Shellable weakly compact subsets of $C[0,1]$

TL;DR

The paper characterizes when weakly compact subsets of C[0,1] can be represented via operators from reflexive Banach lattices, providing a complete answer to a question posed by Talagrand in the 1980s.

Contribution

It establishes conditions under which weakly compact sets in C[0,1] are representable through operators from reflexive Banach lattices, resolving a long-standing open problem.

Findings

Weakly compact sets with finite Cantor-Bendixson rank are representable.

Counterexample exists for certain complex weakly compact sets.

Answers a question posed by Talagrand in the 1980s.

Abstract

We show that for every weakly compact subset $K$ of $C[0,1]$ with finite Cantor-Bendixson rank, there is a reflexive Banach lattice $E$ and an operator $T:E\rightarrow C[0,1]$ such that $K\subseteq T(B_E)$. On the other hand, we exhibit an example of a weakly compact set of $C[0,1]$ homeomorphic to $\omega^\omega+1$ for which such $T$ and $E$ cannot exist. This answers a question of M. Talagrand in the 80's.