Trees in renorming theory

Abstract

Trees are very agreeable objects to work with, offering a diversity of behaviour within a structure that is sufficiently simple to admit precise analysis. Thus we are able to offer fairly satisfactory necessary and sufficient conditions on a tree $\Upsilon $ for the existence of equivalent LUR or strictly convex norms on $\C_0(\Upsilon )$ and for norms with the Kadec Property. In particular, we show that for a {\sl finitely branching} tree $\Upsilon $ the space $\C_0(\Upsilon )$ admits a Kadec renorming. Since some finitely branching trees fail the condition for strictly convex renormability, we obtain an example of a Banach space that is Kadec renormable but not strictly convexifiable. Consideration of specially tailored examples enables us to answer the ``three-space problem'' for strictly convex renorming: there exists a Banach space $X$ with a closed subspace $Y$ such that both $Y$ and the quotient $X/Y$ admit strictly convex norms, while $X$ does not. We also solve a problem about the property of mid-point locally uniform convexity (MLUR), showing that this does not imply LUR renormability.