On the embeddability of countably branching bundle graphs into dual spaces

TL;DR

This paper extends a result on embedding countably branching bundle graphs into Banach spaces to dual spaces with specific asymptotic structures, and explores properties of weak$^*$ asymptotic convexity.

Contribution

It generalizes previous embeddability results from reflexive spaces to dual spaces with weak$^*$ unconditional asymptotic structures.

Findings

Embeddability extends to dual spaces with weak$^*$ unconditional asymptotic structure.

Weak$^*$ asymptotic midpoint uniform convexity is equivalent to weak$^*$ asymptotic uniform convexity after renorming.

The results broaden understanding of geometric properties influencing embeddability in Banach space theory.

Abstract

In this note the result by A. Swift concerning the embeddability of countably branching bundle graphs into Banach spaces is extended from the context of reflexive spaces with an unconditional asymptotic structure to the context of dual spaces with a weak$^*$ unconditional asymptotic structure. We also investigate weak$^*$ asymptotic midpoint uniform convexity in dual spaces which turns out to be equivalent to its weak version in general and to the standard weak$^*$ asymptotic uniform convexity up to renorming in dual spaces with a weak$^*$ unconditional asymptotic structure.