  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.
