Asymptotically isometric copies of \(l_{p}\) \((1 \leq p < \infty)\) and \(c_{0}\) in Banach spaces (Q2495535)

In the last decade, the notions of asymptotically isometric copy of \(\ell_1, c_0, \ell_p\) \((1<p<\infty)\) and \(\ell_\infty\) in Banach spaces has been defined and studied. Typically, the results are of the type `A classical theorem involving copies of such sequence spaces is also true in the asymptotic version'. The paper under review follows this line, and asymptotic versions of two of A.~Pełczyński's now classical results (1. if \(\ell_1\) is isomorphic to a quotient space of \(X\), then \(X\) contains a complemented copy of \(\ell_1\); and 2. every infinite-dimensional closed subspace of \(\ell_1\) contains a complemented subspace of \(\ell_1\) that is isomorphic to \(\ell_1\)) are given. In [\textit{D.-Y. Chen}, J. Math. Anal. Appl. 284, 618--625 (2003; Zbl 1034.46023)], the present author proved that if \(X\) is a separable Banach space such that \(X^\ast\) contains asymptotically isometric copies of \(\ell_1\), then there is a quotient space of \(X\) asymptotically isometric to \(c_0\). This is the asymptotic version of a classical result of Johnson and Rosenthal. In the paper under review, it is shown that the above asymptotic result is true with \(\ell_p\) \((1<p<\infty)\) instead of \(\ell_1\) and \(\ell_q \;(1/p+1/q=1)\) instead of \(c_0\). The first section is a short and concise introduction, the second section contains as main points the results mentioned. The final section is devoted to the task of finding asymptotically isometric copies of \(c_0\) in \(W(X,Y)\), the space of weakly compact operators from \(X\) to \(Y\), by using an asymptotic version of the technique from [\textit{G.~Emmanuele}, J.\ Funct.\ Anal.\ 99, 125--130 (1991; Zbl 0769.46006)]. The third section ends with the following result, which is an asymptotic version of a result also due to G.~Emmanuele: Let \(X\) be a Gelfand-Phillips space. If \(X\) contains an asymptotically isometric copy of \(c_0\), then it has to contain a complemented asymptotically isometric copy of \(c_0\). Here recall that a Banach space is Gelfand-Phillips if any of its limited subsets is weakly compact, so the class of Gelfand--Phillips spaces contains the class of Banach spaces with weak-star sequentially compact dual unit balls.