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

TL;DR

This paper investigates the stability of $ ext{epsilon}$-isometries between Banach spaces, establishing conditions under which spaces are universally stable and characterizing such spaces as injective or cardinality-injective.

Contribution

It characterizes universally-left-stable Banach spaces as injective or cardinality-injective, linking stability of approximate isometries to fundamental space properties.

Findings

Dual Banach spaces that are universally-left-stable are isometric to complemented $w^*$-closed subspaces of $ell_ty()$.

A Banach space is universally-left-stable if and only if it is cardinality-injective.

Universally-left-stability spaces are invariant under certain transformations.

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; and universally-left-stability spaces are invariant.