Abstract
In this paper, we show, among other results, that if A is an archimedean vector lattice, then any orthosymmetric disjointness preserving bilinear map on A × A is order bounded if and only if A is hyper-archimedean.
Finally, we show for a uniformly complete semiprime f-algebra A, that the vector space of all linear operators T from \({\Pi(A) = \{ab; \forall a, b \in A\}}\) into A and the vector space of orthosymmetric bilinear maps \({\Psi: A \times A \rightarrow A}\) are isomorphic if and only if A is hyper-archimedean.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aliprantis C.D., Burkinshaw O.: Positive Operators. Academic Press, Orlando (1985)
Ben Amor F.: On orthosymmetric bilinear maps. Positivity 14, 123–134 (2010)
Bernau S.J., Huijsmans C.B.: Almost f-algebras and d-algebras. Math. Proc. Cambridge Philos. Soc 107, 287–308 (1990)
Boulabiar K.: A relationship between two almost f-algebra products. Algebra Universalis 43, 347–367 (2000)
Boulabiar K.: Products in almost f-algebras. Comment. Math. Univ. Carolinae 41, 747–759 (2000)
Buskes B., van Rooij A.: Almost f-algebras, commutativity and the Cauchy-Schwarz inequality. Positivity 4, 227–331 (2000)
Buskes B., van Rooij A.: Squares of Riesz spaces. Rocky Mountain J. Math 31, 45–56 (2001)
Buskes B., van Rooij A.: Bounded variation and tensor products of Banach lattices. Positivity 7, 47–59 (2003)
Buskes B., Kusraev A.G.: Representation and extension of orthoregular bilinear operators. Vladikavkaz Math J. 9, 16–29 (2007)
Conrad P.: Epi-Archimedean groups. Czech. Math. J 24, 192–218 (1974)
Hager A.W., Kimber C.M.: Uniformly hyperarchimedean lattice-ordered groups. Order 24, 121–131 (2007)
Luxemburg W.A.J., Zaanen A.C.: Riesz spaces I. North-Holland, Amsterdam (1971)
Luxemburg W.A.J., Moore L.C.: Archimedean quotient Riesz spaces. Duke Math. J 34, 725–739 (1967)
Marinacci M., Montrucchio L.: On concavity and supermodularity. J. Math. Anal. Appl 344, 642–654 (2008)
Martínez J., Zenk E.R.: Yosida frames. J. Pure Appl. Algebra 204, 473–492 (2006)
Martínez J.: Dimension in algebraic frames. Czech. Math. J 56, 437–474 (2006)
Mc Polin P.T.N., Wickstead A.W.: The order boundedness of band preserving operators on uniformly complete vector lattices. Math. Proc. Cambridge Philos. Soc 97, 481–487 (1985)
de Pagter, B.: f-algebras and Orthomorphisms. PhD thesis, University of Leiden (1981)
de Pagter B.: A note on disjointness preserving operators. Proc. Amer. Math. Soc 90, 543–549 (1984)
de Pagter B.: The space of extended orthomorphisms in a Riesz space. Pacific J. Math. 112, 193–210 (1984)
Schaefer H.H.: Banach Lattices and Positive Operators. Springer, New York (1974)
Toumi, A., Toumi, M.A., Toumi, N.: When a continuous orthosymmetric bilinear operator is symmetric (2009, preprint)
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by J. Martinez.
Rights and permissions
About this article
Cite this article
Toumi, M.A. Characterization of hyper-archimedean vector lattices via disjointness preserving bilinear maps. Algebra Univers. 67, 29–42 (2012). https://doi.org/10.1007/s00012-012-0166-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00012-012-0166-3