On the density of Banach {$C(K)$} spaces with the Grothendieck property

TL;DR

This paper demonstrates, using forcing, that there can be Banach spaces of continuous functions with the Grothendieck property and small density, challenging previous assumptions about their structure without additional set-theoretic axioms.

Contribution

It proves the consistent existence of a small-density Grothendieck $C(K)$ space via forcing, showing limitations of classical results without extra set-theoretic assumptions.

Findings

Existence of a $C(K)$ space with the Grothendieck property and density less than continuum

Classical non-separability results cannot be strengthened without additional axioms

Boolean algebra separation properties differ significantly from the Grothendieck property

Abstract

Using the method of forcing we prove that consistently there is a Banach space of continuous functions on a compact Hausdorff space with the Grothendieck property and with density less than the continuum. It follows that the classical result stating that ``no nontrivial complemented subspace of a Grothendieck $C(K)$ space is separable'' cannot be strengthened by replacing ``is separable'' by ``has density less than that of $l_\infty$'', without using an additional set-theoretic assumption. Such a strengthening was proved by Haydon, Levy and Odell, assuming Martin's axiom and the negation of the continuum hypothesis. Moreover, our example shows that certain separation properties of Boolean algebras are quite far from the Grothendieck property.