Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Geometry of Banach spaces having shrinking approximations of the identity


Author: Eve Oja
Journal: Trans. Amer. Math. Soc. 352 (2000), 2801-2823
MSC (2000): Primary 46B20, 46B28, 47L05
DOI: https://doi.org/10.1090/S0002-9947-00-02521-6
Published electronically: February 14, 2000
MathSciNet review: 1675226
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $a,c\geq 0$ and let $B$ be a compact set of scalars. We introduce property $M^{\ast }(a,B,c)$ of Banach spaces $X$ by the requirement that \begin{equation*}\limsup _{\nu }\Vert ax_{\nu }^{\ast } +bx^{\ast }+cy^{\ast }\V... ...q \limsup _{\nu }\Vert x_{\nu }^{\ast }\Vert\quad \forall b\in B \end{equation*}whenever $(x_{\nu }^{\ast })$ is a bounded net converging weak$^{\ast }$ to $x^{\ast }$ in $X^{\ast }$ and $\Vert y^{\ast }\Vert\leq \Vert x^{\ast }\Vert$. Using $M^{\ast }(a,B,c)$ with $\max \vert B\vert+c>1$, we characterize the existence of certain shrinking approximations of the identity (in particular, those related to $M$-, $u$-, and $h$-ideals of compact or approximable operators). We also show that the existence of these approximations of the identity is separably determined.


References [Enhancements On Off] (What's this?)

