A simple proof of the Grothendieck theorem on the Dieudonné property of $C_0(T)$
HTML articles powered by AMS MathViewer
- by T. V. Panchapagesan PDF
- 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
Additional Information
- T. V. Panchapagesan
- Affiliation: Departamento de Matemáticas, Facultad de Ciencias, Universidad de los Andes, Mérida, Venezuela
- Email: panchapa@ciens.ula.ve
- Received by editor(s): January 22, 1999
- Received by editor(s) in revised form: May 24, 1999
- Published electronically: September 20, 2000
- Additional Notes: This research was supported by the project C-845-97-05-B of the C.D.C.H.T. of the Universidad de los Andes, Mérida, Venezuela.
- Communicated by: Dale Alspach
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 129 (2001), 823-831
- MSC (1991): Primary 47B38, 46G10; Secondary 28B05
- DOI: https://doi.org/10.1090/S0002-9939-00-05612-4
- MathSciNet review: 1707021
Dedicated: Dedicated to the memory of Professor Ivan Dobrakov