1995/09/27 by Richard Haydon, Haydon, Richard
Mathematics · #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topology and Set Theory
paper · pdf · doi:10.48550/arxiv.math/9509217
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 Υ for the existence of equivalent LUR or strictly convex norms on \C0(Υ) and for norms with the Kadec Property. In particular, we show that for a \sl finitely branching tree Υ the space \C0(Υ) 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.