The dual bade theorem in locally convex spaces and reflexivity of a closed unital subalgebra
DOI:
https://doi.org/10.22199/S07160917.1999.0001.00006Keywords:
Reflexivity, Boolean algebra, QuasicompleteAbstract
The results presented in this paper extend a dual version of the reflexivity theorem of W. Bade to locally convex spaces. Dual versión of the Bade theorem in a Banach C(K)-module was firstly discovered in [1]. It is our aim to extend it to a locally convex C(K)-module. As a consequence, it is proven that each unital w* operator topology closed subalgebra of the w* operator topology closed algebra generated by a Boolean algebra of projections is reflexive.
References
[1] Abramovich Y. A., Arenson E. L. and Kitover A. K., Banach C(K)-modules and operators preserving disjointness, Pit. Res. Not. In Math.Ser., England (1992).
[2] Aliprantis C. D. and Burkinshaw 0., Locally solid Riesz spaces, New York-San Francisco-London, Academic Press (1978).
[3] Aliprantis C. D. and Burkinshaw 0., Positive Operators, New York-San Francisco London, Academic Press (1985).
[4] Alpay ?. and Turan B., On f-modules, preprint.
[5] Bade W.G., On Boolean algebras of projections and algebras of operators, Trans.Amer.Math. Soc. 80, pp. 345- 359 (1955).
[6] Dodds P. G. and Pagter de B., Orthomorphism and Boolean algebras of projections, Math.Z.187, pp. 361- 381 (1984).
[7] Dodds P. G. and Ricker W.J., Spectral measures and the Bade reflexivity theorem, J. Funct. Anal. GL pp. 136- 163, (1985).
[8] Dodds P. G., Ricker W. J. and Pagter de B., Reflexivity and order properties of scalar-type spectral operators in locally convex spaces.Trans. Amer. Math. Soc. 293, pp. 355- 380, (1986).
[9] Gök Ö., On a Bade type reftexive algebra, Ph. D. Thesis, METU, Ankara (1990).
[10] Gillespie T. A., Boolean algebras of projections and reflexive algebras of operators, Proc.Lonclon Math. Soc. 37, pp. 56- 74, (1978).
[11] Hadwin D. and Orhon M., Reflexivity and Approximate Reflexivity for Bounded Boolean Algebras of Projections, J. Funct.Anal.87, pp. 348- 358, (1989).
[12] Rall C., Über Booleshe Algebren von Projektionen, Math.Z.153, pp. 199- 217, (1977).
[13] Ricker W. J., On Boolean algebras of projections and scalar-type spectral operators, Proc. Amer. Math. Soc. 87, pp. 73- 77, (1983).
[14] Ricker W. J., Uniformly closed algebras generated by Boolean algebras of projections in locally c:onvex spaces, Canad. J. Math.39, pp.1123- 1146, (1987).
[15] Zaanen A. C., Riesz Spaces II, Amsterdam-New York-Oxford, (1983).
[2] Aliprantis C. D. and Burkinshaw 0., Locally solid Riesz spaces, New York-San Francisco-London, Academic Press (1978).
[3] Aliprantis C. D. and Burkinshaw 0., Positive Operators, New York-San Francisco London, Academic Press (1985).
[4] Alpay ?. and Turan B., On f-modules, preprint.
[5] Bade W.G., On Boolean algebras of projections and algebras of operators, Trans.Amer.Math. Soc. 80, pp. 345- 359 (1955).
[6] Dodds P. G. and Pagter de B., Orthomorphism and Boolean algebras of projections, Math.Z.187, pp. 361- 381 (1984).
[7] Dodds P. G. and Ricker W.J., Spectral measures and the Bade reflexivity theorem, J. Funct. Anal. GL pp. 136- 163, (1985).
[8] Dodds P. G., Ricker W. J. and Pagter de B., Reflexivity and order properties of scalar-type spectral operators in locally convex spaces.Trans. Amer. Math. Soc. 293, pp. 355- 380, (1986).
[9] Gök Ö., On a Bade type reftexive algebra, Ph. D. Thesis, METU, Ankara (1990).
[10] Gillespie T. A., Boolean algebras of projections and reflexive algebras of operators, Proc.Lonclon Math. Soc. 37, pp. 56- 74, (1978).
[11] Hadwin D. and Orhon M., Reflexivity and Approximate Reflexivity for Bounded Boolean Algebras of Projections, J. Funct.Anal.87, pp. 348- 358, (1989).
[12] Rall C., Über Booleshe Algebren von Projektionen, Math.Z.153, pp. 199- 217, (1977).
[13] Ricker W. J., On Boolean algebras of projections and scalar-type spectral operators, Proc. Amer. Math. Soc. 87, pp. 73- 77, (1983).
[14] Ricker W. J., Uniformly closed algebras generated by Boolean algebras of projections in locally c:onvex spaces, Canad. J. Math.39, pp.1123- 1146, (1987).
[15] Zaanen A. C., Riesz Spaces II, Amsterdam-New York-Oxford, (1983).
Published
2018-04-04
How to Cite
[1]
Ömer Gök, “The dual bade theorem in locally convex spaces and reflexivity of a closed unital subalgebra”, Proyecciones (Antofagasta, On line), vol. 18, no. 1, pp. 77-89, Apr. 2018.
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