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

## TL;DR

This paper investigates Banach spaces with highly symmetric isometry groups, introducing a new property called uniform micro-semitransitivity, and explores their geometric and duality properties, including conditions under which classical spaces like Lp are micro-transitive.

## Contribution

It introduces the concept of uniform micro-semitransitivity, relates it to known properties, and characterizes spaces like Lp in terms of micro-transitivity.

## Key findings

- Micro-transitive and uniformly micro-semitransitive spaces are uniformly convex and smooth.
- Such spaces form a self-dual class.
- L_p spaces are micro-transitive only when p=2.

## Abstract

We study Banach spaces whose group of isometries acts micro-transitively on the unit sphere. We introduce a weaker property, which one-complemented subspaces inherit, that we call uniform micro-semitransitivity. We prove a number of results about both micro-transitive and uniformly micro-semitransitive spaces, including that they are uniformly convex and uniformly smooth, and that they 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-Bollob\'as 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 $p=2$.