Abstract and Applied Analysis

VolumeĀ 2011Ā (2011), Article IDĀ 513128, 12 pages

http://dx.doi.org/10.1155/2011/513128

## Nearly Jordan -Homomorphisms between Unital -Algebras

^{1}Department of Mathematics, Urmia University, Urmia, Iran^{2}Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran^{3}Center of Excellence in Nonlinear Analysis and Applications (CENAA), Semnan University, Semnan, Iran

Received 26 February 2011; Revised 7 April 2011; Accepted 10 April 2011

Academic Editor: IrenaĀ Lasiecka

Copyright Ā© 2011 A. Ebadian et al. 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.

#### Abstract

Let , be two unital -algebras. We prove that every almost unital almost linear mapping : which satisfies for all , all , and all , is a Jordan homomorphism. Also, for a unital -algebra of real rank zero, every almost unital almost linear continuous mapping is a Jordan homomorphism when holds for all (), all , and all . Furthermore, we investigate the Hyers- Ulam-Aoki-Rassias stability of Jordan -homomorphisms between unital -algebras by using the fixed points methods.

#### 1. Introduction

The stability of functional equations was first introduced by Ulam [1] in 1940. More precisely, he proposed the following problem: given a group , a metric group and a positive number , does there exist a such that if a function satisfies the inequality for all , then there exists a homomorphism such that for all ?. As mentioned above, when this problem has a solution, we say that the homomorphisms from to are stable. In 1941, Hyers [2] gave a partial solution of *Ulam’s* problem for the case of approximate additive mappings under the assumption that and are Banach spaces. Hyers theorem was generalized by Aoki [3] for additive mappings and by Th. M. Rassias [4] for linear mappings by considering an unbounded Cauchy difference. This phenomenon of stability is called the Hyers-Ulam-Aoki-Rassias stability.

J. M. Rassias [5–7] established the stability of linear and nonlinear mappings with new control functions.

During the last decades, several stability problems of functional equations have been investigated by many mathematicians. A large list of references concerning the stability of functional equations can be found in [8–10].

Bourgin is the first mathematician dealing with the stability of ring homomorphisms. The topic of approximate ring homomorphisms was studied by a number of mathematicians, see [11–19] and references therein.

Jun and Lee [20] proved the following: Let and be Banach spaces. Denote by a function such that for all . Suppose that is a mapping satisfying for all . Then there exists a unique additive mapping such that for all .

Recently, C. Park and W. Park [21] applied the Jun and Lee's result to the Jensen's equation in Banach modules over a -algebra. Johnson (Theorem 7.2 of [22]) also investigated almost algebra -homomorphisms between Banach -algebras: Suppose that U and B are Banach -algebras which satisfy the conditions of (Theorem 3.1 of [22]). Then for each positive and , there is a positive such that if with and , then there is a -homomorphism with . Here is the space of bounded linear maps from U into B, and . See [22] for details. Throughout this paper, let A be a unital -algebra with unit e, and B a unital -algebra. Let be the set of unitary elements in A, , and . In this paper, we prove that every almost unital almost linear mapping is a Jordan homomorphism when holds for all , all , and all , and that for a unital -algebra of real rank zero (see [23]), every almost unital almost linear continuous mapping is a Jordan homomorphism when holds for all , all , and all . Furthermore, we investigate the Hyers-Ulam-Aoki-Rassias stability of Jordan -homomorphisms between unital -algebras by using the fixed point methods.

Note that a unital -algebra is of real rank zero, if the set of invertible self-adjoint elements is dense in the set of self-adjoint elements (see [23]). We denote the algebric center of algebra by .

#### 2. Jordan -Homomorphisms on Unital -Algebras

By a following similar way as in [24], we obtain the next theorem.

