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Duality for contravariant functors on Banach spaces

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Abstract

The following notion of duality is studied: IfG is a contravariant functor on Ban, then\(G^x (X) = Nat(G,(X\widehat{\widehat \otimes }.)')\). We derive the following results: Thex-reflexive functors are exactly the maximal subfunctors ofH(.,A) with reflexiveA, G x isx-reflexive if and only ifG(I) is a reflexive Banach space,G x=(Ge)x, andG x(X′)=Ge(X)′ ifX′ has the metric approximation property. the last result has the consequence that for the tensor product of functors\(G\begin{array}{*{20}c} {\widehat \otimes } \\ {Ban} \\ \end{array} (X'\widehat{\widehat \otimes }.) = G_e (X)\) holds, ifX′ has the m. A. P.

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References

  1. Buchwalter, H.: topologies, bornologies et compactologies. Thèse Doc. Sc. Math. Fac. Sc. Lyon 1968.

  2. Buchwalter, H.: Espaces de Banach et dualité. Publ. Dép. Math. Lyon3–2, 2–61 (1966).

    MathSciNet  Google Scholar 

  3. Cigler, J.: Funktoren auf Kategorien von Banachräumen. Mh. Math.78, 15–24 (1974).

    Article  MATH  MathSciNet  Google Scholar 

  4. Cigler, J.: Duality for functors on Banach spaces. Preprint. Univ. Wien.

  5. Grothendieck, A.: Produits tensoriels topologiques et espaces nucléaires. Mem. AMS 16. Rhode Island: AMS. 1955.

    Google Scholar 

  6. Gil de Lamadrid, J.: Measures and tensors I. Trans. AMS114, 98–121 (1965).

    Article  MATH  MathSciNet  Google Scholar 

  7. Levin, V. L.: Tensorproducts and functors in categories of Banach spaces defined by KB-lineals. Transl. Moskov. Math. Soc.20, 41–77 (1969).

    Google Scholar 

  8. Linton, F. E. J.: Autonomous categories and duality of functors. J. Algebra2, 315–349 (1965).

    Article  MATH  MathSciNet  Google Scholar 

  9. MacLane, S.: Kategorien. Berlin-Heidelberg-New York: Springer. 1972.

    Book  Google Scholar 

  10. Michor, P.: zum Tensorprodukt von Funktoren auf Kategorien von Banachräumen. Mh. Math.78, 117–130 (1974).

    Article  MATH  MathSciNet  Google Scholar 

  11. Michor, P.: Funktoren zwischen Kategorien von Banach-und Waelbroeck-Räumen. Sitzber. österr. Akad. Wiss., Abt. II,182, 43–65 (1973).

    MathSciNet  Google Scholar 

  12. Michor, P.: Duality of functors on categories of Banachspaces. Preprint.

  13. Mitjagin, B. S., andA. S. Shvarts: Functors on categories of Banachspaces. Russ. Math. Surveys19, 65–127 (1964).

    Article  Google Scholar 

  14. Pietsch, A.: Adjungierte normierte Operatorenideale. Math. Nachr.48, 189–212 (1971).

    Article  MATH  MathSciNet  Google Scholar 

  15. Rieffel, M. A.: Induced Banach representations of Banach algebras and locally compact groups. J. Funct. Anal.1, 443–491 (1967).

    Article  MATH  MathSciNet  Google Scholar 

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  1. Mathematisches Institut der Universität, Strudlhofgasse 4, A-1090, Wien, Austria

    P. Michor

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  1. P. Michor
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Dedicated to Prof. Dr. E. Hlawka on the occasion of his 60th birthday

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Michor, P. Duality for contravariant functors on Banach spaces. Monatshefte für Mathematik 82, 177–186 (1976). https://doi.org/10.1007/BF01526324

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