The scalar-plus-compact property in spaces without reflexive subspaces
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- by Spiros A. Argyros and Pavlos Motakis PDF
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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.References
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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
- MR Author ID: 26995
- Email: sargyros@math.ntua.gr
- Pavlos Motakis
- Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Illinois 61801
- MR Author ID: 1037097
- Email: pmotakis@illinois.edu
- 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$\Sigma$TEIA-1082. - © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 1887-1924
- MSC (2010): Primary 46B03, 46B06, 46B25, 46B45
- DOI: https://doi.org/10.1090/tran/7353
- MathSciNet review: 3894038