- Research Article
- Open Access

# Approximately -Jordan Homomorphisms on Banach Algebras

- M. Eshaghi Gordji
^{1}Email author, - T. Karimi
^{2}and - S. Kaboli Gharetapeh
^{3}

**2009**:870843

https://doi.org/10.1155/2009/870843

© M. Eshaghi Gordji et al. 2009

**Received:**29 November 2008**Accepted:**14 January 2009**Published:**27 January 2009

## Abstract

Let , and let be two rings. An additive map is called -Jordan homomorphism if for all . In this paper, we establish the Hyers-Ulam-Rassias stability of -Jordan homomorphisms on Banach algebras. Also we show that (a) to each approximate 3-Jordan homomorphism from a Banach algebra into a semisimple commutative Banach algebra there corresponds a unique 3-ring homomorphism near to , (b) to each approximate -Jordan homomorphism between two commutative Banach algebras there corresponds a unique -ring homomorphism near to for all .

## Keywords

- Real Number
- Exact Solution
- Functional Equation
- Uniqueness Property
- Stability Problem

## 1. Introduction and Preliminaries

Let
be two rings (algebras). An additive map
is called *n*-Jordan homomorphism (*n*-ring homomorphism) if
for all
for all
If
is a linear *n*-ring homomorphism, we say that
is *n*-homomorphism. The concept of *n*-homomorphisms was studied for complex algebras by Hejazian et al. [1] (see also [2, 3]). A 2-Jordan homomorphism is a Jordan homomorphism, in the usual sense, between rings. Every Jordan homomorphism is an *n*-Jordan homomorphism, for all
(e.g., [4, Lemma 6.3.2]), but the converse is false, in general. For instance, let
be an algebra over
and let
be a nonzero Jordan homomorphism on
. Then,
is a 3-Jordan homomorphism. It is easy to check that
is not 2-Jordan homomorphism or 4-Jordan homomorphism. The concept of *n*-Jordan homomorphisms was studied by the first author [5]. A classical question in the theory of functional equations is that "when is it true that a mapping which approximately satisfies a functional equation
must be somehow close to an exact solution of
?" Such a problem was formulated by Ulam [6] in 1940 and solved in the next year for the Cauchy functional equation by Hyers [7]. It gave rise to the *stability theory* for functional equations. Subsequently, various approaches to the problem have been introduced by several authors. For the history and various aspects of this theory we refer the reader to monographs [8–12]. Applying a theorem of Hyers [7], Rassias [13], and Gajda [14], Bourgin [15] proved the stability problem of ring homomorphisms between unital Banach algebras. Badora [16] proved the Hyers-Ulam-Rassias stability of ring homomorphisms, which generalizes the result of Bourgin. Recently, Miura et al. [17] proved the Hyers-Ulam-Rassias stability of Jordan homomorphisms. The stability problem of *n*-homomorphisms between Banach algebras, has been proved by the first author [18]. In this paper, we consider the stability, in the sense of Hyers-Ulam-Rassias, of *n*-Jordan homomorphisms on Banach algebras.

## 2. Main Result

By a following similar way as in [17], we obtain the next theorem.

Theorem 2.1.

for all

Proof.

*n*-Jordan homomorphism. Since it follows from (2.2) that

for all
In other words,
is *n*-Jordan homomorphism. The uniqueness property of
follows from [13, 14].

Theorem 2.2.

*n*-Jordan homomorphism such that

for all

Proof.

which proves , whenever This completes the proof.

By [17, Theorem 1.1] and [5, Theorem 2.5], we have the following theorem.

Theorem 2.3.

Let
be fixed. Suppose
is a Banach algebra, which needs not to be commutative, and suppose
is a semisimple commutative Banach algebra. Then, each *n*-Jordan homomorphism
is a *n*-ring homomorphism.

Let
be fixed. As a direct corollary, we show that to each approximate *n*-Jordan homomorphism
from a Banach algebra into a semisimple commutative Banach algebra there corresponds a unique *n*-ring homomorphism near to
.

Corollary 2.4.

for all

Proof.

It follows from Theorems 2.1, 2.2, and 2.3.

Theorem 2.5.

Let
be fixed,
be two commutative algebras, and let
be a *n*-Jordan homomorphism. Then,
is *n*-ring homomorphism.

Proof.

Hence, is 5-ring homomorphism.

Corollary 2.6.

for all

Proof.

It follows from Theorems 2.1, 2.2, and 2.5.

## Authors’ Affiliations

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## Copyright

This article is published under license to BioMed Central Ltd. 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.