Primariness and the Primary Factorisation Property

TL;DR

This paper explores the relationship between primariness and the primary factorisation property in Banach spaces, developing tools for uncountable sums and establishing new results including the primariness of certain spaces.

Contribution

It introduces methods to transfer primariness and factorisation properties from countable to uncountable Banach space sums, and proves new primariness and UPFP results for complex spaces.

Findings

Primariness of $C[0,1]^*$ under negation of CH.

UPFP results for uncountable sums of ordinal $C(eta)$-spaces.

A uniform primary factorisation theorem for $\, ext{B}(\, ext{ell}_p)$.

Abstract

We study the relation between primariness of Banach spaces and the stronger operator-theoretic notions of the primary factorisation property (PFP) and the uniform primary factorisation property (UPFP). We revisit several classical primariness arguments and isolate the additional information needed to factor the identity through arbitrary operators. In the separable setting, this recovers quantitative factorisation versions of the Casazza--Kottman--Lin method for spaces with symmetric bases and treats the exceptional cases of $\ell_1$ and $\ell_\infty$. We then develop support-reduction and free-selection tools for uncountable direct sums, allowing one to transfer primariness and the PFP/UPFP from countable building blocks to non-separable $\ell_p$-, $c_0$- and more general symmetric sums. As applications, we obtain, among others, the primariness of $C[0,1]^*$ under the negation of the Continuum Hypothesis and UPFP results for uncountable sums of ordinal $C(\alpha)$-spaces. Finally, using the finite-block representation of $\mathcal B(\ell_p)$, we prove a uniform primary factorisation theorem for the Banach space $\mathcal B(\ell_p)$, $1<p<\infty$, and end with open problems concerning the gap between primariness and factorisation.