Surjective isometries on Banach sequence spaces: A survey

References

[1] F. Albiac and N. J. Kalton. Topics in Banach space theory, volume 233 of Graduate Texts in Mathematics. Springer, New York, 2006.Search in Google Scholar

[2] D. E. Alspach and S. A. Argyros. Complexity of weakly null sequences. Dissertationes Math. (Rozprawy Mat.), 321:44, 1992.Search in Google Scholar

[3] G. Androulakis and E. Odell. Distorting mixed Tsirelson spaces. Israel J. Math., 109:125–149, 1999.10.1007/BF02775031Search in Google Scholar

[4] L. Antunes, K. Beanland, and H. V. Chu. On the geometry of higher order Schreier spaces. Illinois J. Math., 65(1):47–69, 2021.10.1215/00192082-8827623Search in Google Scholar

[5] L. Antunes, V. Ferenczi, S. Grivaux, and C. Rosendal. Light groups of isomorphisms of Banach spaces and invariant LUR renormings. Pacific J. Math., 301(1):31–54, 2019.10.2140/pjm.2019.301.31Search in Google Scholar

[6] S. A. Argyros and I. Deliyanni. Examples of asymptotic l₁ Banach spaces. Trans. Amer. Math. Soc., 349(3):973–995, 1997.10.1090/S0002-9947-97-01774-1Search in Google Scholar

[7] S. A. Argyros, P. Motakis, and B. Sarı. A study of conditional spreading sequences. J. Funct. Anal., 273(3):1205–1257, 2017.10.1016/j.jfa.2017.04.009Search in Google Scholar

[8] S. A. Argyros and S. Todorcevic. Ramsey methods in analysis. Advanced Courses in Mathematics. CRM Barcelona. Birkhäuser Verlag, Basel, 2005.10.1007/3-7643-7360-1Search in Google Scholar

[9] S. A. Argyros and A. Tolias. Methods in the theory of hereditarily indecomposable Banach spaces. Mem. Amer. Math. Soc., 170(806):vi+114, 2004.10.1090/memo/0806Search in Google Scholar

[10] A. Baernstein, II. On reflexivity and summability. Studia Math., 42:91–94, 1972.10.4064/sm-42-1-91-94Search in Google Scholar

[11] S. Banach. Théorie des opérations linéaires. Éditions Jacques Gabay, Sceaux, 1993. Reprint of the 1932 original.Search in Google Scholar

[12] S. Banach and S. Saks. Sur la convergence forte dans le champ L^p. Studia Math., 2:51–57, 1930.10.4064/sm-2-1-51-57Search in Google Scholar

[13] T. Banakh. Every 2-dimensional banach space has the mazur-ulam property. preprint https://arxiv.org/pdf/2103.09268.pdf.Search in Google Scholar

[14] K. Beanland, N. Duncan, M. Holt, and J. Quigley. Extreme points for combinatorial Banach spaces. Glasgow Mathematical Journal, 61(2):487–500, 2019.10.1017/S0017089518000319Search in Google Scholar

[15] B. Beauzamy. Banach-Saks properties and spreading models. Math. Scand., 44(2):357–384, 1979.10.7146/math.scand.a-11818Search in Google Scholar

[16] B. Beauzamy and J.-T. Lapresté. Modèles étalés des espaces de Banach. Travaux en Cours. Hermann, Paris, 1984.Search in Google Scholar

[17] S. F. Bellenot. Banach spaces with trivial isometries. Israel J. Math., 56(1):89–96, 1986.10.1007/BF02776242Search in Google Scholar

[18] S. F. Bellenot. Isometries of James space. In Banach space theory (Iowa City, IA, 1987), volume 85 of Contemp. Math., pages 1–18. Amer. Math. Soc., Providence, RI, 1989.10.1090/conm/085/983378Search in Google Scholar

[19] S. F. Bellenot, R. Haydon, and E. Odell. Quasi-reflexive and tree spaces constructed in the spirit of R. C. James. In Banach space theory (Iowa City, IA, 1987), volume 85 of Contemp. Math., pages 19–43. Amer. Math. Soc., Providence, RI, 1989.10.1090/conm/085/983379Search in Google Scholar

[20] A. Bird, G. Jameson, and N. J. Laustsen. The Giesy-James theorem for general index p, with an application to operator ideals on the pth James space. J. Operator Theory, 70(1):291–307, 2013.10.7900/jot.2011aug11.1936Search in Google Scholar

