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

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 it answers a question posed by J. Bourgain. Moreover, the space \mathfrak{X}_{\mathfrak{nr}} satisfies the ``scalar-plus-compact'' property and 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.