[Paper Review] Spaces of continuous functions over Dugundji compacta

Тарас Банах, Kubis, Wieslaw|ArXiv.org|Oct 26, 2006

Advanced Banach Space Theory8 references3 citations

TL;DR

This paper establishes that for every Dugundji compact space $K$ of weight $\aleph_1$, the Banach space $C(K)$ of continuous real-valued functions on $K$ is 1-Plichko, meaning it admits a projectional resolution of the identity with norm-one projections. Consequently, the space of probability measures $P(K)$ on $K$ is Valdivia compact. This resolves a question of Kalenda by showing that $C(K)$ can be 1-Plichko even when $K$ is not in the class $\mathcal{R}$, demonstrating that $P(K)$ being Valdivia does not imply $K$ is a Valdivia compactum.

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.

Motivation & Objective

• To investigate the existence of many norm-one projections in non-separable Banach spaces $C(K)$ for non-metrizable compacta $K$.
• To determine whether the topological property of being Dugundji compact implies structural properties in $C(K)$, such as being 1-Plichko.
• To address a question posed by Ondśej Kalenda regarding the relationship between $P(K)$ being Valdivia and $K$ being in the class $\mathcal{R}$.
• To show that $C(K)$ can be 1-Plichko even when $K$ is not a Valdivia compactum, by constructing a counterexample using a non-Valdivia compact group.

Proposed method

• Use Haydon's theorem to represent a Dugundji compact $K$ of weight $\aleph_1$ as a continuous inverse limit of metrizable compacta $K_\alpha$ with open bonding maps.
• Leverage the existence of regular averaging operators on open surjections between metric compacta to construct norm-one projections on $C(K_{\alpha+1})$ onto $C(K_\alpha)$.
• Construct a projectional resolution of the identity $\{P_\alpha\}_{\alpha<\omega_1}$ on $C(K)$ using transfinite induction and regularity of the projections.
• Apply duality: since each $P_\alpha$ is regular, its adjoint $P_\alpha^*$ yields a retraction on $P(K)$, and by a known criterion, this implies $P(K)$ is Valdivia compact.
• Use the fact that regular operators are closed under composition and pointwise limits to ensure the projectional resolution is well-defined and norm-one.

Experimental results

Research questions

• RQ1 Does every Dugundji compact space $K$ of weight $\aleph_1$ yield a 1-Plichko space $C(K)$?
• RQ2 Can $P(K)$ be Valdivia compact even when $K$ is not in the class $\mathcal{R}$?
• RQ3 Is there a Dugundji compact space $K$ such that $C(K)$ is 1-Plichko but $K$ is not Valdivia?
• RQ4 Does the existence of a regular averaging operator on a bonding map $p^{\alpha+1}_\alpha$ imply that $C(K_\alpha)$ is 1-complemented in $C(K_{\alpha+1})$?
• RQ5 Is the class of 1-Plichko spaces stable under complemented subspaces, and Valdivia compacta under retracts?

Key findings

• For every Dugundji compact $K$ of weight $\aleph_1$, the space $C(K)$ is 1-Plichko, meaning it admits a projectional resolution of the identity with norm-one projections.
• The space of probability measures $P(K)$ on such a $K$ is Valdivia compact, as shown via the dual action of regular projections on $C(K)$.
• The paper provides a counterexample to Kalenda's question: there exists a compact group $K$ of weight $\aleph_1$ such that $K \notin \mathcal{R}$, yet $P(K)$ is Valdivia and $C(K)$ is 1-Plichko.
• The construction relies on the existence of regular averaging operators on open surjections between metric compacta, which induce norm-one projections on $C(K)$.
• The result fails for weight $\aleph_2$, as shown by the counterexample $K = \omega_2 + 1$, where $C(K)$ is not a Plichko space.
• The proof generalizes to inverse limits of metric compacta where each bonding map admits a norm-one averaging operator, not necessarily regular.