Kadec norms on spaces of continuous functions

TL;DR

This paper investigates conditions under which Banach spaces of continuous functions on certain compact spaces admit pointwise Kadec renormings, extending previous results and exploring stability under product and inverse limit constructions.

Contribution

It extends the class of compact spaces for which $C(K)$ admits a pointwise Kadec renorming, including products of compact linearly ordered spaces and certain inverse limit spaces.

Findings

Existence of pointwise Kadec renormings for $C(K)$ when $K$ is a product of compact linearly ordered spaces.

Stability of such renormings under products with spaces obtained via inverse limits.

A three-space property for these renormings.

Abstract

We study the existence of pointwise Kadec renormings for Banach spaces of the form $C(K)$. We show in particular that such a renorming exists when $K$ is any product of compact linearly ordered spaces, extending the result for a single factor due to Haydon, Jayne, Namioka and Rogers. We show that if $C(K_1)$ has a pointwise Kadec renorming and $K_2$ belongs to the class of spaces obtained by closing the class of compact metrizable spaces under inverse limits of transfinite continuous sequences of retractions, then $C(K_1\times K_2)$ has a pointwise Kadec renorming. We also prove a version of the three-space property for such renormings.