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

TL;DR

This paper extends approximation theorems for Banach spaces using dilation techniques, and applies these results to establish uniform ball-covering properties of operator spaces with quantitative estimates.

Contribution

It introduces new dilation-based methods to characterize the uniform ball-covering property in operator spaces and provides explicit quantitative bounds for renormed spaces.

Findings

Separable Banach spaces have $ ext{UBAP}$ iff they embed into complemented subspaces with $1$-UFDD.

Operator spaces with certain approximation properties have the UBCP.

Quantitative estimates for renormed spaces' UBCP are established.

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 $\lambda$-unconditional bounded approximation property ($\lambda$-UBAP) if and only if, for any $\varepsilon>0$, it can be embeded into a $(\lambda+\varepsilon)$-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 $\lambda$-UBAP and block unconditional Schauder frames, we prove that if $X^\ast$, $Y$ are separable and (1) $X$ or $Y$ has the $\lambda$-reverse metric approximation property ($\lambda$-RMAP) for some $\lambda>1$; or (2) $X$ or $Y$ has an approximating sequence $\{S_n\}_{n=1}^\infty$ such that $\lim_n\|id-2S_n\| < 3/2$, 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 $(2-\varepsilon)$-UBAP for any $\varepsilon>0$, then for all $1-\varepsilon/2 < \alpha \leq 1$, the renormed space $Z_\alpha=(B(X,Y),\|\cdot\|_\alpha)$ has the $(2\alpha, 2\alpha+\varepsilon-2-\sigma)$-UBCP for all $0 <\sigma < 2\alpha+\varepsilon-2$. Furthermore, we point out the connections between the UBCP, u-ideals and the ball intersection property (BIP).