On universal left-stability of $\epsilon$-isometries

Abstract

Let $X$, $Y$ be two real Banach spaces, and $\eps\geq0$. A map $f:X\to Y$ 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 $f:X\to Y$ there is a bounded linear operator $T:L(f)\equiv \overline{\mathrm{span}}f(X)\to X$ such that $\|Tf(x)-x\|\leqγ\eps$ 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 $w^{\ast }$-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; and universally-left-stability spaces are invariant.

10 pages, accepted in Acta Mathematica Sinica, English Series, title changed, typo corrected, arXiv admin note: substantial text overlap with arXiv:1301.3374

Cited by in corpus (2)

• Universal stability of Banach spaces for $\epsilon$-isometries
• On Qian's problem for ${\mathcal{L}}_{\infty }$-spaces