On the embeddability of countably branching bundle graphs into dual spaces

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${}^{\ast }$ unconditional asymptotic structure. We also investigate weak${}^{\ast }$ asymptotic midpoint uniform convexity in dual spaces which turns out to be equivalent to its weak version in general and to the standard weak${}^{\ast }$ asymptotic uniform convexity up to renorming in dual spaces with a weak${}^{\ast }$ unconditional asymptotic structure.