On Banach spaces whose group of isometries acts micro-transitively on the unit sphere

Félix Cabello Sánchez, Sheldon Dantas, Vladimir Kadets, Sun Kwang Kim, Han Ju Lee, Miguel Martín

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8 Scopus citations

Abstract

We study Banach spaces whose group of isometries acts micro-transitively on the unit sphere. We introduce a weaker property, inherited by one-complemented subspaces, that we call uniform micro-semitransitivity. We prove a number of results about both micro-transitive and uniformly micro-semitransitive spaces. In particular, they are uniformly convex and uniformly smooth, and form a self-dual class. To this end, we relate the fact that the group of isometries acts micro-transitively with a property of operators called the pointwise Bishop-Phelps-Bollobás property and use some known results on it. Besides, we show that if there is a non-Hilbertian non-separable Banach space with uniform micro-semitransitive (or micro-transitive) norm, then there is a non-Hilbertian separable one. Finally, we show that an Lp(μ) space is micro-transitive or uniformly micro-semitransitive only when p=2.

Original languageEnglish
Article number124046
JournalJournal of Mathematical Analysis and Applications
Volume488
Issue number1
DOIs
StatePublished - 1 Aug 2020

Keywords

  • Banach space
  • Bishop-Phelps-Bollobás property
  • Mazur rotation problem
  • Micro-transitivity
  • Norm attaining operators

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