This is an announcement for the paper “A basis of $\R ^n$ with good isometric properties and some applications to denseness of norm attaining operators” by M.D. Acostahttps://arxiv.org/search/math?searchtype=author&query=Acosta%2C+M+D, J.L. Dávilahttps://arxiv.org/search/math?searchtype=author&query=D%C3%A1vila%2C+J+L.

Abstract: We characterize real Banach spaces $Y$ such that the pair $(\ell_\infty ^n, Y)$ has the Bishop-Phelps-Bollobás property for operators. To this purpose it is essential the use of an appropriate basis of the domain space $\R^n$. As a consequence of the mentioned characterization, we provide examples of spaces $Y$ satisfying such property. For instance, finite-dimensional spaces, uniformly convex spaces, uniform algebras and $L_1(μ)$ ($μ$ a positive measure) satisfy the previous property.

https://arxiv.org/abs/1811.08387