TREES IN RENORMING THEORY

A systematic study is made of the isomorphic properties of the Banach space ${\cal C}_0(\Upsilon)$ of continuous functions, vanishing at infinity, on a tree $\Upsilon$, equipped with its natural locally compact topology. Necessary and sufficient conditions, expressed in terms of the combinatorial structure of $\Upsilon$, are obtained for ${\cal C}_0(\Upsilon)$ to possess equivalent norms with various good properties of smoothness and strict convexity. These characterizations, together with the construction of appropriate trees, lead to counter-examples refuting a number of conjectures about renormings. It is shown that the existence of a Fr\'echet-smooth renorming is not inherited by quotients, that strict convexifiability is not a three-space property and that neither the Kadec property nor the MLUR property implies the existence of an equivalent norm which is locally uniformly rotund. An example is also given of a space with a smooth norm but no equivalent strictly convex norm. Finally, it is shown that ${\cal C}_0(\Upsilon)$ always admits a ${\cal C}^\infty$ `bump-function', even in cases where no good norms exist.