Characterization of separating couples

Toumi, Nedra

doi:10.1007/s12215-008-0012-9

Rend. Circ. Mat. Palermo

2008

DOI: 10.1007/s12215-008-0012-9

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Characterization of separating couples

Nedra Toumi

Abstract: Let A, B be two archimedean -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 -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 or… Show more

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2012

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CONTINUOUS GENERALIZED (θ, ϕ)-SEPARATING DERIVATIONS ON ARCHIMEDEAN ALMOST f-ALGEBRAS

Toumi

2012

Asian-European J. Math.

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Let A be an ℓ-algebra and let θ and ϕ be two endomorphisms of A. The couple (θ, ϕ) is called to be separating if xy = 0 implies θ(x)ϕ(y) = 0. If in addition θ and ϕ are ring endomorphisms of A, then the couple (θ, ϕ) is said to be ring-separating. An additive mapping δ : A → A is called (θ, ϕ)-separating derivation on A if there exists a (θ, ϕ)-separating couple with δ(xy) = δ(x)θ(y) + ϕ(x)δ(y), holds for all x, y ∈ A. If an addition θ, ϕ and δ are continuous, then δ is called a continuous (θ, ϕ)-ring-separating derivation. If in addition the couple (θ, ϕ) is ring-separating then δ is called a continuous (θ, ϕ)-ring-separating derivation. An additive mapping F : A → A is called a continuous generalized (θ, ϕ)-separating derivation on A if F is continuous mapping and if there exists a derivation d : A → A such that θ and ϕ are continuous, (θ, ϕ) is a separating couple and F(xy) = F(x)θ(y) + ϕ(x)d(y), holds for all x, y ∈ A. In this paper, we give a description of continuous (θ, ϕ)-ring-separating derivations on some ℓ-algebras. This generalizes a well-known theorem by Colville, Davis, and Keimel [Positive derivations on f-rings, J. Austral. Math. Soc23 (1977) 371–375] and generalizes the results of Boulabiar in [Positive derivations on almost f-rings, Order19 (2002) 385–395], Ben Amor [On orthosymmetric bilinear maps, Positivity14(1) (2010) 123–130] and Toumi et al. in [Order bounded derivations on Archimedean almost f-algebras, Positivity14(2) (2010) 239–245]. Moreover, inspiring from [Toumi, Order-bounded generalized derivations on Archimedean almost f-algebras, Commun. Algebra38(1) (2010) 154–164], it is shown that the notion of continuous generalized (θ, ϕ)-separating derivation on an archimedean almost f-algebra A is the concept of generalized θ-multiplier, that is an additive mapping satisfying F(xyz) = F(x)θ(yz), for all x, y, z ∈ A. In the case where A is an archimedean f-algebra, the situation improves. Indeed, the collection of all continuous generalized (θ, ϕ)-separating derivation on A coincides with the concept of θ-multiplier, that is an additive mapping satisfying F(xy) = F(x)θ(y), for all x, y ∈ A. If in addition A is a Dedekind complete vector lattice and θ is a positive mapping, then the set of all order bounded generalized of the form (θ, ϕ)-separating derivations on A, under composition, is an archimedean lattice-ordered algebra.

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CONTINUOUS GENERALIZED (θ, ϕ)-SEPARATING DERIVATIONS ON ARCHIMEDEAN ALMOST f-ALGEBRAS

Toumi

2012

Asian-European J. Math.

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Characterization of separating couples

Cited by 1 publication

References 11 publications

CONTINUOUS GENERALIZED (θ, ϕ)-SEPARATING DERIVATIONS ON ARCHIMEDEAN ALMOST f-ALGEBRAS

CONTINUOUS GENERALIZED (θ, ϕ)-SEPARATING DERIVATIONS ON ARCHIMEDEAN ALMOST f-ALGEBRAS

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