Abstract
Let A, B be two archimedean ℓ-algebras and let U,V be two positive linear maps from A to B. We call that the couple (U,V) is separating with respect to A and B if |a||b| = 0 in A implies |U (a)||V (b)| = 0 in B. In this paper, we prove that if A is an f-algebra with unit elment e, if B is an ℓ-algebra and if (U,V) is a separating couple with respect to A and B then (U ∼∼,V ∼∼), where U ∼∼ (resp V ∼∼) is the bi-adjoint of U (resp of V), is again a separating couple with respect to the order continuous order biduals (A′)′ n and (B′)′ n of A and B respectively furnished with their Arens products respectively. Moreover, in the case where B′ separates the points of B, we give a characterization of any separating couple with respect to A and B.
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
Arens, A.: The adjoint of a bilinear operation, Proc. Amer. Math. Soc., 2 (1951), 839–848
Bernau, S.J, Huijsmans, C.B.: The order bidual of almost f-algebras and d-algebras, Trans. Amer. Math. Soc., 347 (1995), 4259–4275
Bernau, S.J, Huijsmans, C.B.: Almost f-algebras and d-algebras, Math. Proc. Cam. Phil. Soc., 107 (1990), 287–308
Boulabiar, K, Toumi, M.A.: Lattice bimorphisms on f-algebras, Algebra Universalis, 48 (2002), 103–116
Huijsmans, C.B, De Pagter, B.: The order bidual of lattice ordered algebras, J. Funct. Anal., 59 (1984), 61–74
Huijsmans, C.B.: Lattice-Ordered Algebras and f-algebras: A survey in Studies in Economic Theory 2′, Springer Verlag, Berlin-Heidelberg-New York, (1990), 151–169
Grobler, J.J.: Commutativity of the Arens product in lattice ordered algebras, Positivity, 3 (1999), 357–364
Luxemburg, W.A.J, Zaanen, A.C.: Riesz spaces I, North-Holland, Amsterdam, 1971
Meyer-Neiberg, P.: Banach lattices, Springer Verlag, Berlin-Heidelberg-New York, 1974
De Pagter, B.: f-algebras and Orthomorphisms, thesis, Leiden, 1981
Schaefer, H.H.: Banach lattices and positive operators, Springer Verlag, New York, 1974
Toumi, M.A.: On some f-subalgebras of a d-algebra, Math. Rep (Bucur), 4(54), 3 (2002), 303–310
Toumi, M.A.: Extensions of orthosymmetric lattice bimorphisms, Proc. Amer. Math. Soc., 134 (2006), 1615–1621
Toumi, M.A.: A simple proof for a theorem of Luxemburg and Zaanen, J. Math. Anal. Appl., 322 (2006) 1231–1234
Toumi, M.A, Toumi, N.: Laterally closed lattice homomorphisms, J. Math. Anal. Appl., 324 (2006) 1178–1194
Toumi, M.A, Toumi, N.: Structure Theorem for d-algebras, submitted to Demonstratio Mathematica
Zaanen, AC.: Riesz spaces II, North-Holland, Amsterdam, 1983
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Toumi, N. Characterization of separating couples. Rend. Circ. Mat. Palermo 57, 181–192 (2008). https://doi.org/10.1007/s12215-008-0012-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12215-008-0012-9