On Supra-Additive and Supra-Multiplicative Maps

LetA and B be ordered algebras overR, whereA has a generating positive cone and B satisfies the property that b2 > 0 if 0 ̸ = b ∈ B. We give some conditions for a map T : A → B which is supra-additive and supra-multiplicative for all positive and negative elements to be linear and multiplicative; that is, T is a homomorphism of algebras. Our results generalize some known results on supra-additive and supra-multiplicative maps between spaces of real functions.

Let  and  be ordered algebras over R, where  has a generating positive cone and  satisfies the property that  2 > 0 if 0 ̸ =  ∈ .In this short paper, we prove that if a positive map  :  →  is supra-additive and supra-multiplicative for all positive and negative elements in , then  is indeed linear and multiplicative.In particular, if  is squareroot closed, then every map which is supra-additive and supramultiplicative for all positive and negative elements is both linear and multiplicative.From our result, it follows that for arbitrary topological spaces , , a supra-additive and supramultiplicative map  : () → () is indeed both linear and multiplicative.This generalizes the results of Rǎdulescu [1] and Ercan [2].As a special case we consider the supraadditive and supra-multiplicative maps on  0 () and obtain a Banach-Stone type result.
Recall that an ordered real vector space  under a multiplication is said to be an ordered algebra whenever the multiplication makes  an algebra, and in addition it satisfies the following property: if The positive cone of  is said to be generating (or  is positively generated) if  =  + −  + .A map  :  →  between two ordered algebras is called positive whenever ( + ) ⊆  + .Let  − := − + and  ± :=  + ∪  − .Theorem 1.Let  be an ordered algebra which has a generating positive cone.Let  be an Archimedean ordered algebra satisfying the property that  2 > 0 if 0 ̸ =  ∈ .If a positive map  :  →  satisfies the following inequalities: for all  1 ,  2 ∈  ± , then  is both linear and multiplicative.
Remark 2. Let  and  be as in the above theorem.It may be asked whether the positive map  :  →  is linear and multiplicative whenever  :  →  is supra-additive and supramultiplicative only on  + or only on  − .Indeed, this is not the case.For instance, the positive nonlinear map  : R → R defined by () = || is additive and multiplicative on R + (or R − , resp.).
Recall that an ordered algebra  is said to be squareroot closed and that whenever for any  ∈  + there exists  ∈  + , such that  =  2 .When  is square-root closed, we have the following corollary.

Corollary 3.
Let  be a square-root closed ordered algebra with a generating positive cone.Let  be an Archimedean ordered algebra satisfying the property that  2 > 0 if 0 ̸ =  ∈ .If a map  :  →  is supra-additive and supra-multiplicative on  ± , then  is both linear and multiplicative on .
Proof.By the above Theorem, to complete the proof, we need only to verify that  is positive.Since  is square-root closed for each  ≥ 0 in  there exists  ∈  + , such that  =  2 .Hence, by our hypothesis on , we have This implies that  is positive.
Remark 4. It should be noted that the space of all real functions (all real continuous functions) on a nonempty set (a topological space, resp.), with the pointwise algebraic operations and the pointwise ordering, is a square-root closed Archimedean lattice-ordered algebra with the property mentioned in Corollary 3. Thus, the results on supra-additive and supra-multiplicative maps between spaces of real functions obtained by Rǎdulescu [1], Volkmann [4], and Ercan [2] can now follow from Corollary 3. In their earlier proofs, the constant function or the multiplicative unit element plays an essential role.
Let  be a locally compact Hausdorff space, and let  0 () be the Banach lattice of all continuous real functions defined on  and vanishing at infinity.Note that  0 () does not necessarily contain the constant function or a unit element unless  is compact.The following result is an immediate consequence of Corollary 3.

Corollary 5.
Let  and  be locally compact Hausdorff spaces.If  :  0 () →  0 () is supra-additive and supramultiplicative for all elements in  0 () ± , then  is an algebra and lattice homomorphism.