# $\Gamma$-flatness and Bishop-Phelps-Bollob\'as type theorems for operators

## TL;DR

This paper extends Bishop-Phelps-Bollobás theorems to a broader class of operators and spaces by introducing $\Gamma$-flat operators and ACK$_
ho$ structures, providing new approximation results.

## Contribution

It introduces $\Gamma$-flat operators and ACK$_
ho$ structures, broadening the applicability of Bishop-Phelps-Bollobás theorems to more Banach spaces and operators.

## Key findings

- Proves a general BPB-type theorem for $\Gamma$-flat operators and ACK$_
ho$ spaces.
- Shows uniform algebras and spaces with property $eta$ have ACK$_
ho$ structure.
- Identifies new spaces where the BPB property for Asplund operators holds.

## 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 $x_0$, respectively, such that $T_0$ attains its norm at $x_0$. 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 ($X,Y$).