Absolute sums of Banach spaces and some geometric properties related to rotundity and smoothness

TL;DR

This paper investigates the properties of acs, luacs, and uacs Banach spaces, focusing on their geometric characteristics related to rotundity and smoothness, especially in the context of absolute sums and new intermediate classes.

Contribution

It introduces new classes of Banach spaces between acs and uacs and analyzes their behavior under absolute sums, extending the understanding of geometric properties.

Findings

Analysis of absolute sums of acs, luacs, and uacs spaces

Introduction of new intermediate classes of Banach spaces

Behavior of these classes under absolute sum operations

Abstract

We study the notions of acs, luacs and uacs Banach spaces which were introduced by V. Kadets et al. in 2000 and form common generalisations of the usual rotundity and smoothness properties of Banach spaces. In particular, we are interested in (mainly infinite) absolute sums of such spaces. We also introduce some new classes of spaces that lie inbetween those of acs and uacs spaces and study their behaviour under the formation of absolute sums as well.