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

2019/06/21 by Félix Cabello Sánchez, Sheldon Dantas, Sánchez, Félix Cabello +9 · 1 citation
Mathematics · #Advanced Banach Space Theory #Advanced Topics in Algebra #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.1906.09279

Abstract

We study Banach spaces whose group of isometries acts micro-transitively on\nthe unit sphere. We introduce a weaker property, which one-complemented\nsubspaces inherit, that we call uniform micro-semitransitivity. We prove a\nnumber of results about both micro-transitive and uniformly\nmicro-semitransitive spaces, including that they are uniformly convex and\nuniformly smooth, and that they form a self-dual class. To this end, we relate\nthe fact that the group of isometries acts micro-transitively with a property\nof operators called the pointwise Bishop-Phelps-Bollob 'as property and use\nsome known results on it. Besides, we show that if there is a non-Hilbertian\nnon-separable Banach space with uniform micro-semitransitive (or\nmicro-transitive) norm, then there is a non-Hilbertian separable one. Finally,\nwe show that an L_p(\μ) space is micro-transitive or uniformly\nmicro-semitransitive only when p=2.\n

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