The transfer of property $(\beta )$ of Rolewicz by a uniform quotient map
HTML articles powered by AMS MathViewer
- by S. J. Dilworth, Denka Kutzarova and N. Lovasoa Randrianarivony PDF
- Trans. Amer. Math. Soc. 368 (2016), 6253-6270
Abstract:
We provide a Laakso construction to prove that the property of having an equivalent norm with the property $(\beta )$ of Rolewicz is qualitatively preserved via surjective uniform quotient mappings between separable Banach spaces. On the other hand, we show that the $(\beta )$-modulus is not quantitatively preserved via such a map by exhibiting two uniformly homeomorphic Banach spaces that do not have $(\beta )$-moduli of the same power type even under renorming.References
- J. M. Ayerbe, T. Domínguez Benavides, and S. Francisco Cutillas, Some noncompact convexity moduli for the property $(\beta )$ of Rolewicz, Comm. Appl. Nonlinear Anal. 1 (1994), no. 1, 87–98. MR 1268081
- S. Bates, W. B. Johnson, J. Lindenstrauss, D. Preiss, and G. Schechtman, Affine approximation of Lipschitz functions and nonlinear quotients, Geom. Funct. Anal. 9 (1999), no. 6, 1092–1127. MR 1736929, DOI 10.1007/s000390050108
- Florent Baudier, Metrical characterization of super-reflexivity and linear type of Banach spaces, Arch. Math. (Basel) 89 (2007), no. 5, 419–429. MR 2363693, DOI 10.1007/s00013-007-2108-4
- F. Baudier, N. J. Kalton, and G. Lancien, A new metric invariant for Banach spaces, Studia Math. 199 (2010), no. 1, 73–94. 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, The metrical interpretation of superreflexivity in Banach spaces, Israel J. Math. 56 (1986), no. 2, 222–230. MR 880292, DOI 10.1007/BF02766125
- Peter G. Casazza and Thaddeus J. Shura, Tsirel′son’s 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
- Jeff Cheeger and Bruce Kleiner, Realization of metric spaces as inverse limits, and bilipschitz embedding in $L_1$, Geom. Funct. Anal. 23 (2013), no. 1, 96–133. MR 3037898, DOI 10.1007/s00039-012-0201-8
- S. J. Dilworth, Denka Kutzarova, G. Lancien, and N. L. Randrianarivony, Asymptotic geometry of Banach spaces and uniform quotient maps, Proc. Amer. Math. Soc. 142 (2014), no. 8, 2747–2762. MR 3209329, DOI 10.1090/S0002-9939-2014-12001-6
- S. J. Dilworth, Denka Kutzarova, N. Lovasoa Randrianarivony, J. P. Revalski, and N. V. Zhivkov, Compactly uniformly convex spaces and property $(\beta )$ of Rolewicz, J. Math. Anal. Appl. 402 (2013), no. 1, 297–307. MR 3023259, DOI 10.1016/j.jmaa.2013.01.039
- G. Godefroy, N. J. Kalton, and G. Lancien, Szlenk indices and uniform homeomorphisms, Trans. Amer. Math. Soc. 353 (2001), no. 10, 3895–3918. MR 1837213, DOI 10.1090/S0002-9947-01-02825-2
- R. Huff, Banach spaces which are nearly uniformly convex, Rocky Mountain J. Math. 10 (1980), no. 4, 743–749. MR 595102, DOI 10.1216/RMJ-1980-10-4-743
- Robert C. James, Weak compactness and reflexivity, Israel J. Math. 2 (1964), 101–119. MR 176310, DOI 10.1007/BF02759950
- William B. Johnson, Factoring compact operators, Israel J. Math. 9 (1971), 337–345. MR 290133, DOI 10.1007/BF02771684
- William B. Johnson, Joram Lindenstrauss, David Preiss, and Gideon Schechtman, Almost Fréchet differentiability of Lipschitz mappings between infinite-dimensional Banach spaces, Proc. London Math. Soc. (3) 84 (2002), no. 3, 711–746. MR 1888429, DOI 10.1112/S0024611502013400
- William B. Johnson and Gideon Schechtman, Diamond graphs and super-reflexivity, J. Topol. Anal. 1 (2009), no. 2, 177–189. MR 2541760, DOI 10.1142/S1793525309000114
