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Journal of Convex Analysis 21 (2014), No. 3, 663--680
Copyright Heldermann Verlag 2014

Order Asymptotically Isometric Copies of co in the Subspaces of Order Continuous Elements in Orlicz Spaces

Yunan Cui
Dept. of Mathematics, Harbin University of Science and Technology, Harbin 150080, PR China

Henryk Hudzik
Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, 61-614 Poznan, Poland

Grzegorz Lewicki
Dept. of Mathematics, Jagiellonian University, Lojasiewicza 6, 30-348 Krakow, Poland


Necessary and sufficient conditions in order that the subspace of order continuous elements of Orlicz sequence space contain an order asymptotically isometric copy of $c_0$ are given for both, the Luxemburg and the Amemiya-Orlicz norm. In case of a non-atomic, complete and $\sigma-$finite measure space $(T,\Sigma,\mu)$ and the Luxemburg norm (the Amemiya-Orlicz norm) such criteria are obtained under the additional assumption that the space $L^\Phi(T,\Sigma,\mu)$ is a dual space (resp. the space $L^\Phi_A(T,\Sigma,\mu)$ is a dual and non-square space). In both cases, the Luxemburg and the Amemiya-Orlicz norm the criteria are given under the necessary assumption that the spaces $E^\Phi(T,\Sigma,\mu)$ and $E^\Phi_A(T,\Sigma,\mu)$ are non-trivial. The asymptotically isometric copies of $c_0$ that are built in our theorems are order copies.

Keywords: Orlicz space, subspace of order continuous elements, Luxemburg norm, Amemiya-Orlicz-norm, condition Delta-2, asymptotically isometric copy of c-sub-o, the fixed point property.

MSC: 46B04, 46E30

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