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

A hereditarily indecomposable Banach space ${\mathfrak{X}}_{\mathfrak{n}\mathfrak{r}}$ is constructed that is the first known example of a ${\mathcal{L}}_{\mathrm{\infty }}$-space not containing $c_0$, ${\mathrm{\ell }}_1$, or reflexive subspaces and answers a question posed by J. Bourgain. Moreover, the space ${\mathfrak{X}}_{\mathfrak{n}\mathfrak{r}}$ 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 ${\mathfrak{X}}_{\mathfrak{n}\mathfrak{r}}$ has a shrinking finite dimensional decomposition and does not contain a boundedly complete sequence.