Spaces of continuous functions over Dugundji compacta

TL;DR

This paper proves that for certain compact spaces called Dugundji compacta of weight aleph one, their associated Banach spaces have specific structural properties, and the space of probability measures is Valdivia compact, answering a known open question.

Contribution

It establishes that $C(K)$ is 1-Plichko and $P(K)$ is Valdivia compact for Dugundji compacta of weight aleph one, linking topology and Banach space theory.

Findings

$C(K)$ is 1-Plichko for such $K$

$P(K)$ is Valdivia compact for such $K$

Answer to Kalenda's question about non-Valdivia compact groups

Abstract

We show that for every Dugundji compact $K$ of weight aleph one the Banach space $C(K)$ is 1-Plichko and the space $P(K)$ of probability measures on $K$ is Valdivia compact. Combining this result with the existence of a non-Valdivia compact group, we answer a question of Kalenda.