Hostname: page-component-89b8bd64d-z2ts4 Total loading time: 0 Render date: 2026-05-06T20:17:26.262Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE STRUCTURE OF SEPARABLE ${\mathcal{L}}_{\infty }$ -SPACES

Part of: Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  07 March 2016

Spiros A. Argyros ,
Ioannis Gasparis  and
Pavlos Motakis
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
Ioannis Gasparis
Affiliation:
National Technical University of Athens, Faculty of Applied Sciences, Department of Mathematics, Zografou Campus, 157 80, Athens, Greece email ioagaspa@math.ntua.gr
Pavlos Motakis
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, U.S.A. email pavlos@math.tamu.edu
Get access

Abstract

Based on a construction method introduced by Bourgain and Delbaen, we give a general definition of a Bourgain–Delbaen space and prove that every infinite-dimensional separable ${\mathcal{L}}_{\infty }$ -space is isomorphic to such a space. Furthermore, we provide an example of a ${\mathcal{L}}_{\infty }$ and asymptotic $c_{0}$ space not containing $c_{0}$ .

MSC classification

Primary: 46B03: Isomorphic theory (including renorming) of Banach spaces 46B06: Asymptotic theory of Banach spaces 46B07: Local theory of Banach spaces

Information

Type
Research Article
Information
Mathematika , Volume 62 , Issue 3 , 2016 , pp. 685 - 700
Copyright
Copyright © University College London 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Alspach, D. E., The dual of the Bourgain–Delbaen space. Israel J. Math. 117 2000, 239–259.CrossRefGoogle Scholar
Alspach, D. E. and Benyamini, Y., C (K) quotients of separable L ∞ spaces. Israel J. Math. 32 1979, 145–160.Google Scholar
Argyros, S. A., Beanland, K. and Motakis, P., The strictly singular operators in Tsirelson like reflexive spaces. Illinois J. Math. 57(4) 2013, 1173–1217.Google Scholar
Argryros, S., Freeman, D., Haydon, R., Odell, E., Raikoftsalis, Th., Schlumprecht, Th. and Zisimopoulou, D. Z., Embedding Banach spaces into spaces with very few operators. J. Funct. Anal. 262(3) 2012, 825–849.CrossRefGoogle Scholar
Argyros, S. A. and Haydon, R. G., A hereditarily indecomposable L ∞ -space that solves the scalar-plus-compact problem. Acta Math. 206 2011, 1–54.Google Scholar
Bourgain, J. and Delbaen, F., A class of special L ∞ spaces. Acta Math. 145 1980, 155–176.Google Scholar
Freeman, D., Odell, E. and Schlumprecht, Th., The universality of ℓ 1 as a dual space. Math. Ann. 351(1) 2011, 149–186.Google Scholar
Godefroy, G. and Li, D., Some natural families of M-ideals. Math. Scand. 66 1990, 249–263.CrossRefGoogle Scholar
Johnson, W. B., Rosenthal, H. P. and Zippin, M., On bases, finite dimensional decompositions and weaker structures in Banach spaces. Israel J. Math. 9 1971, 488–506.Google Scholar
Lewis, D. and Stegall, C., Banach spaces whose duals are isomorphic to ℓ 1(𝛤). J. Funct. Anal. 12 1973, 177–187.Google Scholar
Lindenstrauss, J. and Pełczyński, A., Absolutely summing operators in L p -spaces and their applications. Studia Math. 29 1968, 275–326.Google Scholar
Lindenstrauss, J. and Rosenthal, H. P., The L p spaces. Israel J. Math. 7 1969, 325–349.Google Scholar
Pełczynśki, A., On Banach spaces containing L 1(𝜇). Studia Math. 30 1968, 231–246.Google Scholar
Stegall, C., Banach spaces whose duals contain ℓ 1(𝛤) with applications to the study of dual L 1(𝜇) spaces. Trans. Amer. Math. Soc. 176 1973, 463–477.Google Scholar
Tsirelson, B. S., Not every Banach space contains ℓ p or c 0 . Funct. Anal. Appl. 8 1974, 138–141.Google Scholar