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

Argyros, Spiros; Motakis, Pavlos

doi:10.1090/tran/7353

The scalar-plus-compact property in spaces without reflexive subspaces
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by Spiros A. Argyros and Pavlos Motakis PDF

Trans. Amer. Math. Soc. 371 (2019), 1887-1924 Request permission

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 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.

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