Abstract
The theory of dual functors in the category
of Banach spaces is applied to the study of tensor norms in the sense of Grothendieck. The dual functors of the tensor norms arising from the projective and inductive tensor product as well as from more general tensor norms, such as the norms dp of Saphar, are identified as various spaces of operators, which include p-integral and absolutely p-summing operators. Properties of these operators are then easily derived by categorical means. Applications of the methods provide simplified proofs of composition theorems and the characterization of dual spaces of type (L).
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References
Amemiya, I., and Shiga, K.: On tensor products of Banach spaces. Kodai Mathematical Seminar Reports 9 (1957). 161–178.
Cohen, J.S.: Absolutely p-summing, p-nuclear operators, and their conjugates. Math. Ann. 201 (1973). 177–200.
Diestel, J., and Uhl, J.J. Jr.: The Radon-Nikodym Theorem for Banach space valued measures. Rocky Mountain J. Math. 6 (1976). 1–46.
Gilbert, J.E., and Leih, T.J.: Factorization, tensor products, and bilinear forms in Banach space theory. Preprint.
Gordon, Y., Lewis, D.R., and Retherford, J.R.: Banach ideals of operators with applications. J. Functional Analysis (1973). 85–129.
Grothendieck, A.: Produits tensoriels topologiques et espaces nucléaires. A.M.S. Memoir 16 (1955).
Grothendieck, A.: Résumé de la théorie métrique des produits tensoriels topologiques. Boletin da Sociedade de Mathematica de Sao-Paulo 8 (1956). 1–79.
Grothendieck, A.: Une caracterisation vectoriellemetrique des espaces L1. Can. J. Math. 7 (1955). 552–561.
Herz, C., and Pelletier, J. Wick.: Dual functors and integral operators in the category of Banach spaces. J. Pure and Applied Alg. 8 (1976). 5–22.
Johnson, G.E.: Book review of Banach modules and functors on categories of Banach spaces by Cigler, Losert, and Michor. Bull. A.M.S. (New Series) 3 (1980). 885–886.
Kwapien, S.: On operators factorizable through Lp space. Bull. Soc. Math. France, Mémoire 31–32 (1972). 215–225.
Lewis, D.R.: Duals of tensor products. Proc. Pelczyński conference on Banach spaces of analytic functions (Kent, Ohio 1976). Lecture Notes in Mathematics 604, Springer-Verlag (1977).
Lindenstrauss, J., and Pelczyński, A.: Absolutely summing operators in LP-spaces and applications. Studia Math. 29 (1968). 275–326.
MacLane, S.: Categories for the working mathematician. New York, Springer-Verlag 1971.
Michor, P.: Functors and Categories of Banach Spaces. Lecture Notes in Mathematics 651, Springer-Verlag (1978).
Mityagin, B.S., and Švarc, A.S.: Functors in categories of Banach spaces. Russian Math. Surveys 19, No. 2 (1964). 65–127.
Pelletier, J. Wick.: Dual functors and the Radon-Nikodym property in the category of Banach spaces. J. Australian Math. Soc., (Series A) 27 (1979). 479–494.
Persson, A., and Pietsch, A.: p-nukleare und p-integrale Abbildungen in Banachräumen. Studia Math. 33 (1969). 19–62.
Pietsch, A.: Quasinukleare Abbildungen in normierten Räumen. Math. Ann. 165 (1966). 76–90.
Pietsch, A.: Absolute p-summierende Abbildungen in normierten Räumen. Studia Math. 28 (1967). 333–353.
Saphar, P.: Produits tensoriels d'espaces de Banach et classes d'applications linéaires. Studia Math. 38 (1970). 71–100.
Schatten, R.: A Theory of Cross-spaces. Princeton Univ. Press, 1950.
Stegall, C.: Characterizations of Banach spaces whose duals are L1 spaces. Israel J. Math. 11 (1972). 299–308.
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The author acknowledges the hospitality of the University of Massachusetts at Amherst during the year when the first draft of this paper was written as well as support from the Natural Sciences and Engineering Research Council of Canada.
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Pelletier, J.W. Tensor norms and operators in the category of Banach spaces. Integr equ oper theory 5, 85–113 (1982). https://doi.org/10.1007/BF01694031
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DOI: https://doi.org/10.1007/BF01694031