Complementation in Spaces of Continuous Functions on Compact Lines

TL;DR

This paper characterizes order-preserving surjections with averaging operators between compact line spaces and explores their implications for the structure of continuous function spaces, including properties like separable complementation.

Contribution

It provides a characterization of certain surjections and applies these results to analyze advanced structural properties of spaces of continuous functions on compact lines.

Findings

Characterization of order-preserving surjections with averaging operators.

Estimates of the norm of such operators.

Applications to separable complementation properties in function spaces.

Abstract

We characterize order preserving continuous surjections between compact linearly ordered spaces which admit an averaging operator, together with estimates of the norm of such an operator. This result is used to the study of strengthenings of the separable complementation property in spaces of continuous functions on compact lines. These properties include in particular continuous separable complementation property and existence of a projectional skeleton.