Approximation properties and quantitative estimation for uniform ball-covering property of operator spaces

Abstract

In this paper, by dilation technique on Schauder frames, we extend Godefroy and Kalton's approximation theorem (1997), and obtain that a separable Banach space has the $\hat{I}»$-unconditional bounded approximation property ($\hat{I}»$-UBAP) if and only if, for any $\epsilon >0$, it can be embeded into a $(\hat{I}»+\epsilon )$-complemented subspace of a Banach space with an $1$-unconditional finite-dimensional decomposition ($1$-UFDD). As applications on ball-covering property (BCP) (Cheng, 2006) of operator spaces, also based on the relationship between the $\hat{I}»$-UBAP and block unconditional Schauder frames, we prove that if $X^{\ast }$, $Y$ are separable and (1) $X$ or $Y$ has the $\hat{I}»$-reverse metric approximation property ($\hat{I}»$-RMAP) for some $\hat{I}»>1$; or (2) $X$ or $Y$ has an approximating sequence $\{S_n{\}}_{n=1}^{\infty }$ such that ${\lim }_n\Vert id-2S_n\Vert <3/2$, then the space of bounded linear operators $B(X,Y)$ has the uniform ball-covering property (UBCP). Actually, we give uniformly quantitative estimation for the renormed spaces. We show that if $X^{\ast }$, $Y$ are separable and $X$ or $Y$ has the $(2-\epsilon )$-UBAP for any $\epsilon >0$, then for all $1-\epsilon /2<\hat{I}\pm \le 1$, the renormed space $Z_{\hat{I}}\pm =(B(X,Y),\Vert \cdot {\Vert }_{\hat{I}}\pm )$ has the $(2\hat{I}\pm ,2\hat{I}\pm +\epsilon -2-\ddot{I}ƒ)$-UBCP for all $0<\ddot{I}ƒ<2\hat{I}\pm +\epsilon -2$. Furthermore, we point out the connections between the UBCP, u-ideals and the ball intersection property (BIP).