Skip to main content

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2132))

  • 959 Accesses

Abstract

In this chapter we present and discuss in some detail problems that we encountered in the course of our work. Some of them have already been mentioned in previous chapters, others have appeared under different disguises and a few are new. The contents of the sections may freely overlap.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. D. Amir, J. Lindenstrauss, The structure of weakly compact sets in Banach spaces. Ann. Math. 88, 35–46 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  2. S.A. Argyros, R. Haydon, A hereditarily indecomposable \(\mathcal{L}_{\infty }\)-space that solves the scalar-plus-compact problem. Acta Math. 206, 1–54 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. S.A. Argyros, J.M.F. Castillo, A.S. Granero, M. Jiménez, J.P. Moreno, Complementation and embeddings of c 0(I) in Banach spaces. Proc. Lond. Math. Soc. 85, 742–772 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. S.A. Argyros, I. Gasparis, P. Motakis. On the structure of separable \(\mathcal{L}_{\infty }\)-spaces. arXiv: 1504.08223v1 (2015)

    Google Scholar 

  5. A. Avilés, Y. Moreno, Automorphisms in spaces of continuous functions on Valdivia compacta. Top. Appl. 155, 2027–2030 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. A. Avilés, F. Cabello Sánchez, J.M.F. Castillo, M. González, Y. Moreno, Banach spaces of universal disposition. J. Funct. Anal. 261, 2347–2361 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. A. Avilés, F. Cabello Sánchez, J.M.F. Castillo, M. González, Y. Moreno. On separably injective Banach spaces. Adv. Math. 234, 192–216 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. A. Avilés, F. Cabello Sánchez, J.M.F. Castillo, M. González, Y. Moreno, On ultraproducts of Banach space of type \(\mathcal{L}_{\infty }\). Fundam. Math. 222, 195–212 (2013)

    Article  MATH  Google Scholar 

  9. W.G. Bade, Complementation problems for the Baire classes. Pac. J. Math. 45, 1–11 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  10. S. Baratella, S.-A. Ng, Isometry games in Banach spaces. Bull. Belg. Math. Soc. Simon Stevin 15, 509–521 (2008)

    MathSciNet  MATH  Google Scholar 

  11. Y. Benyamini, J. Lindenstrauss, A predual of 1 which is not isomorphic to a C(K) space. Isr. J. Math. 13, 246–254 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  12. Y. Benyamini, M.E. Rudin, M. Wage, Continuous images of weakly compact subsets of Banach spaces. Pac. J. Math. 70(2), 309–324 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  13. J. Bourgain, A result on operators on C[0, 1]. J. Oper. Theory 3, 275–289 (1980)

    MathSciNet  MATH  Google Scholar 

  14. J. Bourgain, New Classes of \(\mathcal{L}^{p}\) -Spaces. Lecture Notes in Mathematics, vol. 889 (Springer, Berlin, 1981)

    Google Scholar 

  15. J. Bourgain, F. Delbaen, A class of special \(\mathcal{L}_{\infty }\) spaces. Acta Math. 145, 155–176 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  16. J. Bourgain, G. Pisier, A construction of \(\mathcal{L}_{\infty }\)-spaces and related Banach spaces. Bol. Soc. Bras. Mat. 14, 109–123 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  17. C. Brech, On the density of Banach spaces C(K) with the Grothendieck property. Proc. Am. Math. Soc. 134, 3653–3663 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. F. Cabello Sánchez, J.M.F. Castillo, Uniform boundedness and twisted sums of Banach spaces. Houst. J. Math. 30, 523–536 (2004)

    MathSciNet  MATH  Google Scholar 

  19. F. Cabello Sánchez, J.M.F. Castillo, N.J. Kalton, D.T. Yost, Twisted sums with C(K)-spaces. Trans. Am. Math. Soc. 355, 4523–4541 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. L. Candido, P. Koszmider, On complemented copies of c 0(ω 1) in C(K n) spaces. arXiv: 1501.01785v2 (2015)

