Hostname: page-component-76fb5796d-dfsvx Total loading time: 0 Render date: 2024-04-28T12:23:28.859Z Has data issue: false hasContentIssue false
  • English
  • Français

Homological characterizations of the approximation property for Banach spaces

Published online by Cambridge University Press:  18 May 2009

Yu. V. Selivanov
Yu. V. Selivanov
Affiliation:
Chair of Higher Mathematics, Moscow Aircraft Technological Institute n.a., Tsiolkovsky, Petrovka 27, Moscow K-31, 103767, Russia.
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 E be a Banach space, and let N(E) be the Banach algebra of all nuclear operators on E. In this work, we shall study the homological properties of this algebra. Some of these properties turn out to be equivalent to the (Grothendieck) approximation property for E. These include:

(i) biprojectivity of N(E);

(ii) biflatness of N(E);

(iii) homological finite-dimensionality of N(E);

(iv) vanishing of the three-dimensional cohomology group, H3(N(E), N(E)).

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 34 , Issue 2 , May 1992 , pp. 229 - 239
Copyright
Copyright © Glasgow Mathematical Journal Trust 1992

References

REFERENCES

1.Helemskii, A. Ya., The homology of Banach and topological algebras (Kluwer, Dordrecht, 1989). (Russian original 1986).CrossRefGoogle Scholar
2.Grothendieck, A, Produits tensoriels topologiques et espaces nucléaires, Mem. Arner. Math. Soc., 16, 1955.Google Scholar
3.Enflo, P., A counterexample to the approximation problem, Acta Math., 130 (1973), 309–317.CrossRefGoogle Scholar
4.Lindenstrauss, J. and Tzafriri, L., Classical Banach spaces, Springer-Verlag, 1977.CrossRefGoogle Scholar
5.Helemskii, A. Ya., Banach and polynormed algebras: general theory, representations, homology, Nauka, Moscow, 1989 (Russian). To appear in English, Oxford Univ. Press, 1992.Google Scholar
6.Selivanov, Yu. V., Biprojective Banach algebras, Izv. Akad. Nauk SSSR ser. matem., 43 (1979), 1159–1174 (Russian); Math. USSR-Izv., 15 (1980), 387–399.Google Scholar
7.Grosser, M., Bidualraüme und Vervollstdndigungen von Banachmoduln, Lecture Notes in Math., 717, Springer-Verlag, 1979.CrossRefGoogle Scholar
8.Selivanov, Yu. V., Biprojective Banach algebras and their structure, cohomology, and relation with nuclear operators, Funkc. anal, i pril., 10 (1976), 89–90 (Russian); Fund. Analysis Appi, 10 (1976).Google Scholar
9.Kaliman, Sh. I. and Selivanov, Yu. V., On the cohomology of operator algebras, Vest. Mosk. Univ. ser. mat. meh., 29 (5) (1974), 24–27 (Russian); Moscow Univ. Math. Bull., 29 (1974).Google Scholar
10Johnson, B. E., Perturbations of Banach algebras, Proc. London Math. Soc., 34 (1977), 439–458.Google Scholar