This is an announcement for the paper “Prescribed Szlenk index of iterated duals” by Ryan M. Causeyhttps://arxiv.org/find/math/1/au:+Causey_R/0/1/0/all/0/1, Gilles Lancienhttps://arxiv.org/find/math/1/au:+Lancien_G/0/1/0/all/0/1.
Abstract: In a previous work, the first named author described the set $ \mathbb{P}$ of all values of the Szlenk indices of separable Banach spaces. We complete this result by showing that for any integer $n$ and any ordinal $\alpha$ in $\mathbb{P}$, there exists a separable Banach space $X$ such that the Szlenk of the dual of order $k$ of $X$ is equal to the first infinite ordinal $\omega$ for all $k$ in ${0,…, n-1}$ and equal to $\alpha$ for $k=n$. One of the ingredients is to show that the Lindenstrauss space and its dual both have a Szlenk index equal to $\omega$.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1710.01638