Uniform homeomorphisms of Banach spaces and asymptotic structure
HTML articles powered by AMS MathViewer
- by N. J. Kalton PDF
- Trans. Amer. Math. Soc. 365 (2013), 1051-1079 Request permission
Abstract:
We give a general result on the behavior of spreading models in Banach spaces which coarse Lipschitz-embed into asymptotically uniformly convex spaces. We use this result to study the uniqueness of the uniform structure in $\ell _p$-sums of finite-dimensional spaces for $1<p<\infty$; in particular we give some new examples of spaces with unique uniform structure.References
- Fernando Albiac and Nigel J. Kalton, Topics in Banach space theory, Graduate Texts in Mathematics, vol. 233, Springer, New York, 2006. MR 2192298
- F. Baudier, N. J. Kalton, and G. Lancien, A new metric invariant for Banach spaces, Studia Math. 199 (2010), no.Β 1, 73β94. MR 2652598, DOI 10.4064/sm199-1-5
- Yoav Benyamini and Joram Lindenstrauss, Geometric nonlinear functional analysis. Vol. 1, American Mathematical Society Colloquium Publications, vol. 48, American Mathematical Society, Providence, RI, 2000. MR 1727673, DOI 10.1090/coll/048
- J. Bourgain, Remarks on the extension of Lipschitz maps defined on discrete sets and uniform homeomorphisms, Geometrical aspects of functional analysis (1985/86), Lecture Notes in Math., vol. 1267, Springer, Berlin, 1987, pp.Β 157β167. MR 907692, DOI 10.1007/BFb0078143
- J. Bourgain and H. P. Rosenthal, Martingales valued in certain subspaces of $L^{1}$, Israel J. Math. 37 (1980), no.Β 1-2, 54β75. MR 599302, DOI 10.1007/BF02762868
- Antoine Brunel and Louis Sucheston, On sequences invariant under spreading in Banach spaces, Probability in Banach spaces (Proc. First Internat. Conf., Oberwolfach, 1975) Lecture Notes in Math., Vol. 526, Springer, Berlin, 1976, pp.Β 21β31. MR 0461100
- P. G. Casazza and N. J. Kalton, Notes on approximation properties in separable Banach spaces, Geometry of Banach spaces (Strobl, 1989) London Math. Soc. Lecture Note Ser., vol. 158, Cambridge Univ. Press, Cambridge, 1990, pp.Β 49β63. MR 1110185
- Peter G. Casazza and Thaddeus J. Shura, Tsirelβ²sonβs space, Lecture Notes in Mathematics, vol. 1363, Springer-Verlag, Berlin, 1989. With an appendix by J. Baker, O. Slotterbeck and R. Aron. MR 981801, DOI 10.1007/BFb0085267
- Yves Dutrieux, Quotients of $c_0$ and Lipschitz-homeomorphisms, Houston J. Math. 27 (2001), no.Β 3, 593β599. MR 1864800
- G. Godefroy, N. Kalton, and G. Lancien, Subspaces of $c_0(\mathbf N)$ and Lipschitz isomorphisms, Geom. Funct. Anal. 10 (2000), no.Β 4, 798β820. MR 1791140, DOI 10.1007/PL00001638
- G. Godefroy, N. J. Kalton, and G. Lancien, Szlenk indices and uniform homeomorphisms, Trans. Amer. Math. Soc. 353 (2001), no.Β 10, 3895β3918. MR 1837213, DOI 10.1090/S0002-9947-01-02825-2
- E. Gorelik, The uniform nonequivalence of $L_p$ and $l_p$, Israel J. Math. 87 (1994), no.Β 1-3, 1β8. MR 1286810, DOI 10.1007/BF02772978
- R. Haydon, M. Levy, and Y. Raynaud, Randomly normed spaces, Travaux en Cours [Works in Progress], vol. 41, Hermann, Paris, 1991. MR 1760188
- Stefan Heinrich, Ultraproducts in Banach space theory, J. Reine Angew. Math. 313 (1980), 72β104. MR 552464, DOI 10.1515/crll.1980.313.72
- S. Heinrich and P. Mankiewicz, Applications of ultrapowers to the uniform and Lipschitz classification of Banach spaces, Studia Math. 73 (1982), no.Β 3, 225β251. MR 675426, DOI 10.4064/sm-73-3-225-251
- William B. Johnson, Factoring compact operators, Israel J. Math. 9 (1971), 337β345. MR 290133, DOI 10.1007/BF02771684
- W. B. Johnson, On quotients of $L_{p}$ which are quotients of $l_{p}$, Compositio Math. 34 (1977), no.Β 1, 69β89. MR 454595
- William B. Johnson and Joram Lindenstrauss, Basic concepts in the geometry of Banach spaces, Handbook of the geometry of Banach spaces, Vol. I, North-Holland, Amsterdam, 2001, pp.Β 1β84. MR 1863689, DOI 10.1016/S1874-5849(01)80003-6
- William B. Johnson, Joram Lindenstrauss, David Preiss, and Gideon Schechtman, Almost FrΓ©chet differentiability of Lipschitz mappings between infinite-dimensional Banach spaces, Proc. London Math. Soc. (3) 84 (2002), no.Β 3, 711β746. MR 1888429, DOI 10.1112/S0024611502013400
- W. B. Johnson, J. Lindenstrauss, and G. Schechtman, Banach spaces determined by their uniform structures, Geom. Funct. Anal. 6 (1996), no.Β 3, 430β470. MR 1392325, DOI 10.1007/BF02249259
- W. B. Johnson and E. Odell, Subspaces of $L_{p}$ which embed into $l_{p}$, Compositio Math. 28 (1974), 37β49. MR 352938
- W. B. Johnson and M. Zippin, On subspaces of quotients of $(\sum G_{n})_{lp}$ and $(\sum G_{n})_{c_{0}}$, Israel J. Math. 13 (1972), 311β316 (1973). MR 331023, DOI 10.1007/BF02762805
- W. B. Johnson and M. Zippin, Subspaces and quotient spaces of $(\sum G_{n})_{l_{p}}$ and $(\sum G_{n})_{c_{0}}$, Israel J. Math. 17 (1974), 50β55. MR 358296, DOI 10.1007/BF02756824
- M. I. Kadets and A. PeΕczyΕski, Bases, lacunary sequences and complemented subspaces in the spaces $L_{p}$, Studia Math. 21 (1961/1962), 161β176.
