Kadec Norms on Spaces of Continuous Functions

2000 Mathematics Subject Classification: Primary: 46B03, 46B26. Secondary: 46E15, 54C35.

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(K1) has a pointwise Kadec renorming and K2 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(K1×K2) has a pointwise Kadec renorming. We also prove a version of the three-space property for such renormings.