[21] A. Bird, N. J. Laustsen, and A. Zsák. Some remarks on James-Schreier spaces. J. Math. Anal. Appl., 371(2):609–613, 2010.10.1016/j.jmaa.2010.05.067Search in Google Scholar

[22] C. Brech, V. Ferenczi, and A. Tcaciuc. Isometries of combinatorial Banach spaces. Proc. Amer. Math. Soc., 148(11):4845–4854, 2020.10.1090/proc/15122Search in Google Scholar

[23] C. Brech and C. Piña. Banach-Stone-like results for combinatorial Banach spaces. Ann. Pure Appl. Logic, 172(8):Paper No. 102989, 13, 2021.Search in Google Scholar

[24] P. G. Casazza and T. J. Shura. Tsirel′son’s space, volume 1363 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1989. With an appendix by J. Baker, O. Slotterbeck and R. Aron.10.1007/BFb0085267Search in Google Scholar

[25] R. Cheng, J. Mashreghi, and W. T. Ross. Function theory and ℓ^p spaces, volume 75 of University Lecture Series. American Mathematical Society, Providence, RI, [2020] ©2020.10.1090/ulect/075Search in Google Scholar

[26] W. J. Davis. Separable Banach spaces with only trivial isometries. Rev. Roumaine Math. Pures Appl., 16:1051–1054, 1971.Search in Google Scholar

[27] G. Diestel. Sobolev spaces with only trivial isometries. II. Positivity, 13(4):621–630, 2009.10.1007/s11117-008-2242-7Search in Google Scholar

[28] G. Diestel and A. Koldobsky. Sobolev spaces with only trivial isometries. Positivity, 10(1):135–144, 2006.10.1007/s11117-005-4703-6Search in Google Scholar

[29] S. J. Dilworth and B. Randrianantoanina. On an isomorphic Banach-Mazur rotation problem and maximal norms in Banach spaces. J. Funct. Anal., 268(6):1587–1611, 2015.10.1016/j.jfa.2014.11.021Search in Google Scholar

[30] G.-G. Ding. The isometric extension of the into mapping from a ℒ ^∞ (Γ)-type space to some Banach space. Illinois J. Math., 51(2):445–453, 2007.10.1215/ijm/1258138423Search in Google Scholar

[31] X. Fang and J. Wang. Extension of isometries on the unit sphere of l^p(Γ) space. Sci. China Math., 53(4):1085–1096, 2010.10.1007/s11425-010-0028-4Search in Google Scholar

[32] V. Ferenczi and E. M. Galego. Countable groups of isometries on Banach spaces. Trans. Amer. Math. Soc., 362(8):4385–4431, 2010.10.1090/S0002-9947-10-05034-8Search in Google Scholar

[33] V. Ferenczi and C. Rosendal. Displaying Polish groups on separable Banach spaces. Extracta Math., 26(2):195–233, 2011.Search in Google Scholar

[34] V. Ferenczi and C. Rosendal. On isometry groups and maximal symmetry. Duke Math. J., 162(10):1771–1831, 2013.10.1215/00127094-2322898Search in Google Scholar

[35] T. Figiel and W. B. Johnson. A uniformly convex Banach space which contains no l_p. Compositio Math., 29:179–190, 1974.Search in Google Scholar

[36] R. J. Fleming and J. E. Jamison. Isometries on Banach spaces: function spaces, volume 129 of Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics. Chapman & Hall/CRC, Boca Raton, FL, 2003.Search in Google Scholar

[37] W. Gowers. Must an “explicitly defined” Banach space contain c₀ or ℓ_p?, Feb. 17, 2009. Gowers’s Weblog: Mathematics related discussions.Search in Google Scholar

[38] W. T. Gowers and B. Maurey. The unconditional basic sequence problem. J. Amer. Math. Soc., 6(4):851–874, 1993.10.1090/S0894-0347-1993-1201238-0Search in Google Scholar

[39] R. C. James. A non-reflexive Banach space isometric with its second conjugate space. Proc. Nat. Acad. Sci. U. S. A., 37:174–177, 1951.10.1073/pnas.37.3.174Search in Google Scholar PubMed PubMed Central

[40] K. Jarosz. Any Banach space has an equivalent norm with trivial isometries. Israel J. Math., 64(1):49–56, 1988.10.1007/BF02767369Search in Google Scholar

