This is an announcement for the paper “Octahedral norms in tensor products of Banach spaces” by Johann Langemetshttps://arxiv.org/find/math/1/au:+Langemets_J/0/1/0/all/0/1, Vegard Limahttps://arxiv.org/find/math/1/au:+Lima_V/0/1/0/all/0/1, Abraham Rueda Zocahttps://arxiv.org/find/math/1/au:+Zoca_A/0/1/0/all/0/1.
Abstract: We continue the investigation of the behaviour of octahedral norms in tensor products of Banach spaces. Firstly, we will prove the existence of a Banach space Y such that the injective tensor products $\ell_1\hat{\otimes_{\epsilon}} Y$ and $L_1\hat{\otimes_{\epsilon}} Y$ both fail to have an octahedral norm. Secondly, we will show that in the presence of the metric approximation property octahedrality is preserved from one of the factors by taking projective tensor products with an arbitrary Banach space. These results show how octahedrality is preserved by injective and projective tensor products and solve open problems from the literature.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1609.02062