Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



An approximation property related to $ M$-ideals of compact operators

Authors: Rafael Payá and Wend Werner
Journal: Proc. Amer. Math. Soc. 111 (1991), 993-1001
MSC: Primary 46B20; Secondary 47B07, 47D15, 47D30
MathSciNet review: 1039261
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We investigate a variant of the compact metric approximation property which, for subspaces $ X$ of $ {c_0}$, is known to be equivalent to $ K(X)$, the space of compact operators on $ X$, being an $ M$-ideal in the space of bounded operators on $ X,L(X)$. Among other things, it is shown that an arbitrary Banach space $ X$ has this property iff $ K(Y,X)$ is an $ M$-ideal in $ L(Y,X)$ for all Banach spaces $ Y$ and, furthermore, that $ X$ must contain a copy of $ {c_0}$. The proof of the central theorem of this note uses a characterization of those Banach spaces $ X$ for which $ K(X)$ is an $ M$-ideal in $ L(X)$ obtained earlier by the second author, as well as some techniques from Banach algebra theory.

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

  • [1] E. M. Alfsen and E. G. Effros, Structure in real Banach spaces. Parts I and II. Ann of Math. 96 (1972), 98-173. MR 0352946 (50:5432)
  • [2] E. Behrends, $ M$-structure and the Banach-Stone theorem, Lecture Notes in Math. 736. Springer, Berlin, 1979. MR 547509 (81b:46002)
  • [3] F. F. Bonsall and J. Duncan, Complete normed algebras, Springer, Berlin, 1973. MR 0423029 (54:11013)
  • [4] C.-M. Cho and W. B. Johnson, A characterization of subspaces $ X$ of $ {l^p}$ for which $ K(X)$ is an $ M$-ideal in $ L(X)$. Proc. Amer. Math. Soc. 93 (1985), 466-470. MR 774004 (86h:46026)
  • [5] J. Diestel, Sequences and series in Banach spaces, Springer, Berlin, 1984. MR 737004 (85i:46020)
  • [6] J. Dixmier, Les fonctionnelles linéaires sur l'ensemble des opérateurs bornés d'un espace de Hilbert, Ann. of Math. 51 (1950), 387-408. MR 0033445 (11:441e)
  • [7] J. Duncan and S. A. R. Hosseiniun, The second dual of a Banach algebra, Proc. Royal Soc. Edinburgh 84 (1979), 309-325. MR 559675 (81f:46057)
  • [8] P. Harmand and Å. Lima, Banach spaces which are $ M$-ideals in their biduals, Trans. Amer. Math. Soc. 283 (1983), 253-264.
  • [9] D. Li, Quantitative unconditionality of Banach spaces $ E$ for which $ K(E)$ is an $ M$-ideal in $ L(E)$, Stud. Math. (to appear).
  • [10] Å. Lima, Intersection properties of balls and subspaces in Banach spaces, Trans. Amer. Math. Soc. 227 (1977), 1-62. MR 0430747 (55:3752)
  • [11] -, On $ M$-ideals and best approximation, Indiana Univ. Math. J. 31 (1982), 27-36. MR 642613 (83b:46021)
  • [12] D. Werner, Remarks on $ M$-ideals of compact operators, Quart. J. Math. Oxford (to appear).
  • [13] W. Werner, Inner $ M$-ideals in Banach-algebras, 1988, submitted.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46B20, 47B07, 47D15, 47D30

Retrieve articles in all journals with MSC: 46B20, 47B07, 47D15, 47D30

Additional Information

Article copyright: © Copyright 1991 American Mathematical Society

American Mathematical Society