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Transactions of the American Mathematical Society

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


Authors: Spiros A. Argyros and Pavlos Motakis
Journal: Trans. Amer. Math. Soc. 371 (2019), 1887-1924
MSC (2010): Primary 46B03, 46B06, 46B25, 46B45
DOI: https://doi.org/10.1090/tran/7353
Published electronically: September 13, 2018
MathSciNet review: 3894038
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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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Additional Information

Spiros A. Argyros
Affiliation: National Technical University of Athens, Faculty of Applied Sciences, Department of Mathematics, Zografou Campus, 157 80, Athens, Greece
Email: sargyros@math.ntua.gr

Pavlos Motakis
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Illinois 61801
Email: pmotakis@illinois.edu

DOI: https://doi.org/10.1090/tran/7353
Keywords: Bourgain--Delbaen spaces, separable $\mathscr{L}_\infty$-spaces, isomorphic $\ell_1$-preduals, asymptotic $c_0$ spaces.
Received by editor(s): December 31, 2016
Received by editor(s) in revised form: July 10, 2017
Published electronically: September 13, 2018
Additional Notes: The second author’s research was supported by NSF DMS-1600600.
This research was supported by program API$Σ$TEIA-1082.
Article copyright: © Copyright 2018 American Mathematical Society