Additive mappings on -algebras sub-preserving absolute values of products
© Taghavi; licensee Springer 2012
Received: 1 January 2012
Accepted: 2 July 2012
Published: 19 July 2012
Let be a -algebra of real rank zero and be a -algebra with unit I. It is shown that if is an additive mapping which satisfies for every and for some with , then the restriction of mapping ϕ to is a Jordan homomorphism, where denotes the set of all self-adjoint elements. We will also show that if ϕ is surjective preserving the product and an absolute value, then ϕ is a -linear or -antilinear ∗-homomorphism on .
MSC:47B49, 46L05, 47L30.
Keywords-algebra -linear -antilinear homomorphism linear preserver problem real rank zero
1 Introduction and preliminaries
In recent years, the subject of linear preserver problems is the focus of attention of many mathematicians, and much research has been going on in this area. Here we refer to the articles [1–6, 9–16].
In what follows, let and be two -algebras with unit I. We say that a mapping is preserving (resp. sub-preserving) absolute values of a product if (resp. ) for every , where . By a ∗-homomorphism we just mean a map which preserves the ring structure and for which for every . A map is said to be a Jordan ∗-homomorphism if it is -linear, and for all . We also say a map is unital if . The class of all self-adjoint elements in a -algebra is denoted by . We define for all . It is known that is a -algebra which is called a Jordan algebra.
It is the aim of this paper to continue this work by studying additive mappings ϕ from a -algebra of real rank zero into a -algebra that sub-preserve product and absolute value. In fact, we show that if the mapping ϕ which is an additive sub-preserving product absolute value map from a -algebra into a -algebra then ϕ is a contraction. Moreover, if is a -algebra of real rank zero and for some A in the closed unit ball , then the restriction of mapping ϕ to is a Jordan homomorphism. We will also show that if ϕ is a surjective preserving product and absolute value and is a -algebra of real rank zero, then ϕ is a -linear or -antilinear ∗-homomorphism on . All we need about -algebras and von Neumann algebras can be found in [7, 8].
2 The main results
Firstly, we need some auxiliary lemmas to prove our main result.
Proof If A is a positive element in , then . This means is positive and preserves positive elements.
Let and . By the assumption . Thus and hence . □
Lemma 2.2 Letandbe two unital-algebras with unit I. Ifis an additive mapping satisfying (2.1), then ϕ is order preserving and contraction (i.e. ) on.
Proof By Lemma 2.1 ϕ preserves positive elements. Hence additivity of ϕ implies that ϕ is order preserving. And also, since every self-adjoint element is the difference of two positive elements, ϕ preserves self-adjoint elements. Indeed, we show that ϕ maps the part of positive (resp. negative) of A to the part of positive (resp. negative) of . In fact, and , where, and . We just need to show because the decomposition of is unique and ϕ preserves positives. Applying Lemma 2.1 and the equation , we get the assertion.
The proof of -linearity of ϕ is similar to the first step of the proof of , Theorem 1]. The details are omitted.
since . Taking square root, we obtain , which yields ϕ is a contraction on . □
then ϕ is unital.
because ϕ is contraction by Lemma 2.2. Therefore, . □
The following example shows that the condition (2.3) in Lemma 2.3 is necessary.
- (i)for every self-adjoint operator , we have(2.4)
is a closed ideal of .
Proof (i) Let E and F be mutually orthogonal projections. By Lemma 2.1 , in particular, . That is, .
Let be a self-adjoint element such that . We show that .
Since , by using the order preserving property of ϕ we yield . So because is a self-adjoint element.
It follows N is a closed ideal of by the step 4 of , Theorem 2.1]. □
In the following theorem we would like to characterize the Jordan homomorphisms ϕ which are additive mappings sub-preserving a product and an absolute value.
Theorem 2.6 Letbe a-algebra of real rank zero andbe a unital-algebra with unit I. Ifis an additive mapping satisfying (2.1) and (2.3), then the restriction of the map ϕ tois a Jordan homomorphism.
Proof According to Lemma 2.3 and Lemma 2.5(i) we yield the statement. □
Now we also show that the condition (2.3) in Theorem 2.6 is necessary.
where is invertible, and . Obviously, ϕ is an additive mapping that satisfies in (2.1), but no nonzero multiple of ϕ is a Jordan homomorphism, because if with is a Jordan homomorphism, then we obtain , that is a contradiction.
In the following theorem we show that if is an injective operator and ϕ is an additive map which satisfies in (2.1), then the restriction of ϕ is a Jordan homomorphism multiplied by .
Theorem 2.8 Letbe a-algebra of real rank zero andbe a unital-algebra with unit I. Ifis an additive mapping satisfyingfor everyandis an injective operator, then the restriction of mappingis defined by, tois a Jordan homomorphism.
This completes the proof. □
where, ψ is either a linear or conjugate linear ∗-algebra isomorphism and is a scalar function of Modulus 1. Below, we present the result where we do not assume injectivity but ϕ is an additive map from -algebra onto a -algebra of real rank zero which preserves a product and an absolute value, and it is shown ϕ is a -linear or -antilinear ∗-homomorphism.
Theorem 2.9 Letandbe two unital-algebra with unit I. Ifis an additive mapping satisfyingfor everyandfor some, then ϕ is unital and the restriction of mapping ϕ tois a Jordan homomorphism. Moreover, if ϕ is surjective andbe a-algebra of real rank zero then, ϕ is a -linear or -antilinear ∗-homomorphism.
So ϕ is a unital map.
Therefore, ϕ is a -linear or -antilinear ∗-homomorphism on by , Theorem 2.5]. □
This research is partially supported by the Research Center in Algebraic Hyperstructures and Fuzzy Mathematics, University of Mazandaran, Babolsar, Iran.
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