Abstract
We establish that, in a universally complete complex K-space with a fixed multiplicative structure, the σ-distributivity of the base is equivalent to each of the following assertions: (1) every band preserving linear operator is order bounded; (2) there are no nonzero derivations; (3) every band preserving endomorphism is a band projection; (4) there are no nontrivial band preserving automorphisms.
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References
Gutman A. E., “Locally one-dimensional K-spaces and σ-distributive Boolean algebras,” Siberian Adv. Math., 5, No.2, 99–121 (1995).
Kusraev A. G., Dominated Operators, Kluwer Academic Publishers, Dordrecht (2001).
Abramovich Yu. A., Veksler A. I., and Koldunov A. V., “On disjointness preserving operators,” Dokl. Akad. Nauk SSSR, 248, No.5, 1033–1036 (1979).
Abramovich Yu. A., Veksler A. I., and Koldunov A. V., “Disjointness preserving operators, their continuity, and multiplicative representation,” in: Linear Operators and Their Applications [in Russian], Leningrad, Leningrad Ped. Inst., 1981, pp. 3–34.
Kusraev A. G., “On band preserving operators,” Vladikavkazsk. Mat. Zh., 6, No.3, 48–58 (2004).
Kusraev A. G. and Kutateladze S. S., Introduction to Boolean Valued Analysis [in Russian], Nauka, Moscow (2005).
Aliprantis C. D. and Burkinshaw O., Positive Operators, Acad. Press, New York (1985).
Schaefer H. H., Banach Lattices and Positive Operators, Springer-Verlag, Berlin etc. (1974).
Kusraev A. G. and Kutateladze S. S., Nonstandard Methods of Analysis [in Russian], Nauka, Novosibirsk (1990).
Bell J. L., Boolean-Valued Models and Independence Proofs in Set Theory, Clarendon Press, New York etc. (1985).
Bourbaki N., Algebra (Polynomials and Fields. Ordered Groups) [Russian translation], Nauka, Moscow (1965).
van der Waerden B. L., Algebra [Russian translation], Nauka, Moscow (1976).
Zariski O. and Samuel P., Commutative Algebra [Russian translation], Izdat. Inostr. Lit., Moscow (1963).
Sikorski R., Boolean Algebras [Russian translation], Mir, Moscow (1969).
Aczel J. and Dhombres J., Functional Equations in Several Variables [Russian translation], Fizmatlit, Moscow (2003).
Gutman A. E., “Disjointness preserving operators,” in: Vector Lattices and Integral Operators (Ed. S. S. Kutateladze), Kluwer Acad. Publ., Dordrecht etc., 1996, pp. 361–454.
Wickstead A. W., “Representation and duality of multiplication operators on Archimedean Riesz spaces,” Compositio Math., 35, No.3, 225–238 (1977).
Goodearl K. R., Von Neumann Regular Rings, Pitman, London (1979).
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To Semen Samsonovich Kutateladze on occasion of his sixtieth birthday.
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Translated from Sibirskii Matematicheskii Zhurnal, Vol. 47, No. 1, pp. 97–107, January–February, 2006.
Original Russian Text Copyright © 2006 Kusraev A. G.
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Kusraev, A.G. Automorphisms and Derivations on a Universally Complete Complex f-Algebra. Sib Math J 47, 77–85 (2006). https://doi.org/10.1007/s11202-006-0010-0
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DOI: https://doi.org/10.1007/s11202-006-0010-0