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Approximately -Jordan Homomorphisms on Banach Algebras

Journal of Inequalities and Applications20092009:870843

DOI: 10.1155/2009/870843

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 [812]. 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.

Let be a normed algebra, let be a Banach algebra, let and be nonnegative real numbers, and let be a real numbers such that or , and that . Assume that satisfies the system of functional inequalities
(2.1)
(2.2)
for all Then, there exists a unique n-Jordan homomorphism such that
(2.3)

for all

Proof.

Put , and for all It follows from [13, 14] that is additive map satisfies (2.3). We will show that is n-Jordan homomorphism. Since it follows from (2.2) that
(2.4)
Hence, we have
(2.5)

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

Theorem 2.2.

Let be a normed algebra, let be a Banach algebra, let and be nonnegative real numbers, and let be real numbers such that and . If is a mapping, with , such that the inequalities (2.1) and (2.2) are valid. Then, there exists a unique n-Jordan homomorphism such that
(2.6)

for all

Proof.

Assume that It follows from [13] that there exists an additive map satisfies (2.6). It suffices to show that for all Since is additive, we get and so the case is omitted. Let be arbitrarily. If then the proof of Theorem 2.1 works well, and Thus we need to consider only the case Since it follows from (2.2), that
(2.7)
Hence, we have
(2.8)
On the other hand, we have
(2.9)
It follows from (2.8) and (2.9) that
(2.10)

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.

Let be fixed. Suppose is a Banach algebra, which needs not to be commutative, and suppose is a semisimple commutative Banach algebra. Let and be nonnegative real numbers and let be a real numbers such that or and . Assume that satisfies the system of functional inequalities
(2.11)
for all Then, there exists a unique n-ring homomorphism such that
(2.12)

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.

For (see [5, Theorem 2.2]). Now suppose Then, is additive and for all Replacing by to get
(2.13)
Now, replacing by in (2.13), we obtain that
(2.14)
By (2.13) and (2.14), we get
(2.15)
By (2.15) it follows that
(2.16)
Replacing by in (2.16), we obtain
(2.17)
Replacing by in (2.17), we get
(2.18)
Hence, we get
(2.19)
By (2.19) it follows that
(2.20)
Replacing by in (2.20), we obtain
(2.21)
Replacing by in (2.21), we get
(2.22)
Replacing by in above equality to get
(2.23)
Replacing by in (2.23), we obtain
(2.24)
Combining (2.23) by (2.24), we get
(2.25)
Replacing by in (2.25) to obtain
(2.26)
replacing by in (2.26), we get
(2.27)
Now, replace by in (2.27), we obtain
(2.28)

Hence, is 5-ring homomorphism.

Corollary 2.6.

Let be fixed. Suppose are commutative Banach algebras. Let and be nonnegative real numbers and let be a real numbers such that or , and . Assume that satisfies the system of functional inequalities
(2.29)
for all Then, there exists a unique n-ring homomorphism such that
(2.30)

for all

Proof.

It follows from Theorems 2.1, 2.2, and 2.5.

Authors’ Affiliations

(1)
Department of Mathematics, Semnan University
(2)
Department of Mathematics, Payame Noor University of Fariman Branch
(3)
Department of Mathematics, Payame Noor University of Mashhad Branch

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