Asymptotically Isometric Copies of $l^\infty$ in Some Banach Lattices

Abstract:

      In this paper, we show that any $\sigma$-complete Banach lattice, with a $\sigma$-order semi-continuous but not $\sigma$-order continuous norm, contains an asymptotically isometric copy of $l^\infty$. We also get that the Fenchel-Orlicz space with the Orlicz norm may not contain an asymptotically isometric copy of $l^\infty$.