Hamiltonian ordered algebras and congruence extension

We introduce the notion of Hamiltonian ordered algebra and explore its relationships with various congruence extension properties for ordered algebras. Furthermore, we give sufficient conditions for extending order-congruences or compatible quasiorders from subalgebras.

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

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.

On Partially Ordering Operator Algebras

In this paper, we consider linear spaces and algebras with real scalars. It is well known that if X is a Banach space and is the set of all bounded linear operators which map X into itself, then is a Banach algebra. In this paper we shall show that can be partially ordered so that it becomes a partially ordered algebra in which norm convergence is equivalent to order convergence. This motivates a study of Banach algebras of operators in which one uses the order structure to obtain various results. In addition, it encourages a study of partially ordered algebras in general, since our result shows that among such algebras one finds all real Banach algebras of operators. Of course, there are many other real algebras which are naturally partially ordered and which have been studied from that point of view.