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Kadec norms on spaces of continuous functions

Max Burke·Wieslaw Kubis·Stevo Todorcevic
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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.

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Max Burke, Wieslaw Kubis, Stevo Todorcevic. 2003-11-29. Kadec norms on spaces of continuous functions. https://arxiv.org/abs/math/0312013

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