Isometric embedding of Banach spaces under optimal projection constants

Abstract

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