An asymptotic analog of a local-to-global phenomenon for uniformly convex renormings

2024/01/30 by Florent P. Baudier, Gilles Lancien, Baudier, Florent P. +1
Mathematics · #Advanced Banach Space Theory #Holomorphic and Operator Theory #Advanced Operator Algebra Research

paper · pdf · doi:10.48550/arxiv.2401.17465

Abstract

In this note, we investigate the renorming theory of Banach spaces with property (β) of Rolewicz. In particular, we give a "coordinate-free" proof of the fact that every Banach space with property (β) admits an equivalent norm that is asymptotically uniformly smooth; a result originally due to Kutzarova for spaces with a Schauder basis. We also show that if a natural modulus associated with a Banach space X with property (β) is positive at some point in the interval (0,1), then X admits an equivalent norm with property (β). This is an asymptotic analog of a profound result from the local geometry of Banach spaces that states that if the modulus of uniform convexity of a Banach space X is positive at some point in the interval (0,2), then X admits an equivalent norm that is uniformly convex.

Related

• Three-space property for asymptotically uniformly smooth renormings
• Equivalent norms with the property (β) of Rolewicz
• ξ-asymptotically uniformly smooth, ξ-asymptotically uniformly convex, and (β) operators
• The transfer of property (β) of Rolewicz by a uniform quotient map
• Asymptotic geometry of Banach spaces and uniform quotient maps