A note on asymptotically isometric copies of \(\ell^1\) and \(c_0\) (Q2701630)

The paper is concerned with the occurrence of asymptotic \(\ell ^1\) and \(c_0\) copies in Banach spaces in connection with the fixed point property and the theory of \(L\)-summands and \(M\)-ideals. Let \(X\) be a Banach space and \((x_n)\) a sequence of nonzero elements in \(X\). One says that \((x_n)\) spans \(\ell ^1\) almost isometrically (respectively asymptotically) provided there exists a tending to zero sequence \((\delta _m)\) in [0,1) such that \((1-\delta _m)\sum_{n=m}^{\infty}|\alpha_n|\leq \|\sum_{n=m}^{\infty} \alpha_nx_n\|\leq \sum_{n=m}^{\infty}|\alpha_n|\) (respectively \(\sum_{n=1}^{\infty} (1-\delta_n)|\alpha_n|\leq \|\sum_{n=1}^{\infty} \alpha_nx_n\|\leq \sum_{n=1}^{\infty}|\alpha_n|\)), for all scalars \(\alpha_n\). The fact that \((x_n)\) spans \(c_0\) almost isometrically (respectively asymptotically) is defined in a similar way using the operator sup instead of \(\sum\) in the extreme sides of the above inequalities. A mapping \(f:C\to C\), \(C\subset X,\) is called contractive if \(\|f(x)-f(y)\|<\|x-y\|,\) for all \(x,y\in C\). The Banach space \(X\) is said to have the Fixed Point Property (FPP) if any contractive mapping on a nonempty closed bounded convex subset of \(X\) has a fixed point. \textit{P. Dowling} and \textit{C. J. Lennard} [Proc. Am. Math. Soc. 125, No. 2, 443-446 (1997; Zbl 0861.47032)], proved that the presence of an asymptotic copy of \(\ell ^1\)-copy makes a Banach space fail the FPP. In particular, every nonreflexive subspace of \(L^1[0,1]\) fails the FPP. A Banach space is called \(M\)-embedded (\(L\)-embedded) if it is an \(M\)-ideal (respectively an \(L\)-summand) in its bidual [see \textit{P. Harmand, D. Werner} and \textit{W. Werner}, ``M-ideals in Banach spaces and Banach algebras'', Lect. Notes Math. 1547, Berlin (1993; Zbl 0789.46011)]. The first result of the present paper (Lemma 1) is a stability result: almost isometric copies of \(\ell^1\) which are subspaces of \(L\)-embedded spaces are \(L\)-embedded. By James' distortion theorem, every isomorphic copy of \(\ell ^1\) (of \(c_0\)) contains an almost isometrical copy of \(\ell^1\) (of \(c_0\)). \textit{P. Dowling, W. B. Johnson, C. J. Lennard} and \textit{B. Turett} [Proc. Am. Math. Soc. 125, No. 1, 167-174 (1997; Zbl 0860.46005)], have shown that an isomorphic \(\ell^1\)-copy does not contain necessarily isometric \(\ell ^1\)-copies. The author proves (Corollary 3) that a sequence spanning \(\ell^1\) almost isometrically in an \(L\)-embedded Banach space admits a subsequence which spans \(\ell ^1\) asymptotically. Also (Corollary 4), every nonreflexive subspace of an \(L\)-embedded Banach space contains an asymptotic copy of \(\ell^1\) and, consequently, it fails the FPP. The questions whether asymptotic \(\ell ^1\)-copies are always \(L\)-embedded or whether asymptotic \(c_0\)-copies are always \(M\)-embedded remain open. The author proves that there are almost isometric copies of \(\ell ^1\) which are not \(L\)-embedded and that every nonreflexive subspace of an \(M\)-embedded Banach space contains an asymptotic copy of \(c_0\).