On universal left-stability of $ϵ$-isometries

Let $X$, $Y$ be two real Banach spaces, and $\text{eps}\ge 0$. A map $f:X\to Y$ is said to be a standard $\text{eps}$-isometry if $\mathrm{\mid }\mathrm{\parallel }f(x)-f(y)\mathrm{\parallel }-\mathrm{\parallel }x-y\mathrm{\parallel }\mathrm{\mid }\le \text{eps}$ 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 $\text{eps}$ and every standard $\text{eps}$-isometry $f:X\rightarrow Y$thereisaboundedlinearoperator$T:L(f)\equiv\overline{{\rm span}}f(X)\rightarrow X$suchthat$|Tf(x)-x|\leq\gamma\eps$forall$x\in X$\mathrm{.}$X (Y)$issaidtobeleft(right)-universallystable,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 ${\mathrm{\ell }}_{\mathrm{\infty }}(\mathrm{\Gamma })$ for some set $\mathrm{\Gamma }$, 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.