Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-19T23:53:53.212Z Has data issue: false hasContentIssue false
  • English
  • Français

ON ASYMPTOTIC UNIFORM SMOOTHNESS AND NONLINEAR GEOMETRY OF BANACH SPACES

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

Published online by Cambridge University Press:  25 March 2019

B. M. Braga
B. M. Braga*
Affiliation:
Department of Mathematics and Statistics, York University, 4700 Keele St., Toronto, Ontario, M3J IP3, Canada ([email protected]) https://sites.google.com/site/demendoncabraga
Get access Rights & Permissions [Opens in a new window]

Abstract

These notes concern the nonlinear geometry of Banach spaces, asymptotic uniform smoothness and several Banach–Saks-like properties. We study the existence of certain concentration inequalities in asymptotically uniformly smooth Banach spaces as well as weakly sequentially continuous coarse (Lipschitz) embeddings into those spaces. Some results concerning the descriptive set theoretical complexity of those properties are also obtained. We finish the paper with a list of open problem.

Keywords

coarse Lipschitz geometryasymptotic uniform smoothnessweak sequential continuityBanach–Saks properties

MSC classification

Primary: 46B80: Nonlinear classification of Banach spaces; nonlinear quotients
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 20 , Issue 1 , January 2021 , pp. 65 - 102
Check for updates
Copyright
© Cambridge University Press 2019

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.)

Footnotes

The author is supported by York Science Research Fellowship.

References

Aharoni, I., Every separable metric space is Lipschitz equivalent to a subset of c 0 + , Israel J. Math. 19 (1974), 284–291.CrossRefGoogle Scholar
Aharoni, I. and Lindenstrauss, J., Uniform equivalence between Banach spaces, Bull. Amer. Math. Soc. (N.S.) 84(2) (1978), 281–283.CrossRefGoogle Scholar
Aharoni, I. and Lindenstrauss, J., An extension of a result of Ribe, Israel J. Math. 52(1–2) (1985), 59–64.10.1007/BF02776080CrossRefGoogle Scholar
Baudier, F., Lancien, G., Motakis, P. and Schlumprecht, T., A new coarsely rigid class of Banach spaces. Preprint, 2018, arXiv:1806.00702.Google Scholar
Baudier, F., Lancien, G. and Schlumprecht, T., The coarse geometry of Tsirelson’s space and applications, J. Amer. Math. Soc. 31(3) (2018), 699–717.CrossRefGoogle Scholar
Beauzamy, B., Banach–Saks properties and spreading models, Math. Scand. 44(2) (1979), 357–384.10.7146/math.scand.a-11818CrossRefGoogle Scholar
Braga, B. M., On the complexity of some classes of Banach spaces and non-universality, Czechoslovak Math. J. 64(139)(4) (2014), 1123–1147.10.1007/s10587-014-0157-yCrossRefGoogle Scholar
Braga, B. M., Asymptotic structure and coarse Lipschitz geometry of Banach spaces, Studia Math. 237(1) (2017), 71–97.CrossRefGoogle Scholar
Braga, B. M., Coarse and uniform embeddings, J. Funct. Anal. 272(5) (2017), 1852–1875.CrossRefGoogle Scholar
Braga, B. M., Nonlinear weakly sequentially continuous embeddings between Banach spaces, Int. Math. Res. Not. IMRN (2018), to appear.Google Scholar
Causey, R. M., Power type asymptotically uniformly smooth and asymptotically uniformly flat norms, Positivity 22(5) (2018), 1197–1221.10.1007/s11117-018-0568-3CrossRefGoogle Scholar
Causey, R. M., Three and a half asymptotic properties. Preprint, 2018, arXiv:1805.02747.Google Scholar
Causey, R. M. and Lancien, G., Prescribed Szlenk index of iterated duals. Preprint, 2017, arXiv:1710.01638.Google Scholar
Dimant, V., Gonzalo, R. and Jaramillo, J. A., Asymptotic structure, l p-estimates of sequences, and compactness of multilinear mappings, J. Math. Anal. Appl. 350(2) (2009), 680–693.CrossRefGoogle Scholar
Dodos, P., Banach Spaces and Descriptive Set Theory: Selected Topics, Lecture Notes in Mathematics, Volume 1993, (Springer, Berlin, 2010).10.1007/978-3-642-12153-1CrossRefGoogle Scholar
Dunford, N. and Schwartz, J. T., Linear Operators. Part I, Wiley Classics Library, (John Wiley & Sons, Inc., New York, 1988). General theory, With the assistance of William G. Bade and Robert G. Bartle, Reprint of the 1958 original, A Wiley-Interscience Publication.Google Scholar
Freeman, D., Odell, E., Sari, B. and Zheng, B., On spreading sequences and asymptotic structures, Trans. Amer. Math. Soc. 370(10) (2018), 6933–6953.CrossRefGoogle Scholar
Guentner, E. and Kaminker, J., Exactness and uniform embeddability of discrete groups, J. Lond. Math. Soc. (2) 70(3) (2004), 703–718.CrossRefGoogle Scholar
James, R. C., Uniformly non-square Banach spaces, Ann. of Math. (2) 80 (1964), 542–550.CrossRefGoogle Scholar
Kalton, N. and Randrianarivony, L., The coarse Lipschitz geometry of l p⊕l q , Math. Ann. 341(1) (2008), 223–237.CrossRefGoogle Scholar
Kalton, N. J., Coarse and uniform embeddings into reflexive spaces, Q. J. Math. 58(3) (2007), 393–414.CrossRefGoogle Scholar
Kalton, N. J., Uniform homeomorphisms of Banach spaces and asymptotic structure, Trans. Amer. Math. Soc. 365(2) (2013), 1051–1079.CrossRefGoogle Scholar
Kechris, A. S., Classical Descriptive Set Theory, Graduate Texts in Mathematics, Volume 156, (Springer, New York, 1995).CrossRefGoogle Scholar
Lancien, G. and Raja, M., Asymptotic and coarse Lipschitz structures of quasi-reflexive Banach spaces, Houston J. Math. 44(3) (2018), 927–940.Google Scholar
Lindenstrauss, J., On James’s paper ‘Separable conjugate spaces’, Israel J. Math. 9 (1971), 279–284.CrossRefGoogle Scholar
Lindenstrauss, J. and Tzafriri, L., On Orlicz sequence spaces, Israel J. Math. 10 (1971), 379–390.10.1007/BF02771656CrossRefGoogle Scholar
Mendel, M. and Naor, A., Metric cotype, Ann. of Math. (2) 168(1) (2008), 247–298.CrossRefGoogle Scholar
Odell, E. and Schlumprecht, T., Embedding into Banach spaces with finite dimensional decompositions, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 100(1–2) (2006), 295–323.Google Scholar
Raja, M., On asymptotically uniformly smooth Banach spaces, J. Funct. Anal. 264(2) (2013), 479–492.10.1016/j.jfa.2012.11.004CrossRefGoogle Scholar
Todorcevic, S., Introduction to Ramsey Spaces, Annals of Mathematics Studies, Volume 174, (Princeton University Press, Princeton, NJ, 2010).CrossRefGoogle Scholar