    Google Scholar 

  21. J.M.F. Castillo, M. González, Three-Space Problems in Banach Space Theory. Lecture Notes in Mathematical, vol. 1667 (Springer, Berlin, 1997)

    Google Scholar 

  22. J.M.F. Castillo, Y. Moreno, On the Lindenstrauss-Rosenthal theorem. Isr. J. Math. 140, 253–270 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  23. J.M.F. Castillo, A. Plichko, Banach spaces in various positions. J. Funct. Anal. 259, 2098–2138 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. J.M.F. Castillo, M. Simões, Property (V ) still fails the 3-space property. Extracta Math. 27, 5–11 (2012)

    MathSciNet  MATH  Google Scholar 

  25. J.M.F. Castillo, J. Suárez, On \(\mathcal{L}_{\infty }\)-envelopes of Banach spaces. J. Math. Anal. Appl. 394, 152–158 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  26. P. Cembranos, C(K, X) contains a complemented copy of c 0. Proc. Am. Math. Soc. 91, 556–558 (1984)

    MathSciNet  MATH  Google Scholar 

  27. F.K. Dashiell Jr., Isomorphism problems for the Baire classes. Pac. J. Math. 52, 29–43 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  28. F.K. Dashiell Jr., Nonweakly compact operators from order-Cauchy complete C(S) lattices, with application to Baire classes. Trans. Am. Math. Soc. 266, 397–413 (1981)

    MathSciNet  MATH  Google Scholar 

  29. F.K. Dashiell Jr., J. Lindenstrauss, Some examples concerning strictly convex norms on C(K) spaces. Isr. J. Math. 16, 329–342 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  30. E.K. van Douwen, The integers and topology, in Handbook of Set-Theoretic Topology (Elsevier, New York, 1984)

    Google Scholar 

  31. A. Dow, J. Vermeer, Extremally disconnected spaces, subspaces and retracts. Top. Appl. 50, 263–282 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  32. A. Dow, H. Jumila, J. Pelant, Chain condition and weak topologies. Top. Appl. 156, 1327–1344 (2009)

    Article  MATH  Google Scholar 

  33. P.N. Dowling, On -subspaces of Banach spaces. Collect. Math. 51, 255–260 (2000)

    MathSciNet  MATH  Google Scholar 

  34. V.P. Fonf, J. Lindenstrauss, R.R. Phelps, Infinite dimensional convexity, in Handbook of the Geometry of Banach Spaces, vol. 1, ed. by W.B. Johnson, J. Lindenstrauss (North-Holland, Amsterdam, 2001), pp. 599–670

    Chapter  Google Scholar 

  35. E. Galego, J. Hagler, Copies of c 0(Γ) in C(K, ;X) spaces. Proc. Am. Math. Soc. 140, 3843–3852 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  36. I. Gasparis, Operators on C[0, F1] preserving copies of asymptotic 1 spaces. Math. Ann. 333, 831–858 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  37. I. Gasparis, On a problem of H.P. Rosenthal concerning operators on C[0, 1]. Adv. Math. 218, 1512–1525 (2008)

    Google Scholar 

  38. I. Gasparis, A new isomorphic 1 predual not isomorphic to a complemented subspace of a C(K) space. Bull. Lond. Math. Soc. 45, 789–799 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  39. I. Gasparis, A note on operators fixing cotype subspaces of C[0, 1]. Proc. Am. Math. Soc. 142, 1633–1639 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  40. R.G. Haydon, A nonreflexive Grothendieck space that does not contain . Isr. J. Math. 40, 65–73 (1981)

    Article  MathSciNet  Google Scholar 

  41. S. Heinrich, C.W. Henson, Banach space model theory. II. Isomorphic equivalence. Math. Nachr. 125, 301–317 (1986)

    MathSciNet  MATH  Google Scholar 

  42. C.W. Henson, Nonstandard hulls of Banach spaces. Isr. J. Math. 25, 108–144 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  43. C.W. Henson, L.C. Moore, Nonstandard analysis and the theory Banach spaces, in Nonstandard Analysis–Recent Developments. Lecture Notes in Mathematics, vol. 983 (Springer, Berlin, 1983), pp. 27–112

