On the compact operators case of the Bishop-Phelps-Bollobás property for numerical radius

Abstract

We study the Bishop–Phelps–Bollobás property for numerical radius restricted to the case of compact operators (BPBp-nu for compact operators in short). We show that C0(L) spaces have the BPBp-nu for compact operators for every Hausdorff topological locally compact space L. To this end, on the one hand, we provide some techniques allowing to pass the BPBp-nu for compact operators from subspaces to the whole space and, on the other hand, we prove some strong approximation property of C0(L) spaces and their duals. Besides, we also show that real Hilbert spaces and isometric preduals of ℓ1 have the BPBp-nu for compact operators.