Abstract
In this paper, using derivation theory, we present some results concerning the automatic order boundedness of band preserving operators on Archimedean semiprime \(f\)-algebras. Finally, inspired by the proof of Bernau and Huijsmans (Math Proc Camb Philos Soc 107:287–308, 1990), we give necessary and sufficient conditions for Archimedean lattice-ordered algebras to be commutative.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abramovich, Y.A., Veksler, A.I., Koldunov, A.V.: On disjointness preserving operators. Dokl. Akad. Nauk SSSR 289(5), 1033–1036 (1979)
Abramovich, Y.A., Veksler, A.I., Koldunov, A.V.: Disjointness-preserving operators, their continuity and multiplicative representation. In: Linear Operators and Their Applications, pp. 13–34. Sb. Nauchn. Trudov, Leningrad (1981)
Basly, A., Triki, A.: \(ff\)-algèbres réticulées. Preprint, Tunis (1988)
Bernau, C.B.: Orthomorphisms of Archimedean vector lattices. Math. Proc. Camb. Philos. Soc. 89, 119–126 (1981)
Bernau, S.J., Huijsmans, C.B.: Almost \(f\)-algebras and \(d\)-algebras. Math. Proc. Camb. Philos. Soc. 107, 287–308 (1990)
Birkhoff, G.: Lattice theory, Third edn. American Mathematical Society (AMS), vol. 25. VI, p. 418. Colloquium Publications, RI (1967)
Birkhoff, G., Pierce, R.S.: Lattice-ordered rings. An. Acad. Brasil. Ciènc. 28, 41–69 (1956)
Buskes, G., Van Rooij, A.: Almost \(f\)-algebras: structure and the Dedekind completion. Positivity 4(3), 233–243 (2000)
Buskes, G., Van Rooij, A.: Almost \(f\)-algebras: commutativity and Cauchy–Schwartz inequality. Positivity 4(3), 227–231 (2000)
Gutman, A.E.: Locally one-dimensional K-spaces and -distributive Boolean algebras. Sib. Adv. Math. 5(2), 99–121 (1995)
Gutman, A.E., Kusraev, A.G., Kutateladze, S.S.: The Wickstead problem. Sib. Electron. Math. Rep. 5, 293–333 (2008)
McPolin, P.T.N., Wickstead, A.W.: The order boundedness of band preserving operators on uniformly complete vector lattices. Math. Proc. Camb. Philos. Soc. 97(3), 481–487 (1985)
de Pagter, B.: A note on disjointness preserving operators. Proc. Am. Math. Soc. 90(4), 543–549 (1984)
de Pagter, B.: \(f\)-Algebras and orthomorphisms. Thesis, Leiden (1981)
Scheffold, E.: \(ff\)-Banachverbandsalgebren. Math. Z. 177, 183–205 (1981)
Kudláček, V.: O nĕkterych typech l-okruhu (on some types of \(\ell \)-rings). Sborni Vysokého Učeni Techn. v Brnĕ 1–2, 179–181 (1962)
Toumi, M.A.: Commutativity of almost F-algebras. Positivity 11(2), 357–368 (2007)
Toumi, M.A., Toumi, N.: The Wickstead problem on Dedekind \(\sigma \)-complete vector lattices. Positivity 14(1), 135–144 (2010)
Wickstead, A.W.: Representation and duality of multiplication operators on Archimedean Riesz spaces. Compos. Math. 35(3), 225–238 (1977)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kouki, N., Toumi, M.A. & Toumi, N. Commutativity of some Archimedean ordered algebras. Positivity 18, 805–821 (2014). https://doi.org/10.1007/s11117-014-0277-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11117-014-0277-5