This is an announcement for the paper “$\Gamma$-flatness and Bishop-Phelps-Bollobás type theorems for operators” by Bernardo Cascaleshttps://arxiv.org/find/math/1/au:+Cascales_B/0/1/0/all/0/1, Antonio J. Guiraohttps://arxiv.org/find/math/1/au:+Guirao_A/0/1/0/all/0/1, Vladimir Kadetshttps://arxiv.org/find/math/1/au:+Kadets_V/0/1/0/all/0/1, Mariia Soloviovahttps://arxiv.org/find/math/1/au:+Soloviova_M/0/1/0/all/0/1.
Abstract: The Bishop-Phelps-Bollob'{a}s property deals with simultaneous approximation of an operator $T$ and a vector x at which $T$ nearly attains its norm by an operator $T_0$ and a vector x0, respectively, such that $T_0$ attains its norm at x0. In this note we extend the already known results about {the} Bishop-Phelps-Bollob'{a}s 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: $\Gamma$-flat operators and Banach spaces with ACK$_{\rho}$ structure. In particular, we prove a general BPB-type theorem for $\Gamma$-flat operators acting to a space with ACK$_{\rho}$ structure and show that uniform algebras and spaces with the property $\beta$ have ACK$_{\rho}$ structure. We also study the stability of the ACK$_{\rho}$ structure under some natural Banach space theory operations. As a consequence, we discover many new examples of spaces $Y$ such that the Bishop-Phelps-Bollob'{a}s property for Asplund operators is valid for all pairs of the form $(X, Y)$.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1704.01768