Γ-flatness and Bishop-Phelps-Bollobás type theorems for operators

The Bishop-Phelps-Bollobás property deals with simultaneous approximation of an operator T and a vector x at which T nearly attains its norm by an operator T0 and a vector x0, respectively, such that T0 attains its norm at x0. In this note we extend the already known results about the Bishop-Phelps-Bollobás property for Asplund operators to a wider class of Banach spaces and to a wider class of operators. Instead of proving a BPB-type theorem for each space separately we isolate two main notions: Γ-flat operators and Banach spaces with ACKρ structure. In particular, we prove a general BPB-type theorem for Γ-flat operators acting to a space with ACKρ structure and show that uniform algebras and spaces with the property β have ACKρ structure. We also study the stability of the ACKρ structure under some natural Banach space theory operations. As a consequence, we discover many new examples of spaces Y such that the Bishop-Phelps-Bollobás property for Asplund operators is valid for all pairs of the form (X,Y).