Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-27T16:18:08.193Z Has data issue: false hasContentIssue false
  • English
  • Français

Factoring absolutely summing operators through Hilbert-Schmidt operators

Published online by Cambridge University Press:  18 May 2009

Hans Jarchow
Hans Jarchow
Affiliation:
Mathematisches Institut, Universität Zürich, Rämistrasse 74, CH 8001 Zürich, Switzerland
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let K be a compact Hausdorff space, and let C(K) be the corresponding Banach space of continuous functions on K. It is well-known that every 1-summing operator S:C(K)→l2 is also nuclear, and therefore factors S = S1S2, with S1:l2→l2 a Hilbert–Schmidt operator and S1:C(K)→l2 a bounded operator. It is easily seen that this latter property is preserved when C(K) is replaced by any quotient, and that a Banach space X enjoys this property if and only if its second dual, X**, does. This led A. Pełczyński [15] to ask if the second dual of a Banach space X must be isomorphic to a quotient of a C(K)-space if X has the property that every 1-summing operator X-→l2 factors through a Hilbert-Schmidt operator. In this paper, we shall first of all reformulate the question in an appropriate manner and then show that counter-examples are available among super-reflexive Tsirelson-like spaces as well as among quasi-reflexive Banach spaces.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 31 , Issue 2 , May 1989 , pp. 131 - 135
Copyright
Copyright © Glasgow Mathematical Journal Trust 1989

References

1.Akemann, C. A., Dodds, P. G., and Gamlen, J. L. B., Weak compactness in the dual space of a C*-algebra. J. Functional Analysis 10 (1972), 446–450.CrossRefGoogle Scholar
2.Aldous, D., Subspaces of L 1 via random measures. Trans. Amer. Math. Soc. 258 (1981), 445–463.Google Scholar
3.Casazza, P. G. and Shura, T. J., Tsirelson's space. Lecture Notes in Mathematics, to appear.Google Scholar
4.Figiel, T., Lindenstrauss, J. and Milman, V. D., The dimension of almost spherical sections of convex bodies. Ada Math. 139 (1977), 53–94.Google Scholar
5.Gordon, Y. and Lewis, D. R., Absolutely summing operators and local unconditional structure. Acta Math. 133 (1974), 27–48.CrossRefGoogle Scholar
6.Guerre, S. and Lévy, M., Espaces lp dans les sous-espaces de L 1. Trans. Amer. Math. Soc. 279 (1983), 611–616.Google Scholar
7.James, R. C., A non-reflexive Banach space isometric with its second conjugate space. Proc. Nat. Acad. Sci. U.S.A. 37 (1951), 174–177.CrossRefGoogle ScholarPubMed
8.Jarchow, H., On weakly compact operators on C*-algebras. Math. Ann. 273 (1986), 341–343.CrossRefGoogle Scholar
9.Johnson, W. B., Banach spaces all of whose subspaces have the approximation property. Special Topics of Applied Mathematics, Bonn 1979; (North-Holland, 1980), 15–26. See also Séminaire d'Analyse Fonctionelle 79/80, Ecole Polytechn. Palaiseau, Exp. no 16.Google Scholar
10.Kühn, T., γ-summing operators in Banach spaces of type p, 1 <p≤2, and cotype q, 2≤q< ∞, Theory Prob. Appl. 26 (1981), 118–129.CrossRefGoogle Scholar
11.Linde, W. and Pietsch, A., Mappings of Gaussian measures of cylindrical sets in Banach spaces, Theory Prob. Appl. 19 (1974), 445–460.CrossRefGoogle Scholar
12.Lindenstrauss, J. and Tzafriri, L., Classical Banach spaces I. Sequence spaces (Springer-Verlag, 1977).Google Scholar
13.Lindenstrauss, J. and Tzafriri, L., Classical Banach spaces II. Function spaces (Springer-Verlag, 1979).CrossRefGoogle Scholar
14.Maurey, B. and Pisier, G., Séries de variables aléatoires vectorielles indépendantes et propriétés geomériques des espaces de Banach. Studia Math. 58 (1976), 45–90.CrossRefGoogle Scholar
15.Petczyński, A., Compactness of absolutely summing operators. Linear and Complex Analysis Problem Book; Lecture Notes in Mathematics 1043 (1984), 7–9.Google Scholar
16.Pietsch, A., Operator Ideals. (VEB Deutscher Verlag der Wissenschaften 1978/North-Holland 1980).Google Scholar
17.Pisier, G., Some results on Banach spaces without local unconditional structure. Compositio Math. 37 (1978), 3–19.Google Scholar
18.Pisier, G., Factorization of linear operators and geometry of Banach spaces, Amer. Math. Soc. CBMS 60 (1986).CrossRefGoogle Scholar
19.Pisier, G., Weak Hilbert spaces, Proc. London Math. Soc. 56 (1988), 547–579.CrossRefGoogle Scholar
20.Pisier, G., The dual J* of the James space has cotype 2 and the Gordon-Lewis property. Preprint.Google Scholar