A simple proof of the Grothendieck theorem on the Dieudonné property of 𝐶₀(𝑇)

Panchapagesan, T.

doi:10.1090/S0002-9939-00-05612-4

A simple proof of the Grothendieck theorem on the Dieudonné property of $C_0(T)$
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Proc. Amer. Math. Soc. 129 (2001), 823-831 Request permission

Abstract:

Let $T$ be a locally compact Hausdorff space and let $C_0(T)$ be the Banach space of all complex valued continuous functions vanishing at infinity in $T$, provided with the supremum norm. Let $X$ be a locally convex Hausdorff space (briefly, an lcHs) which is quasicomplete. A simple proof of the Grothendieck theorem on the Dieudonné property of $C_0(T)$ is given. The present proof is much simpler than that given in an earlier work of the author (Characterizations of weakly compact operators on $C_0(T)$, Trans. Amer. Math. Soc. 350 (1998), 4849-4867).

References

Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
J. Diestel and J. J. Uhl Jr., Vector measures, Mathematical Surveys, No. 15, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis. MR 0453964
N. Dinculeanu, Vector measures, Hochschulbücher für Mathematik, Band 64, VEB Deutscher Verlag der Wissenschaften, Berlin, 1966. MR 0206189
N. Dinculeanu and I. Kluvanek, On vector measures, Proc. London Math. Soc. (3) 17 (1967), 505–512. MR 214722, DOI 10.1112/plms/s3-17.3.505
I. Dobrakov and T. V. Panchapagesan, A simple proof of the theorem on Borel extension and weak compactness of operators, submitted.
K. A. Hirsch, On skew-groups, Proc. London Math. Soc. 45 (1939), 357–368. MR 0000036, DOI 10.1112/plms/s2-45.1.357
Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
Morgan Ward and R. P. Dilworth, The lattice theory of ova, Ann. of Math. (2) 40 (1939), 600–608. MR 11, DOI 10.2307/1968944
Igor Kluvánek, Characterization of Fourier-Stieltjes transforms of vector and operator valued measures, Czechoslovak Math. J. 17(92) (1967), 261–277 (English, with Russian summary). MR 230872
Charles W. McArthur, On a theorem of Orlicz and Pettis, Pacific J. Math. 22 (1967), 297–302. MR 213848
T. V. Panchapagesan, Applications of a theorem of Grothendieck to vector measures, J. Math. Anal. Appl. 214 (1997), no. 1, 89–101. MR 1645515, DOI 10.1006/jmaa.1997.5589
T. V. Panchapagesan, Baire and $\sigma$-Borel characterizations of weakly compact sets in $M(T)$, Trans. Amer. Math. Soc. 350 (1998), no. 12, 4839–4847. MR 1615946, DOI 10.1090/S0002-9947-98-02359-9
T. V. Panchapagesan, Characterizations of weakly compact operators on $C_0(T)$, Trans. Amer. Math. Soc. 350 (1998), no. 12, 4849–4867. MR 1615942, DOI 10.1090/S0002-9947-98-02358-7
T. V. Panchapagesan, On the limitations of the Grothendieck techniques, to appear in Rev. Real Acad. Cienc. Exact. Fis. Natur. Madrid.
Maurice Sion, Outer measures with values in a topological group, Proc. London Math. Soc. (3) 19 (1969), 89–106. MR 239039, DOI 10.1112/plms/s3-19.1.89
Hans Weber, Fortsetzung von Massen mit Werten in uniformen Halbgruppen, Arch. Math. (Basel) 27 (1976), no. 4, 412–423 (German). MR 425070, DOI 10.1007/BF01224694