Banach spaces of polynomials without copies of $l^ 1$

Abstract:

Let X be a Banach space. For a positive integer m, let ${\mathcal {P}_{{w^ \ast }}}{(^m}{X^ \ast })$ denote the Banach space formed by all m-homogeneous polynomials defined on ${X^ \ast }$ whose restrictions to the closed unit ball $B({X^ \ast })$ of ${X^ \ast }$ are continuous for the weak-star topology. For each one of such polynomials, its norm will be the supremum of the absolute value in $B({X^ \ast })$. In this paper the bidual of ${\mathcal {P}_{{w^ \ast }}}{(^m}{X^ \ast })$ is constructed when this space does not contain a copy of ${l^1}$. It is also shown that, whenever X is an Asplund space, ${\mathcal {P}_{{w^ \ast }}}{(^m}{X^ \ast })$ is also Asplund.