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

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\}...

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Bibliographic Details
Published in:arXiv.org 2020-11
Main Author: Zarikian, Vrej
Format: Article
Language:English
Subjects:
Algebra
Inclusions
Metric space
Online Access:Get full text
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Summary: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).
ISSN:2331-8422