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ISOMETRIC REFLECTIONS IN TWO DIMENSIONS AND DUAL L1-STRUCTURES

  • Garcia-Pacheco, Francisco J. (Department of Mathematics Texas A&M University)
  • Received : 2011.04.01
  • Published : 2012.11.30

Abstract

In this manuscript we solve in the positive a question informally proposed by Enflo on the measure of the set of isometric reflection vectors in non-Hilbert 2-dimensional real Banach spaces. We also reformulate equivalently the separable quotient problem in terms of isometric reflection vectors. Finally, we give a new and easy example of a real Banach space whose dual has a non-trivial L-summand that does not come from an M-ideal in the predual.

References

  1. A. Aizpuru and F. J. Garcia-Pacheco, A short note about exposed points in real Banach spaces, Acta Math. Sci. Ser. B Engl. Ed. 28 (2008), no. 4, 797-800.
  2. A. Aizpuru, F. J. Garcia-Pacheco, and F. Rambla, Isometric reflection vectors in Banach spaces, J. Math. Anal. Appl. 299 (2004), no. 1, 40-48. https://doi.org/10.1016/j.jmaa.2004.06.004
  3. J. Becerra-Guerrero and A. Rodriguez-Palacios, Isometric reflections on Banach spaces after a paper of A. Skorik and M. Zaidenberg, Rocky Mountain J. Math. 30 (2000), no. 1, 63-83. https://doi.org/10.1216/rmjm/1022008976
  4. E. Behrends et al., $L_p$-Structure in Real Banach Spaces, Lecture Notes in Mathematics, 613, Springer-Verlag, Berlin-Heidelberg-New York, 1977.
  5. J. Diestel, Sequences and Series in Banach Spaces, Springer-Verlag, New York, 1984.
  6. F. J. Garcia-Pacheco, Geometry of isometric reflection vectors, Math. Slovaca 61 (2010), no. 5, 807-816.
  7. F. J. Garcia-Pacheco, Rotundity and connectedness in two dimensions, Bol. Soc. Mat. Mexicana (3) 15 (2009), no. 2, 165-173.
  8. P. Hadamard, D. Werner, and W. Werner, M-Ideals in Banach Spaces and Banach Algebras, Springer-Verlag, Berlin, 1993.
  9. G. J. O. Jameson, The weak star closure of the unit ball in a hyperplane, Proc. Edinburgh Math. Soc. (2) 18 (1972), no. 2, 7-11.
  10. J. Mujica, Separable quotients of Banach spaces, Rev. Mat. Univ. Complut. Madrid 10 (1998), 299-330.
  11. H. P. Rosenthal, On quasi-complemented subspaces of Banach spaces, Proc. Natl. Acad. Sci. USA 59 (1968), 361-364. https://doi.org/10.1073/pnas.59.2.361
  12. S. A. Saxon and A. Wilansky, The equivalence of some Banach space problems, Colloq. Math. 37 (1977), no. 2, 217-226. https://doi.org/10.4064/cm-37-2-217-226
  13. A. Skorik and M. Zaidenberg, On isometric reflections in Banach spaces, Mat. Fiz. Anal. Geom. 4 (1997), no. 1-2, 212-247.

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  2. A simple equivalent reformulation of the separable quotient problem vol.148, pp.1, 2016, https://doi.org/10.1007/s10474-015-0555-0