- N. J. Kalton, Examples of uniformly homeomorphic Banach spaces, Israel J. Math. 194 (2013), no. 1, 151–182. MR 3047066, DOI 10.1007/s11856-012-0080-6
- D. N. Kutzarova, On condition $(\beta )$ and $\Delta$-uniform convexity, C. R. Acad. Bulgare Sci. 42 (1989), no. 1, 15–18. MR 991453
- Denka Kutzarova, An isomorphic characterization of property $(\beta )$ of Rolewicz, Note Mat. 10 (1990), no. 2, 347–354. MR 1204212
- Denka Kutzarova, $k$-$\beta$ and $k$-nearly uniformly convex Banach spaces, J. Math. Anal. Appl. 162 (1991), no. 2, 322–338. MR 1137623, DOI 10.1016/0022-247X(91)90153-Q
- T. J. Laakso, Ahlfors $Q$-regular spaces with arbitrary $Q>1$ admitting weak Poincaré inequality, Geom. Funct. Anal. 10 (2000), no. 1, 111–123. MR 1748917, DOI 10.1007/s000390050003
- Tomi J. Laakso, Plane with $A_\infty$-weighted metric not bi-Lipschitz embeddable to ${\Bbb R}^N$, Bull. London Math. Soc. 34 (2002), no. 6, 667–676. MR 1924353, DOI 10.1112/S0024609302001200
- Vegard Lima and N. Lovasoa Randrianarivony, Property $(\beta )$ and uniform quotient maps, Israel J. Math. 192 (2012), no. 1, 311–323. MR 3004085, DOI 10.1007/s11856-012-0025-0
- Manor Mendel and Assaf Naor, Markov convexity and local rigidity of distorted metrics, J. Eur. Math. Soc. (JEMS) 15 (2013), no. 1, 287–337. MR 2998836, DOI 10.4171/JEMS/362
- V. D. Milman, Geometric theory of Banach spaces. II. Geometry of the unit ball, Uspehi Mat. Nauk 26 (1971), no. 6(162), 73–149 (Russian). MR 0420226
- Edward W. Odell and Thomas Schlumprecht, Embedding into Banach spaces with finite dimensional decompositions, RACSAM. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 100 (2006), no. 1-2, 295–323 (English, with English and Spanish summaries). MR 2267413
- Mikhail I. Ostrovskii, On metric characterizations of the Radon-Nikodým and related properties of Banach spaces, J. Topol. Anal. 6 (2014), no. 3, 441–464. MR 3217866, DOI 10.1142/S1793525314500186
- Stanisław Prus, Nearly uniformly smooth Banach spaces, Boll. Un. Mat. Ital. B (7) 3 (1989), no. 3, 507–521 (English, with Italian summary). MR 1010520
- S. Rolewicz, On drop property, Studia Math. 85 (1986), no. 1, 27–35 (1987). MR 879413, DOI 10.4064/sm-85-1-17-35
- S. Rolewicz, On $\Delta$-uniform convexity and drop property, Studia Math. 87 (1987), no. 2, 181–191. MR 928575, DOI 10.4064/sm-87-2-181-191
- Jeremy T. Tyson, Bi-Lipschitz embeddings of hyperspaces of compact sets, Fund. Math. 187 (2005), no. 3, 229–254. MR 2213936, DOI 10.4064/fm187-3-3
Additional Information
- S. J. Dilworth
- Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
- MR Author ID: 58105
- Email: dilworth@math.sc.edu
- Denka Kutzarova
- Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801
- MR Author ID: 108570
- Email: denka@math.uiuc.edu
- N. Lovasoa Randrianarivony
- Affiliation: Department of Mathematics and Computer Science, Saint Louis University, St. Louis, Missouri 63103
- Email: nrandria@slu.edu
- Received by editor(s): August 23, 2013
- Received by editor(s) in revised form: June 1, 2014, and August 11, 2014
- Published electronically: December 9, 2015
- Additional Notes: The first author was partially supported by NSF grant DMS–1101490. The third author was partially supported by NSF grant DMS–1301591.
- © Copyright 2015 by the authors
- Journal: Trans. Amer. Math. Soc. 368 (2016), 6253-6270
- MSC (2010): Primary 46B80, 46B20, 46T99, 51F99
- DOI: https://doi.org/10.1090/tran/6553
- MathSciNet review: 3461033