Characterization of Banach spaces $Y$ satisfying that the pair $({\ell }_{\infty }^4,Y)$ has the Bishop-Phelps-Bollobás property for operators

Abstract

We study the Bishop-Phelps-Bollobás property for operators from ${\ell }_{\infty }^4$ to a Banach space. For this reason we introduce an appropiate geometric property, namely the AHSp-${\ell }_{\infty }^4$. We prove that spaces $Y$satisfying AHSp-${\ell }_{\infty }^4$ are precisely those spaces $Y$ such that $({\ell }_{\infty }^4,Y)$ has the Bishop-Phelps-Bollobás property. We also provide classes of Banach spaces satisfying this condition. For instance, finite-dimensional spaces, uniformly convex spaces, $C_0(L)$ and $L_1(\mu )$ satisfy AHSp-${\ell }_{\infty }^4$.

28 pages