On orthomorphism elements in ordered algebra

ISSN: 1300-0098
Yayın Aralığı: Yılda 6 Sayı
Yayıncı: TÜBİTAK

Let C be an ordered algebra with a unit e. The class of orthomorphism elements Orthe C of C was introduced and studied by Alekhno in ”The order continuity in ordered algebras”. If C = L G , where G is a Dedekind complete Riesz space, this class coincides with the band Orth G of all orthomorphism operators on G. In this study, the properties of orthomorphism elements similar to properties of orthomorphism operators are obtained. Firstly, it is shown that if C is an ordered algebra such that Cr , the set of all regular elements of C , is a Riesz space with the principal projection property and Orthe C is topologically full with respect to Ie , then Be = Orthe C holds, where Be is the band generated by e in Cr . Then, under the same hypotheses, it is obtained that Orthe C is an f -algebra with a unit e.

Keywords:

Ordered algebra, orthomorphism elements orthomorphism, f -algebra,

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[1] Alekhno EA. The order continuity in ordered algebras. Positivity 2017; 21 (2): 539-574. doi: 10.1007/s11117-016- 0406-4
[2] Alekhno EA. The irreducibility in ordered Banach algebras. Positivity 2012; 16 (1): 143-176. doi: 10.1007/s11117- 011-0117-9
[3] Aliprantis CD, Burkinshaw O. Positive Operators. London, United Kingdom: Academic Press, 1985. doi: 10.1007/978-1-4020-5008-4
[4] Alpay Ş, Turan B. On the commutant of the ideal centre. Note Di Matematica 1999; 18 (1): 63-69.
[5] Luxemburg WAJ, Zaanen AC. Riesz Space I. Amsterdam, Holland: North Holland Publishing Company, 1971.
[6] Luxemburg WAJ, Schep AR. Radon-Nikodym type theorem for positive operators and a dual. Indagationes Mathematicae 1978; 41: 145-154.
[7] Schaefer HH. Banach Lattices and Positive Operators. Berlin, Germany: Springer, 1991. doi: 10.1007/978-3-642- 65970-6
[8] Turan B. On f -linearity and f -orthomorphisms. Positivity 2000; 4: 293-301.
[9] Zaanen AC. Riesz Spaces II. Amsterdam, Holland: North Holland Publishing Company, 1983.