Trees in Renorming Theory

Abstract

A systematic study is made of the isomorphic properties of the Banach space C0(ϒ) of continuous functions, vanishing at infinity, on a tree ϒ, equipped with its natural locally compact topology. Necessary and sufficient conditions, expressed in terms of the combinatorial structure of ϒ, are obtained for C0(ϒ) 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 Fréchet-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 C0(ϒ) always admits a C∞ ‘bump-function’, even in cases where no good norms exist. 1991 Mathematics Subject Classification: 11D25, 11G05, 14G05.