  • 1. E. Behrends, A generalization of the principle of local reflexivity, Rev. Roumaine Math. Pures Appl. 31 (1986), 293-296. MR 88a:46011
  • 2. J. C. Cabello, E. Nieto, and E. Oja, On ideals of compact operators satisfying the $M(r,s)$-inequality, J. Math. Anal. Appl. 220 (1998), 334-348. MR 99f:46018
  • 3. P. G. Casazza and H. Jarchow, Self-induced compactness in Banach spaces, Proc. Roy. Soc. Edinburgh Sect. A 126 (1996), 355-362. MR 97a:46010
  • 4. P. G. Casazza and N. J. Kalton, Notes on approximation properties in separable Banach spaces, Geometry of Banach Spaces. Proc. Conf. Strobl (1989) (P. F. X. Müller and W. Schachermayer, eds.), London Math. Soc. Lecture Note Series, vol. 158, Cambridge Univ. Press, 1990, pp. 49-63. MR 92d:46022
  • 5. Ch.-M. Cho and W. B. Johnson, A characterization of subspaces $X$ of $\ell _{p}$ for which $K(X)$ is an $M$-ideal in $L(X)$, Proc. Amer. Math. Soc. 93 (1985), 466-470. MR 86h:46026
  • 6. J. Diestel and J. J. Uhl, Vector Measures, Mathematical Surveys 15, Amer. Math. Soc., Providence, Rhode Island, 1977. MR 56:12216
  • 7. D. van Dulst and I. Singer, On Kadec-Klee norms on Banach spaces, Studia Math. 54 (1975), 205-211. MR 52:14937
  • 8. G. Emmanuele and K. John, Some remarks on the position of the space $K(X,Y)$ inside the space $W(X,Y)$, New Zealand J. Math. 26 (1997), 183-189. MR 98m:46024
  • 9. G. Emmanuele and T. S. S. R. K. Rao, Spaces of Bochner integrable functions and spaces of representable operators as $\mathfrak{U}$-ideals, Quart. J. Math. Oxford (2) 48 (1997), 467-478. MR 99a:46066
  • 10. T. Figiel and W. B. Johnson, The approximation property does not imply the bounded approximation property, Proc. Amer. Math. Soc. 41 (1973), 197-200. MR 49:5782
  • 11. C. Finet and W. Schachermayer, Equivalent norms on separable Asplund spaces, Studia Math. 92 (1989), 275-283. MR 90d:46026
  • 12. G. Godefroy and N. J. Kalton, Approximating sequences and bidual projections, Quart. J. Math. Oxford (2) 48 (1997), 179-202. CMP 97:15
  • 13. G. Godefroy, N. J. Kalton, and D. Li, Propriété d'approximation métrique inconditionnelle et sous-espaces de $L^{1}$ dont la boule est compact en mesure, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), 1069-1073. MR 96d:46015
  • 14. G. Godefroy, N. J. Kalton, and D. Li, On subspaces of $L^{1}$which embed into $\ell _{1}$, J. Reine Angew. Math. 471 (1996), 43-75. MR 97d:46017
  • 15. G. Godefroy, N. J. Kalton, and P. D. Saphar, Idéaux inconditionnels dans les espaces de Banach, C. R. Acad. Sci. Paris Sér. I 313 (1991), 845-849. MR 93h:46013
  • 16. G. Godefroy, N. J. Kalton, and P. D. Saphar, Unconditional ideals in Banach spaces, Studia Math. 104 (1993), 13-59. MR 94k:46024
  • 17. G. Godefroy and P. D. Saphar, Duality in spaces of operators and smooth norms on Banach spaces, Illinois J. Math. 32 (1988), 672-695. MR 89j:47026
  • 18. N. Grønbæk and G. A. Willis, Approximate identities in Banach algebras of compact operators, Canad. Math. Bull. 36 (1993), 45-53. MR 94b:46076
  • 19. P. Habala, P. Hájek, and V. Zizler, Introduction to Banach spaces, I, Charles University, Prague, 1996.
  • 20. P. Harmand, D. Werner, and W. Werner, $M$-ideals in Banach spaces and Banach algebras, Lecture Notes in Math., vol. 1547, Springer, Berlin-Heidelberg-New York, 1993. MR 94k:46022
  • 21. K. John, $u$-ideals of factorable operators, Czechoslovak. Math. J. (to appear).
  • 22. J. Johnson, Remarks on Banach spaces of compact operators, J. Funct. Anal. 32 (1979), 304-311. MR 82k:47060
  • 23. W. B. Johnson, Finite-dimensional Schauder decompositions in $\pi _{\lambda }$ and dual $\pi _{\lambda }$ spaces, Illinois J. Math. 14 (1970), 642-647. MR 42:826
  • 24. W. B. Johnson, A complementary universal conjugate Banach space and its relation to the approximation problem, Israel J. Math. 13 (1972), 301-310. MR 48:4700
  • 25. W. B. Johnson, H. P. Rosenthal, and M. Zippin, On bases, finite dimensional decompositions and weaker structures in Banach spaces, Israel J. Math. 9 (1971), 488-506. MR 43:6702
  • 26. N. J. Kalton, Banach spaces for which the ideal of compact operators is an $M$-ideal, C. R. Acad. Sci. Paris Sér. I 313 (1991), 509-513. MR 93a:46027
  • 27. N. J. Kalton, $M$-ideals of compact operators, Illinois J. Math. 37 (1993), 147-169. MR 94b:46028
  • 28. N. J. Kalton and D. Werner, Property $(M)$, $M$-ideals, and almost isometric structure of Banach spaces, J. Reine Angew. Math. 461 (1995), 137-178. MR 96m:46022
  • 29. D. Li, Complex unconditional metric approximation property for $\mathcal{C}_{\Lambda }(\mathbf{T})$-spaces, Studia Math. 121 (1996), 231-247. MR 97k:43008
  • 30. Å. Lima, On $M$-ideals and best approximation, Indiana Univ. Math. J. 31 (1982), 27-36. MR 83b:46021
  • 31. Å. Lima, Property $(wM^{\ast })$ and the unconditional metric compact approximation property, Studia Math. 113 (1995), 249-263. MR 96c:46019
  • 32. J. Lindenstrauss, On nonseparable reflexive Banach spaces, Bull. Amer. Math. Soc. 72 (1966), 967-970. MR 34:4875
  • 33. J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, I, Sequence Spaces, Springer, Berlin-Heidelberg, 1977. MR 58:17766
  • 34. E. Oja, A note on $M$-ideals of compact operators, Tartu Ül. Toimetised 960 (1993), 75-92. MR 95a:46026
  • 35. E. Oja, $HB$-subspaces and Godun sets of subspaces in Banach spaces, Mathematika 44 (1997), 120-132. MR 98j:46008
  • 36. E. Oja, $M$-ideals of compact operators are separably determined, Proc. Amer. Math. Soc. 126 (1998), 2747-2753. MR 98k:46032
  • 37. E. Oja and M. Põldvere, On subspaces of Banach spaces where every functional has a unique norm-preserving extension, Studia Math. 117 (1996), 289-306. MR 97d:46016
  • 38. R. R. Phelps, Convex Functions, Monotone Operators and Differentiability, Lecture Notes in Math., vol. 1364, Springer, Berlin-Heidelberg, 1993. MR 94f:46055
  • 39. S. Simons, A convergence theorem with boundary, Pacific J. Math. 40 (1972), 703-708. MR 47:755
  • 40. S. Simons, An eigenvector proof of Fatou's lemma for continuous functions, Math. Intelligencer 17 (1995), 67-70. MR 96e:26003
  • 41. I. Singer, Bases in Banach spaces II, Editura Acad. R. S. România, Springer, Bucharest, 1981. MR 82k:46024
  • 42. D. Werner, $M$-ideals and the ``basic inequality'', J. Approx. Theory 76 (1994), 21-30. MR 95i:47080
  • 43. W. Werner, Inner $M$-ideals in Banach algebras, Math. Ann. 291 (1991), 205-223. MR 93b:46094
  • 44. G. Willis, The compact approximation property does not imply the approximation property, Studia Math. 103 (1992), 99-108. MR 93i:46035
  • 45. M. Zippin, Banach spaces with separable duals, Trans. Amer. Math. Soc. 310 (1988), 371-379. MR 90b:46028

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 46B20, 46B28, 47L05

Retrieve articles in all journals with MSC (2000): 46B20, 46B28, 47L05


Additional Information

Eve Oja
Affiliation: Faculty of Mathematics, Tartu University, Vanemuise 46, 51014 Tartu, Estonia
Email: eveoja@math.ut.ee

DOI: https://doi.org/10.1090/S0002-9947-00-02521-6
Received by editor(s): March 26, 1998
Published electronically: February 14, 2000
Additional Notes: This work was completed during a visit of the author to Freie Universität Berlin in 1997, supported by a grant from the Deutscher Akademischer Austauschdienst. The research was also partially supported by Estonian Science Foundation Grants 1820 and 3055.
The author is grateful to D. Werner for his hospitality and useful conversations.
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society