# Approximately -Jordan Homomorphisms on Banach Algebras

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

**2009**:870843

**DOI: **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 .

## 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.

*n*-Jordan homomorphism such that

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.

*n*-ring homomorphism such that

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.

*n*-ring homomorphism such that

for all

Proof.

It follows from Theorems 2.1, 2.2, and 2.5.

## Authors’ Affiliations

## References

- Hejazian S, Mirzavaziri M, Moslehian MS:
**-homomorphisms.***Bulletin of the Iranian Mathematical Society*2005,**31**(1):13–23.MathSciNetMATHGoogle Scholar - Bračič J, Moslehian MS:
**On automatic continuity of 3-homomorphisms on Banach algebras.***Bulletin of the Malaysian Mathematical Sciences Society. Second Series*2007,**30**(2):195–200.MathSciNetMATHGoogle Scholar - Park E, Trout J:
**On the nonexistence of nontrivial involutive -homomorphisms of -algebras.**to appear in*Transactions of the American Mathematical Society*to appear in Transactions of the American Mathematical Society - Palmer ThW:
*Banach Algebras and the General Theory of ∗-Algebras. Vol. I: Algebras and Banach Algebra, Encyclopedia of Mathematics and Its Applications*.*Volume 49*. Cambridge University Press, Cambridge, Mass, USA; 1994:xii+794.View ArticleMATHGoogle Scholar - Eshaghi Gordji M:
**-Jordan homomorphisms.**preprint - Ulam SM:
*Problems in Modern Mathematics*. John Wiley & Sons, New York, NY, USA; 1940.MATHGoogle Scholar - Hyers DH:
**On the stability of the linear functional equation.***Proceedings of the National Academy of Sciences of the United States of America*1941,**27**(4):222–224. 10.1073/pnas.27.4.222MathSciNetView ArticleMATHGoogle Scholar - Czerwik S:
*Functional Equations and Inequalities in Several Variables*. World Scientific, River Edge, NJ, USA; 2002:x+410.View ArticleMATHGoogle Scholar - Faĭziev VA, Rassias ThM, Sahoo PK:
**The space of (**,**)-additive mappings on semigroups.***Transactions of the American Mathematical Society*2002,**354**(11):4455–4472. 10.1090/S0002-9947-02-03036-2MathSciNetView ArticleMATHGoogle Scholar - Hyers DH, Isac G, Rassias ThM:
*Stability of Functional Equations in Several Variables, Progress in Nonlinear Differential Equations and Their Applications, 34*. Birkhäuser, Boston, Mass, USA; 1998:vi+313.View ArticleMATHGoogle Scholar - Jung S-M:
*Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis*. Hadronic Press, Palm Harbor, Fla, USA; 2001:ix+256.MATHGoogle Scholar - Rassias ThM:
**On the stability of functional equations and a problem of Ulam.***Acta Applicandae Mathematicae*2000,**62**(1):23–130. 10.1023/A:1006499223572MathSciNetView ArticleMATHGoogle Scholar - Rassias ThM:
**On the stability of the linear mapping in Banach spaces.***Proceedings of the American Mathematical Society*1978,**72**(2):297–300. 10.1090/S0002-9939-1978-0507327-1MathSciNetView ArticleMATHGoogle Scholar - Gajda Z:
**On stability of additive mappings.***International Journal of Mathematics and Mathematical Sciences*1991,**14**(3):431–434. 10.1155/S016117129100056XMathSciNetView ArticleMATHGoogle Scholar - Bourgin DG:
**Approximately isometric and multiplicative transformations on continuous function rings.***Duke Mathematical Journal*1949,**16:**385–397. 10.1215/S0012-7094-49-01639-7MathSciNetView ArticleMATHGoogle Scholar - Badora R:
**On approximate derivations.***Mathematical Inequalities & Applications*2006,**9**(1):167–173.MathSciNetView ArticleMATHGoogle Scholar - Miura T, Takahasi S-E, Hirasawa G:
**Hyers-Ulam-Rassias stability of Jordan homomorphisms on Banach algebras.***Journal of Inequalities and Applications*2005,**2005**(4):435–441. 10.1155/JIA.2005.435MathSciNetView ArticleMATHGoogle Scholar - Eshaghi Gordji M:
**On approximate -ring homomorphisms and -ring derivations.**preprint

## 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.