# Ideals of finite rank operators, intersection properties of balls, and the approximation property

Studia Mathematica (1999)

- Volume: 133, Issue: 2, page 175-186
- ISSN: 0039-3223

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topLima, Åsvald, and Oja, Eve. "Ideals of finite rank operators, intersection properties of balls, and the approximation property." Studia Mathematica 133.2 (1999): 175-186. <http://eudml.org/doc/216612>.

@article{Lima1999,

abstract = {We characterize the approximation property of Banach spaces and their dual spaces by the position of finite rank operators in the space of compact operators. In particular, we show that a Banach space E has the approximation property if and only if for all closed subspaces F of $c_0$, the space ℱ(F,E) of finite rank operators from F to E has the n-intersection property in the corresponding space K(F,E) of compact operators for all n, or equivalently, ℱ(F,E) is an ideal in K(F,E).},

author = {Lima, Åsvald, Oja, Eve},

journal = {Studia Mathematica},

keywords = {approximation property; finite rank operators; -intersection property; compact operators},

language = {eng},

number = {2},

pages = {175-186},

title = {Ideals of finite rank operators, intersection properties of balls, and the approximation property},

url = {http://eudml.org/doc/216612},

volume = {133},

year = {1999},

}

TY - JOUR

AU - Lima, Åsvald

AU - Oja, Eve

TI - Ideals of finite rank operators, intersection properties of balls, and the approximation property

JO - Studia Mathematica

PY - 1999

VL - 133

IS - 2

SP - 175

EP - 186

AB - We characterize the approximation property of Banach spaces and their dual spaces by the position of finite rank operators in the space of compact operators. In particular, we show that a Banach space E has the approximation property if and only if for all closed subspaces F of $c_0$, the space ℱ(F,E) of finite rank operators from F to E has the n-intersection property in the corresponding space K(F,E) of compact operators for all n, or equivalently, ℱ(F,E) is an ideal in K(F,E).

LA - eng

KW - approximation property; finite rank operators; -intersection property; compact operators

UR - http://eudml.org/doc/216612

ER -

## References

top- [1] N. Aronszajn and P. Panitchpakdi, Extension of uniformly continuous transformations and hyperconvex metric spaces, Pacific J. Math. 6 (1956), 405-439. Zbl0074.17802
- [2] C.-M. Cho, The metric approximation property and intersection properties of balls, J. Korean Math. Soc. 31 (1994), 467-475. Zbl0824.46028
- [3] J. Diestel, Sequences and Series in Banach Spaces, Springer, New York, 1984.
- [4] M. Feder and P. Saphar, Spaces of compact operators and their dual spaces, Israel J. Math. 21 (1975), 38-49. Zbl0325.47028
- [5] T. Figiel, Factorization of compact operators and applications to the approximation problem, Studia Math. 45 (1973), 191-210. Zbl0257.47017
- [6] G. Godefroy, N. J. Kalton and P. D. Saphar, Unconditional ideals in Banach spaces, ibid. 104 (1993), 13-59. Zbl0814.46012
- [7] A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16 (1955).
- [8] P. Harmand, D. Werner and W. Werner, M-Ideals in Banach Spaces and Banach Algebras, Lecture Notes in Math. 1547, Springer, Berlin, 1993. Zbl0789.46011
- [9] H. Jarchow, Locally Convex Spaces, Teubner, Stuttgart, 1981. Zbl0466.46001
- [10] W. B. Johnson, Factoring compact operators, Israel J. Math. 9 (1971), 337-345. Zbl0236.47045
- [11] Å. Lima, Uniqueness of Hahn-Banach extensions and liftings of linear dependences, Math. Scand. 53 (1983), 97-113. Zbl0532.46003
- [12] Å. Lima, The metric approximation property, norm-one projections and intersection properties of balls, Israel J. Math. 84 (1993), 451-475. Zbl0814.46016
- [13] Å. Lima, Property (wM*) and the unconditional metric compact approximation property, Studia Math. 113 (1995), 249-263. Zbl0826.46013
- [14] J. Lindenstrauss, On a problem of Nachbin concerning extension of operators, Israel J. Math. 1 (1963), 75-84. Zbl0145.39003
- [15] J. Lindenstrauss, Extension of compact operators, Mem. Amer. Math. Soc. 48 (1964). Zbl0141.12001
- [16] J. Lindenstrauss, On projections with norm 1-an example, Proc. Amer. Math. Soc. 15 (1964), 403-406. Zbl0125.06601
- [17] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Ergeb. Math. Grenzgeb. 92, Springer, Berlin, 1977. Zbl0362.46013
- [18] 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. Zbl0854.46014
- [19] R. R. Phelps, Convex Functions, Monotone Operators and Differentiability, Lecture Notes in Math. 1364, Springer, Berlin, 1993. Zbl0921.46039
- [20] W. M. Ruess and C. P. Stegall, Extreme points in duals of operator spaces, Math. Ann. 261 (1982), 535-546. Zbl0501.47015
- [21] I. Singer, Bases in Banach Spaces II, Springer, Berlin, 1981.

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