Unconditionally saturated Banach space with the scalar-plus-compact property

TL;DR

This paper constructs a specialized Banach space with the scalar-plus-compact property, demonstrating that all bounded operators are compact perturbations of scalar multiples of the identity, while maintaining a rich unconditional structure.

Contribution

It introduces a new Bourgain-Delbaen $ ext{L}_ ext{infty}$-space with unique operator and structural properties not previously known.

Findings

All bounded operators are scalar multiples plus compact perturbations.

The space is saturated with unconditional basic sequences.

The space exhibits a strongly heterogeneous structure.

Abstract

We construct a Bourgain-Delbaen $\mathscr{L}_\infty$-space $\mathfrak{X}_{Kus}$ with strongly heterogenous structure: any bounded operator on $\mathfrak{X}_{Kus}$ is a compact perturbation of a multiple of the identity, whereas the space $\mathfrak{X}_{Kus}$ is saturated with unconditional basic sequences.