The scalar-plus-compact property in spaces without reflexive subspaces

TL;DR

This paper constructs a unique hereditarily indecomposable Banach space that lacks reflexive subspaces, contains the scalar-plus-compact property, and answers longstanding questions in Banach space theory.

Contribution

It introduces the first example of a $ olinebreak ext{L}_ ext{infty}$-space without reflexive subspaces that has the scalar-plus-compact property, using advanced construction methods.

Findings

The space does not contain $c_0$, $ ext{ell}_1$, or reflexive subspaces.

It has a shrinking finite dimensional decomposition.

It does not contain a boundedly complete sequence.

Abstract

A hereditarily indecomposable Banach space $\mathfrak{X}_{\mathfrak{nr}}$ is constructed that is the first known example of a $\mathscr{L}_\infty$-space not containing $c_0$, $\ell_1$, or reflexive subspaces and answers a question posed by J. Bourgain. Moreover, the space $\mathfrak{X}_{\mathfrak{nr}}$ satisfies the "scalar-plus-compact" property and it is the first known space without reflexive subspaces having this property. It is constructed using the Bourgain-Delbaen method in combination with a recent version of saturation under constraints in a mixed-Tsirelson setting. As a result, the space $\mathfrak{X}_{\mathfrak{nr}}$ has a shrinking finite dimensional decomposition and does not contain a boundedly complete sequence.