Isometric embeddings of Banach spaces under optimal projection constants

TL;DR

This paper demonstrates that Banach spaces with separable duals can be isometrically embedded into spaces with nearly monotone bases, preserving projection constants and unconditionality, and applies this to fixed point properties.

Contribution

It introduces a method to embed Banach spaces into spaces with nearly monotone bases while controlling projection constants and unconditionality, using renorming and decomposition techniques.

Findings

Embedding of Banach spaces with separable duals into spaces with nearly monotone bases.

Control of projection constants and unconditionality in embeddings.

Application to weak fixed point property for certain Banach spaces.

Abstract

Let $X$ be a Banach space with separable dual. It is proved that for every $\varepsilon\in (0,1)$, $X$ embeds isometrically into a Banach space $W$ with a shrinking basis $(w_n)$ which is $(1+ \varepsilon)$-monotone. Moreover, if $X$ has further an FDD $(E_n)$ whose strong bimonotonicity projection constant is not larger than $\mathcal{D}$, then $(w_n)$ has strong bimonotonicity projection constant not exceeding $\mathcal{D}(1 +\varepsilon)$. Further, if $(E_n)$ is $\mathcal{C}$-unconditional then $(w_n)$ is $\mathcal{C}(1 + \varepsilon)$-unconditional. The proof uses renorming and skipped blocking decomposition techniques. As an application, we prove that every Banach space having a shrinking $\mathcal{D}$-unconditional basis with $\mathcal{D}<\sqrt{6}-1$, has the weak fixed point property.