- N. J. Kalton, Spaces of compact operators, Math. Ann. 208 (1974), 267β278. MR 341154, DOI 10.1007/BF01432152
- N. J. Kalton, $M$-ideals of compact operators, Illinois J. Math. 37 (1993), no.Β 1, 147β169. MR 1193134, DOI 10.1215/ijm/1255987254
- N. J. Kalton, On subspaces of $c_0$ and extension of operators into $C(K)$-spaces, Q. J. Math. 52 (2001), no.Β 3, 313β328. MR 1865904, DOI 10.1093/qjmath/52.3.313
- N. J. Kalton, Coarse and uniform embeddings into reflexive spaces, Q. J. Math. 58 (2007), no.Β 3, 393β414. MR 2354924, DOI 10.1093/qmath/ham018
- N. J. Kalton, Examples of uniformly homeomorphic Banach spaces, to appear.
- Nigel J. Kalton and N. Lovasoa Randrianarivony, The coarse Lipschitz geometry of $l_p\oplus l_q$, Math. Ann. 341 (2008), no.Β 1, 223β237. MR 2377476, DOI 10.1007/s00208-007-0190-3
- Nigel J. Kalton and Dirk Werner, Property $(M)$, $M$-ideals, and almost isometric structure of Banach spaces, J. Reine Angew. Math. 461 (1995), 137β178. MR 1324212, DOI 10.1515/crll.1995.461.137
- H. Knaust, E. Odell, and Th. Schlumprecht, On asymptotic structure, the Szlenk index and UKK properties in Banach spaces, Positivity 3 (1999), no.Β 2, 173β199. MR 1702641, DOI 10.1023/A:1009786603119
- M. Levy and Y. Raynaud, Ultrapuissances de $L^p(L^q)$, Seminar on functional analysis, 1983/1984, Publ. Math. Univ. Paris VII, vol. 20, Univ. Paris VII, Paris, 1984, pp.Β 69β79 (French). MR 825306
- J. Lindenstrauss and L. Tzafriri, The uniform approximation property in Orlicz spaces, Israel J. Math. 23 (1976), no.Β 2, 142β155. MR 399806, DOI 10.1007/BF02756794
- Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92, Springer-Verlag, Berlin-New York, 1977. Sequence spaces. MR 0500056, DOI 10.1007/978-3-642-66557-8
- V. D. Milman, Geometric theory of Banach spaces. II. Geometry of the unit ball, Uspehi Mat. Nauk 26 (1971), no.Β 6(162), 73β149 (Russian). MR 0420226
- E. Odell and Th. Schlumprecht, Trees and branches in Banach spaces, Trans. Amer. Math. Soc. 354 (2002), no.Β 10, 4085β4108. MR 1926866, DOI 10.1090/S0002-9947-02-02984-7
- A. PeΕczyΕski, Projections in certain Banach spaces, Studia Math. 19 (1960), 209β228., DOI 10.4064/sm-19-2-209-228
- A. PeΕczyΕski and H. P. Rosenthal, Localization techniques in $L^{p}$ spaces, Studia Math. 52 (1974/75), 263β289. MR 361729
- M. Ribe, On uniformly homeomorphic normed spaces, Ark. Mat. 14 (1976), no.Β 2, 237β244. MR 440340, DOI 10.1007/BF02385837
- M. Ribe, On uniformly homeomorphic normed spaces. II, Ark. Mat. 16 (1978), no.Β 1, 1β9. MR 487402, DOI 10.1007/BF02385979
- Haskell P. Rosenthal, A characterization of Banach spaces containing $l^{1}$, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 2411β2413. MR 358307, DOI 10.1073/pnas.71.6.2411
Additional Information
- N. J. Kalton
- Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
- Received by editor(s): February 7, 2011
- Received by editor(s) in revised form: July 5, 2011
- Published electronically: September 13, 2012
- Additional Notes: The author acknowledges the support of NSF grant DMS-0555670.
- © Copyright 2012 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 365 (2013), 1051-1079
- MSC (2010): Primary 46B80, 46B20
- DOI: https://doi.org/10.1090/S0002-9947-2012-05665-0
- MathSciNet review: 2995383