This is an announcement for the paper "Quantifying (weak) injectivity of a Banach space and its second dual" by Duanxu Dai.

Abstract: Let $X$, $Y$ be two Banach spaces. Let $\varepsilon\geq 0$. A mapping $f: X\rightarrow Y$ is said a standard $\varepsilon-$ isometry if $f(0)=0$ and $|f(x)-f(y)|-|x-y|\leq \eps$. In this paper, we first show that if $X$ is a separable Banach space and $Y^*$ has the point of $w^*$-norm continuity property(in short,$w^*$-PCP), then for every standard $\varepsilon-$ isometry $f:X\rightarrow Y$ there exists a $w^*$-dense $G_\delta$ subset $\Omega$ of $ExtB_{X^*}$ such that there is a bounded linear operator $T: Y\rightarrow C(\Omega,\tau_{w^*})$ with $|T|=1$ such that $Tf-Id$ is uniformly bounded by $4\eps$ on $X$. More general results are also given. As a corollary, we obtain quantitative characterizations of injectivity, cardinality injectivity and separably injectivity of a Banach space and its second dual which turn out to give a positive answer to Qian's problem of 1995 in the sense of universality. We also discuss Qian's problem in a $\mathcal{L}_{\infty,\lambda}$-space, $C(K)$-space for a compact Hausdorff space $K$. Moreover, by using some results from Avil$\acute{e}$s-S$\acute{a}$nchez-Castillo-Gonz$\acute{a}$lez- Moreno, Cheng-Dong-Zhang, Johnson-Oikhberg, Rosenthal and Lindenstrauss, estimates for several separably injective Banach spaces are given. Finally, we show a more sharp quantitative and generalized Sobczyk 's theorem.

Archive classification: math.FA

Mathematics Subject Classification: Primary 46B04, 46B20, 47A58, Secondary 26E25, 54C60, 54C65, 46A20

Remarks: 21 page

Submitted from: dduanxu@163.com

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1402.2123

or

http://arXiv.org/abs/1402.2123