This is an announcement for the paper "On universally-left-stability of Banach spaces for $\varepsilon$-isometries" by Lingxin Bao, Lixin Cheng, Qingjin Cheng and Duanxu Dai.
Abstract: Let $X$, $Y$ be two real Banach spaces, and $\eps\geq0$. A map $f:X\rightarrow Y$ is said to be a standard $\eps$-isometry if $||f(x)-f(y)|-|x-y||\leq\eps$ for all $x,y\in X$ and with $f(0)=0$. We say that a pair of Banach spaces $(X,Y)$ is stable if there exists $\gamma>0$ such that for every such $\eps$ and every standard $\eps$-isometry $f:X\rightarrow Y$ there is a bounded linear operator $T:L(f)\equiv\overline{{\rm span}}f(X)\rightarrow X$ such that $|Tf(x)-x|\leq\gamma\eps$ for all $x\in X$. $X (Y)$ is said to be left (right)-universally stable, if $(X,Y)$ is always stable for every $Y (X)$. In this paper, we show that if a dual Banach space $X$ is universally-left-stable, then it is isometric to a complemented $w^*$-closed subspace of $\ell_\infty(\Gamma)$ for some set $\Gamma$, hence, an injective space; and that a Banach space is universally-left-stable if and only if it is a cardinality injective space.
Archive classification: math.FA
Mathematics Subject Classification: 46B04, 46B20, 47A58 (Primary) 26E25, 46A20, 46A24 (Secondary)
Remarks: 10 pages, submitted to Acta Mathematica Sinica, English Series
Submitted from: dduanxu@163.com
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1301.3656
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