Extension property and complementation of isometric copies of continuous functions spaces

TL;DR

This paper investigates conditions under which isometric copies of continuous function spaces are complemented, focusing on spaces with the extension property and their relation to compact Hausdorff spaces.

Contribution

It proves that isometric copies of C(L) are complemented in C(K) under specific conditions and explores the properties of spaces with the extension property.

Findings

Isometric copies of C(L) are complemented in C(K) when L is finite height and K has the extension property.

Spaces with the extension property form a closed class and relate to other classes of compact spaces.

The subspace C(L|F) also admits similar complementability results.

Abstract

In this article we prove that every isometric copy of C(L) in C(K) is complemented if L is compact Hausdorff of finite height and K is a compact Hausdorff space satisfying the extension property, i.e., every closed subset of K admits an extension operator. The space C(L) can be replaced by its subspace C(L|F) consisting of functions that vanish on a closed subset F of L. We also study the class of spaces having the extension property, establishing some closure results for this class and relating it to other classes of compact spaces.