Theorem 2.1. *Let be a mapping such that and that
**
for all , all , and all . If there exists a function such that for all and all and that
**
for all and all . If , then the mapping is a Jordan -homomorphism.*

*Proof. *Put , in (2.2), it follows from of [20, Theorem 1] that there exists a unique additive mapping such that
for all . This mapping is given by
for all . By the same reasoning as the proof of [24, Theorem 1], is -linear and -preserving. It follows from (2.1) that
for all , all . Since is additive, then by (2.5), we have
for all and all . Hence,
for all and all . By the assumption, we have
hence, it follows by (2.5) and (2.7) that
for all . Since is invertible, then for all . We have to show that is Jordan homomorphism. To this end, let . By Theorem of [25], is a finite linear combination of unitary elements, that is, , and then it follows from (2.7) that
for all . And this completes the proof of theorem.

Corollary 2.2. *Let , be real numbers. Let be a mapping such that and that
**
for all , all , and all . Suppose that
**
for all and all . If , then the mapping is a Jordan -homomorphism.*

*Proof. *Setting for all . Then by Theorem 2.1, we get the desired result.

Theorem 2.3. *Let be a -algebra of real rank zero. Let be a continuous mapping such that and that
**
for all , all , and all . Suppose that there exists a function satisfying (2.2) and for all and all . If , then the mapping is a Jordan -homomorphism.*

*Proof. *By the same reasoning as the proof of Theorem 2.1, there exists a unique involutive -linear mapping satisfying (2.3). It follows from (2.13) that
for all , and all . By additivity of and (2.14), we obtain that
for all and all . Hence,
for all and all . By the assumption, we have
Similar to the proof of Theorem 2.1, it follows from (2.14) and (2.16) that on . So is continuous. On the other hand, is real rank zero. One can easily show that is dense in . Let . Then there exists a sequence in such that . Since is continuous, it follows from (2.16) that
for all . Now, let . Then we have , where and are self adjoint.

First consider . Since is -linear, it follows from (2.18) that
for all .

If , , then by (2.18), we have
for all .

Finally, consider the case that , . Then it follows from (2.18) that
for all . Hence, for all , and is Jordan -homomorphism.

Corollary 2.4. *Let be a -algebra of rank zero. Let , be real numbers. Let be a mapping such that and that
**
for all , all , and all . Suppose that
**
for all and all . If , then the mapping is a Jordan -homomorphism.*

*Proof. *Setting for all . Then by Theorem 2.3, we get the desired result.

#### 3. Stability of Jordan -Homomorphisms: A Fixed Point Approach

We investigate the generalized Hyers-Ulam-Aoki-Rassias stability of Jordan -homomorphisms on unital -algebras by using the alternative fixed point.

Recently, Cdariu and Radu applied the fixed point method to the investigation of the functional equations. (See also [18, 26–43]).

Theorem 3.1. *Let be a mapping with for which there exists a function satisfying
**
for all , and all . If there exists an such that for all , then there exists a unique Jordan -homomorphism such that
**
for all .*

*Proof. *It follows from that
for all .

Put and in (3.1) to obtain
for all . Hence,
for all .

Consider the set and introduce the generalized metric on X:
It is easy to show that is complete. Now we define the linear mapping by
for all . By Theorem 3.1 of [44],
for all .

It follows from (3.5) that
Now, from the fixed point alternative [45], has a unique fixed point in the set . Let be the fixed point of . is the unique mapping with
for all satisfying there exists such that
for all . On the other hand, we have . It follows that
for all . It follows from , that
This implies the inequality (3.2). It follows from (3.1), (3.3), and (3.12) that
for all . So
for all . Put , and in above equation, we get for all . Hence, is Cauchy additive. By putting , in (3.1), we have
for all and all . It follows that
for all , and all . One can show that the mapping is -linear. By putting in (3.1), it follows that
for all . By the same reasoning as the proof of Theorem 2.1, we can show that is -preserving.

Since is -linear, by putting in (3.1), it follows that
for all . Thus is Jordan -homomorphism satisfying (3.2), as desired.

We prove the following Hyers-Ulam-Aoki-Rassias stability problem for Jordan -homomorphisms on unital -algebras.

Corollary 3.2. *Let , be real numbers. Suppose satisfies
**
for all and all . Then there exists a unique Jordan -homomorphism such that such that
**
for all .*

*Proof. *Setting all . Then by in Theorem 3.2, one can prove the result.

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