This is an announcement for the paper “Garling sequence spaces” by Ben Wallishttps://arxiv.org/find/math/1/au:+Wallis_B/0/1/0/all/0/1.
Abstract: By generalizing a construction of Garling, for each $1\leq p<\infty$ and each normalized, nonincreasing sequence of positive numbers $w\in c_0-\ell_1$ we exhibit an $\ell_p$-saturated, complementably homogeneous Banach space $g(w,p)$ related to the Lorentz sequence space $d(w,p)$. Using methods originally developed for studying $d(w,p)$, we show that $g(w,p)$ admits a unique (up to equivalence) subsymmetric basis, although when $w=(n^{-\theta})_{n=1}^{\infty}$ for some $0<\theta<1$, it does not admit a symmetric basis. We then discuss some additional properties of $g(w,p)$ related to uniform convexity and superreflexivity.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1612.01145