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

Abstract

A hereditarily indecomposable Banach space ${\mathfrak{X}}_{\mathfrak{nr}}$ is constructed that is the first known example of a ${\mathcal{L}}_{\infty }$-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.

42 pages