A Generalization of Whyburn's Theorem, and Aperiodicity for Abelian C*-Inclusions

TL;DR

This paper generalizes Whyburn's theorem to characterize the existence of unique minimal closed sets mapping onto a space, linking it to properties of abelian C*-algebra inclusions and their pseudo-expectations.

Contribution

It extends Whyburn's theorem to a broader setting, establishing a connection between minimal closed sets and the aperiodicity of abelian C*-algebra inclusions.

Findings

Existence of a unique minimal closed set corresponds to the density of points with singleton preimages.

Unique pseudo-expectation exists if and only if the inclusion has the almost extension property.

The result applies to both separable and non-separable abelian C*-algebra inclusions.

Abstract

Let $j:Y \to X$ be a continuous surjection of compact metric spaces. Whyburn proved that $j$ is irreducible, meaning that $j(F) \subsetneq X$ for any proper closed subset $F \subsetneq Y$, if and only if $j$ is almost one-to-one, in the sense that \[ \overline{\{y \in Y: j^{-1}(j(y)) = y\}} = Y. \] In this note we prove the following generalization: There exists a unique minimal closed set $K \subseteq Y$ such that $j(K) = X$ if and only if \[ \overline{\{x \in X: card(j^{-1}(x)) = 1\}} = X. \] Translated to the language of operator algebras, this says that if $A \subseteq B$ is a unital inclusion of separable abelian $C^*$-algebras, then there exists a unique pseudo-expectation (in the sense of Pitts) if and only if the almost extension property of Nagy-Reznikoff holds. More generally, we prove that a unital inclusion of (not necessarily separable) abelian $C^*$-algebras has a unique pseudo-expectation if and only if it is aperiodic (in the sense of Kwa\'{s}niewski-Meyer).