[Paper Review] Kadec norms on spaces of continuous functions

Max Burke, Wiesław Kubiś+1 more|arXiv (Cornell University)|Nov 29, 2003

Advanced Banach Space Theory13 references3 citations

TL;DR

This paper establishes the existence of pointwise Kadec renormings for Banach spaces of continuous functions $C(K)$, proving that such renormings exist when $K$ is a product of compact linearly ordered spaces, and extending this to products $K_1 \times K_2$ when $C(K_1)$ has a pointwise Kadec renorming and $K_2$ belongs to a class closed under inverse limits of transfinite retractions. It also proves a three-space property for the Kadec property, resolving a question from [LZ].

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 $C(K_1 imes K_2)$ has a pointwise Kadec renorming. We also prove a version of the three-space property for such renormings.

Motivation & Objective

• To investigate the existence of pointwise Kadec renormings on $C(K)$ for compact spaces $K$.
• To extend known results on Kadec renormings from single compact linearly ordered spaces to arbitrary products of such spaces.
• To establish conditions under which $C(K_1 \times K_2)$ admits a pointwise Kadec renorming when $C(K_1)$ does and $K_2$ lies in a specific class of compact spaces.
• To resolve a problem from [LZ] by proving a three-space property for the Kadec property, replacing the Kadec-Klee condition with the weaker Kadec condition.
• To clarify the relationship between norm-SLD and pointwise Kadec renormings in product spaces and dual spaces.

Proposed method

• Uses the characterization of $\tau_p$-Kadec renormings via countable covers by convex sets with small local norm-diameter.
• Applies transfinite inverse limits of retractions to construct and analyze the class of compact spaces $K_2$ for which $C(K_1 \times K_2)$ admits a pointwise Kadec renorming.
• Employs the notion of $\tau_p$-lsc LUR norms and their compatibility with the pointwise topology to extend renormings from subspaces.
• Leverages the $({\tau},{\tau}')$-continuity of quotient maps and the LUR property to prove convergence of projections in the norm topology.
• Applies the three-space property framework to show that if $C_0(X)$ has a $\tau_p$-Kadec renorming and $C(K\setminus X)$ has a $\tau_p$-lsc LUR renorming, then $C(K)$ has a $\tau_p$-Kadec renorming.
• Uses the equivalence between $\tau_p$-Kadec norms and the existence of a countable network formed by intersections of $\tau_p$-open sets with convex sets in the unit sphere.

Experimental results

Research questions

• RQ1 Does $C(K)$ admit a $\tau_p$-Kadec renorming when $K$ is a product of compact linearly ordered spaces?
• RQ2 Under what conditions on $K_1$ and $K_2$ does $C(K_1 \times K_2)$ admit a $\tau_p$-Kadec renorming if $C(K_1)$ does?
• RQ3 Is the existence of a $\tau_p$-Kadec renorming a three-space property, and can the Kadec-Klee condition be replaced by the Kadec condition in such results?
• RQ4 Can the norm-SLD property be extended to $\tau_p$-Kadec renormings in product spaces?
• RQ5 What is the role of $\tau_p$-lsc LUR norms in extending renormings from subspaces to larger function spaces?

Key findings

• For any product $K$ of compact linearly ordered spaces, $C(K)$ admits a $\tau_p$-Kadec renorming, extending Haydon, Jayne, Namioka, and Rogers' result from countable products.
• If $C(K_1)$ has a $\tau_p$-Kadec renorming and $K_2$ belongs to the class obtained by closing compact metrizable spaces under inverse limits of transfinite continuous sequences of retractions, then $C(K_1 \times K_2)$ admits a $\tau_p$-Kadec renorming.
• The three-space property for the Kadec property holds: if $Y \subseteq X$ is a subspace such that $Y$ has the $\tau_p$-Kadec property and $X/Y$ has an LUR renorming, then $X$ has the $\tau_p$-Kadec property.
• The existence of a $\tau_p$-Kadec renorming for $C_0(X)$ and a $\tau_p$-lsc LUR renorming for $C(K\setminus X)$ implies that $C(K)$ has a $\tau_p$-Kadec renorming.
• The paper resolves an open problem from [LZ] by showing that the Kadec-Klee condition in the three-space property can be replaced by the weaker Kadec condition.
• The authors establish that $\tau_p$-Kadec renormings are equivalent to the existence of a countable cover of the unit sphere by convex sets such that $\tau_p$-open sets intersected with them form a network for the norm topology.