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Jin Xi Chen, Zi Li Chen, "On Supra-Additive and Supra-Multiplicative Maps", Journal of Function Spaces, vol. 2013, Article ID 108535, 3 pages, 2013. https://doi.org/10.1155/2013/108535
On Supra-Additive and Supra-Multiplicative Maps
Let and be ordered algebras over , where has a generating positive cone and satisfies the property that if . We give some conditions for a map which is supra-additive and supra-multiplicative for all positive and negative elements to be linear and multiplicative; that is, is a homomorphism of algebras. Our results generalize some known results on supra-additive and supra-multiplicative maps between spaces of real functions.
Let be a compact Hausdorff space. Rǎdulescu  proved that whenever a map satisfies the conditions (supra-additive) and (supra-multiplicative) for all , then is linear and multiplicative. Ercan  generalized this result to arbitrary topological spaces. For arbitrary topological spaces , , a supra-additive and supra-multiplicative map is both linear and multiplicative if and only if for each and [2, Theorem 2]. Similar results were obtained for rings (see [3, 4]). In  Dhombres showed that if is a ring and is an ordered ring in which nonzero elements have nonzero positive squares, the supra-additive and supra-multiplicative map is both additive and multiplicative, that is, a homomorphism of rings. Recently, Gusić  considered supra-additive and supra-multiplicative maps between ordered fields. It was proved that on ordered fields every supra-additive and supra-multiplicative nonzero map is an injective homomorphism of fields.
Let and be ordered algebras over , where has a generating positive cone and satisfies the property that if . 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 supra-multiplicative 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 supra-multiplicative map is indeed both linear and multiplicative. This generalizes the results of Rǎdulescu  and Ercan . As a special case we consider the supra-additive and supra-multiplicative maps on 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 , then . is called Archimedean if and for all implies that . The set is called the positive cone of . 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 if . If a positive map satisfies the following inequalities: (1) (supra-additive), (2) (supra-multiplicative), for all , then is both linear and multiplicative.
Proof. From the inequality , it follows that . On the other hand, . Therefore, . It should be noted that for every . By the supra-additivity of , we have
for every . Thus, for every , from the following inequalities:
it follows that for every . By our hypothesis on , we have for all . Now, for all , we have
That is, for all . Similarly, for all , from
it follows that for all .
Next, because is positively generated and is Archimedean, by the positive additivity of on and the Kantorovich theorem (cf. [6, Proposition 1, page 150] or [7, Theorem 1.7]) there exists a unique positive linear map such that on , where is defined by for all . For every , there exist satisfying since is positively generated. From it follows that on . This implies that is linear. On the other hand, let and be arbitrary elements in . Since is positively generated, there exist , such that and . We have That is, is multiplicative, as desired.
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 supra-multiplicative only on or only on . Indeed, this is not the case. For instance, the positive nonlinear map defined by is additive and multiplicative on (or , resp.).
Recall that an ordered algebra is said to be squareroot closed and that whenever for any there exists , such that. Whenis square-root closed, we have the following corollary.
Corollary 3. Letbe a square-root closed ordered algebra with a generating positive cone. Letbe an Archimedean ordered algebra satisfying the property thatif. If a map is supra-additive and supra-multiplicative on, thenis 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 in there exists , such that . 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 , Volkmann , and Ercan  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 be the Banach lattice of all continuous real functions defined on and vanishing at infinity. Note that 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 is supra-additive and supra-multiplicative for all elements in , then is an algebra and lattice homomorphism.
Recall that for any is said to be separating or disjointness preserving if, for any , implies that . Clearly, every multiplicative map is disjointness preserving. Combining Corollary 5 with Theorem 2 of  or Theorem 8 in , we obtain the following Banach-Stone type result.
Corollary 6. Let , be locally compact Hausdorff spaces. If there exists a bijection from onto which is supra-additive and supra-multiplicative on , then is topologically homeomorphic to .
The authors would like to thank the reviewer for his/her kind comments and valuable suggestions which have improved this paper. The authors were supported in part by the Fundamental Research Funds for the Central Universities (SWJTU11CX154, SWJTU12ZT13) and NSFC (No. 11301285).
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Copyright © 2013 Jin Xi Chen and Zi Li Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.