    Google Scholar 

  44. H. Jarchow, The three space problem and ideals of operators. Math. Nachr. 119, 121–128 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  45. T. Jech, Set Theory. The third millennium edition, revised and expanded. Springer Monographs in Mathematics (Springer, Berlin, 2003)

    Google Scholar 

  46. W.B. Johnson, M. Zippin, Separable L 1 preduals are quotients of C(Δ). Isr. J. Math. 16, 198–202 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  47. W.B. Johnson, H.P. Rosenthal, M. Zippin, On bases, finite dimensional decompositions and weaker structures in Banach spaces. Isr. J. Math. 9, 488–506 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  48. M.I. Kadec, On complementably universal Banach spaces. Stud. Math. 40, 85–89 (1971)

    MathSciNet  Google Scholar 

  49. P. Koszmider, Banach spaces of continuous functions with few operators. Math. Ann. 330, 151–184 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  50. P. Koszmider, The interplay between compact spaces and the Banach spaces of their continuous functions. Section 52 in Open Problems in Topology II (Elsevier, Amsterdam, 2007)

    Google Scholar 

  51. D.H. Leung, F. Räbiger, Complemented copies of c 0 in l -sums of Banach spaces. Ill. J. Math. 34 (1990) 52–58.

    MathSciNet  MATH  Google Scholar 

  52. J. Lindenstrauss, On the extension of operators with range in a C(K) space. Proc. Am. Math. Soc. 15, 218–225 (1964)

    MathSciNet  MATH  Google Scholar 

  53. J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces I (Springer, Berlin, 1977)

    Book  MATH  Google Scholar 

  54. W. Marciszewski, On Banach spaces C(K) isomorphic to c 0(Γ). Stud. Math. 156, 295–302 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  55. W. Marciszewski, R. Pol, On Banach spaces whose norm-open sets are F σ sets in the weak topology. J. Math. Anal. Appl. 350, 708–722 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  56. Y. Moreno, A. Plichko, On automorphic Banach spaces. Isr. J. Math. 169, 29–45 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  57. V.M. Onieva, Notes on Banach space ideals. Math. Nachr. 126, 27–33 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  58. J.R. Partington, Subspaces of certain Banach sequence spaces. Bull. Lond. Math. Soc. 13, 162–166 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  59. A. Pełczyński, Universal bases. Stud. Math. 32, 247–268 (1969)

    MathSciNet  MATH  Google Scholar 

  60. T.S.S.R.K. Rao, Isometries of A C (K). Proc. Am. Math. Soc. 85, 544–546 (1982)

    Google Scholar 

  61. H.P. Rosenthal, On relatively disjoint families of measures, with some applications to Banach space theory. Stud. Math. 37, 13–36 (1970)

    MathSciNet  MATH  Google Scholar 

  62. W. Schachermayer, On some classical measure-theoretic theorems for non-sigma-complete Boolean algebras. Dissertationes Math. 214, 33 pp. (1982)

    Google Scholar 

  63. T. Schlumprecht, Limited sets in Banach spaces, Dissertation, Munich, 1987

    MATH  Google Scholar 

  64. M. Talagrand, Un nouveau C(K) qui possède la propriété de Grothendieck. Isr. J. Math. 37, 181–191 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  65. M. Tarbard, Hereditarily indecomposable, separable \(\mathcal{L}_{\infty }\) Banach spaces with 1 dual having few but not very few operators. J. Lond. Math. Soc. 85, 737–764 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  66. S. Ulam, Zur Masstheorie in der allgemeinen Mengenlehre. Fund. Math. 16, 140–150 (1930)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Avilés, A., Sánchez, F.C., Castillo, J.M.F., González, M., Moreno, Y. (2016). Open Problems. In: Separably Injective Banach Spaces. Lecture Notes in Mathematics, vol 2132. Springer, Cham. https://doi.org/10.1007/978-3-319-14741-3_6

Download citation

Publish with us

Policies and ethics