentationes Mathematicae
PY - 2016
VL - 56
IS - 1
SP - null
AB - 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 $\Vert \Vert $ 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 $\varphi _n (\infty ) = \lim _{t \rightarrow \infty } \varphi _n (t)$. (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 $\mathcal {M}_{\infty } (s) = \lim _{n \rightarrow \infty } \mathcal {M} (n, s)$, $s\in S$. 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 $L_{\varphi } (\mu )$.
LA - eng
KW - F-space; F-lattice; Musielak-Orlicz space; sequence space $\omega $;
UR - http://eudml.org/doc/292477
ER -