An isometrically universal Banach space induced by a non-universal Boolean algebra

TL;DR

This paper constructs a specific non-universal Boolean algebra that induces an isometrically universal Banach space, revealing differences in universality properties between separable and nonseparable cases.

Contribution

It demonstrates the existence of a non-universal Boolean algebra that still induces an isometrically universal Banach space, highlighting a distinction in universality phenomena.

Findings

Constructed a Boolean algebra with no uncountable well-ordered chains

Embedded $C(K_A)$ into $C(K_B)$ isometrically

Showed the non-equivalence of universality for Boolean algebras and Banach spaces in nonseparable cases

Abstract

Given a Boolean algebra $A$, we construct another Boolean algebra $B$ with no uncountable well-ordered chains such that the Banach space of real valued continuous functions $C(K_A)$ embeds isometrically into $C(K_B)$, where $K_A$ and $K_B$ are the Stone spaces of $A$ and $B$ respectively. As a consequence we obtain the following: If there exists an isometrically universal Banach space for the class of Banach spaces of a given uncountable density $\kappa$, then there is another such space which is induced by a Boolean algebra which is not universal for Boolean algebras of cardinality $\kappa$. Such a phenomenon cannot happen on the level of separable Banach spaces and countable Boolean algebras. This is related to the open question if the existence of an isometrically universal Banach space and of a universal Boolean algebra are equivalent on the nonseparable level (both are true on the separable level).