This is an announcement for the paper “Duality problem for disjointly homogeneous rearrangement invariant spaces” by Sergey V. Astashkinhttps://arxiv.org/search?searchtype=author&query=Astashkin%2C+S+V. Abstract: Let $1\leq p<\infty$. A Banach lattice $E$ is said to be disjointly homogeneous (resp. $p$-disjointly homogeneous) if two arbitrary normalized disjoint sequences from $E$ contain equivalent in $E$ subsequences (resp. every normalized disjoint sequence contains a subsequence equivalent in $E$ to the unit vector basis of $\ell_p$). Answering a question raised in 2014 by Flores, Hernandez, Spinu, Tradacete, and Troitsky, for each $1< p<\infty$, we construct a reflexive $p$-disjointly homogeneous rearrangement invariant space on $[0,1]$ whose dual is not disjointly homogeneous. Employing methods from interpolation theory, we provide new examples of disjointly homogeneous rearrangement invariant spaces; in particular, we show that there is a Tsirelson type disjointly homogeneous rearrangement invariant space, which contains no subspace isomorphic to $\ell_p$, $1\leq p<\infty$, or $c_0$.

The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1805.00691