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

Abstract

We study Banach spaces whose group of isometrics 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 isometrics acts micro-transitively with a property of operators called the pointwise Bishop-Phelps-Bollobas 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 L-p(mu) space is micro-transitive or uniformly micro-semitransitive only when p = 2. (C) 2020 Elsevier Inc. All rights reserved.