[41] V. Kadets and M. Martín. Extension of isometries between unit spheres of finite-dimensional polyhedral Banach spaces. J. Math. Anal. Appl., 396(2):441–447, 2012.10.1016/j.jmaa.2012.06.031Search in Google Scholar

[42] J. Lamperti. On the isometries of certain function-spaces. Pacific J. Math., 8:459–466, 1958.10.2140/pjm.1958.8.459Search in Google Scholar

[43] P. D. Lax. Functional analysis. Pure and Applied Mathematics (New York). Wiley-Interscience [John Wiley & Sons], New York, 2002.Search in Google Scholar

[44] J. Lindenstrauss and R. R. Phelps. Extreme point properties of convex bodies in reflexive Banach spaces. Israel J. Math., 6:39–48, 1968.10.1007/BF02771604Search in Google Scholar

[45] J. Lindenstrauss and L. Tzafriri. Classical Banach spaces. I. Springer-Verlag, Berlin, 1977. Sequence spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 92.10.1007/978-3-642-66557-8Search in Google Scholar

[46] W. Lusky. A note on rotations in separable Banach spaces. Studia Math., 65(3):239–242, 1979.10.4064/sm-65-3-239-242Search in Google Scholar

[47] A. Manoussakis and A. Pelczar-Barwacz. Strictly singular non-compact operators on a class of HI spaces. Bull. Lond. Math. Soc., 45(3):463–482, 2013.10.1112/blms/bds111Search in Google Scholar

[48] T. Nishiura and D. Waterman. Reflexivity and summability. Studia Math., 23:53–57, 1963.10.4064/sm-23-1-53-57Search in Google Scholar

[49] T. Schlumprecht. An arbitrarily distortable Banach space. Israel J. Math., 76(1-2):81–95, 1991.10.1007/BF02782845Search in Google Scholar

[50] J. Schreier. Ein gegenbeispiel zur theorie der schwachen konvergenz. Studia Math., 2:58–62, 1930.10.4064/sm-2-1-58-62Search in Google Scholar

[51] I. Schur. Einige Bermerkungen zur determinanten theorie. S. B. Preuss. Akad. Wiss. Berlin, 25:454–463, 1925.Search in Google Scholar

[52] P. V. Semënov and A. I. Skorik. Isometries of James spaces. Mat. Zametki, 38(4):537–544, 635, 1985.10.1007/BF01158406Search in Google Scholar

[53] A. Sersouri. On James’ type spaces. Trans. Amer. Math. Soc., 310(2):715–745, 1988.10.1090/S0002-9947-1988-0973175-2Search in Google Scholar

[54] D. Sherman. A new proof of the noncommutative Banach-Stone theorem. In Quantum probability, volume 73 of Banach Center Publ., pages 363–375. Polish Acad. Sci. Inst. Math., Warsaw, 2006.10.4064/bc73-0-29Search in Google Scholar

[55] T. J. Shura and D. Trautman. The λ-property in Schreier’s space S and the Lorentz space d(a, 1). Glasgow Math. J., 32(3):277–284, 1990.10.1017/S0017089500009368Search in Google Scholar

[56] F. C. Sánchez, V. Ferenczi, and B. Randrianantoanina. On mazur rotations problem and its multidimensional versions. preprint https://arxiv.org/abs/2012.08344.Search in Google Scholar

[57] J. Stern. Le groupe des isométries d’un espace de Banach. Studia Math., 64(2):139–149, 1979.10.4064/sm-64-2-139-149Search in Google Scholar

[58] M. H. Stone. Applications of the theory of Boolean rings to general topology. Trans. Amer. Math. Soc., 41(3):375–481, 1937.10.1090/S0002-9947-1937-1501905-7Search in Google Scholar

[59] D.-N. Tan. Isometries of the unit spheres of the Tsirelson space T and the modified Tsirelson space T_M. Houston J. Math., 38(2):571–581, 2012.Search in Google Scholar

[60] D.-N. Tan. Some new properties and isometries on the unit spheres of generalized James spaces J_p. J. Math. Anal. Appl., 393(2):457–469, 2012.10.1016/j.jmaa.2012.03.024Search in Google Scholar

[61] D. Tingley. Isometries of the unit sphere. Geom. Dedicata, 22(3):371–378, 1987.10.1007/BF00147942Search in Google Scholar

[62] B. S. Tsirelson. It is impossible to imbed ℓ_p of c₀ into an arbitrary Banach space. Funkcional. Anal. i Priložen., 8(2):57–60, 1974.Search in Google Scholar