Skip to main content

Approximately -Jordan Homomorphisms on Banach Algebras

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.

References

  1. Hejazian S, Mirzavaziri M, Moslehian MS: -homomorphisms. Bulletin of the Iranian Mathematical Society 2005,31(1):13–23.

    MathSciNet  MATH  Google Scholar 

  2. 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.

    MathSciNet  MATH  Google Scholar 

  3. 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

  4. 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.

    Book  MATH  Google Scholar 

  5. Eshaghi Gordji M: -Jordan homomorphisms. preprint

  6. Ulam SM: Problems in Modern Mathematics. John Wiley & Sons, New York, NY, USA; 1940.

    MATH  Google Scholar 

  7. 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.27.4.222

    MathSciNet  Article  MATH  Google Scholar 

  8. Czerwik S: Functional Equations and Inequalities in Several Variables. World Scientific, River Edge, NJ, USA; 2002:x+410.

    Book  MATH  Google Scholar 

  9. 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/S0002-9947-02-03036-2

    MathSciNet  Article  MATH  Google Scholar 

  10. 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.

    Book  MATH  Google Scholar 

  11. Jung S-M: Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor, Fla, USA; 2001:ix+256.

    MATH  Google Scholar 

  12. Rassias ThM: On the stability of functional equations and a problem of Ulam. Acta Applicandae Mathematicae 2000,62(1):23–130. 10.1023/A:1006499223572

    MathSciNet  Article  MATH  Google Scholar 

  13. 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/S0002-9939-1978-0507327-1

    MathSciNet  Article  MATH  Google Scholar 

  14. Gajda Z: On stability of additive mappings. International Journal of Mathematics and Mathematical Sciences 1991,14(3):431–434. 10.1155/S016117129100056X

    MathSciNet  Article  MATH  Google Scholar 

  15. Bourgin DG: Approximately isometric and multiplicative transformations on continuous function rings. Duke Mathematical Journal 1949, 16: 385–397. 10.1215/S0012-7094-49-01639-7

    MathSciNet  Article  MATH  Google Scholar 

  16. Badora R: On approximate derivations. Mathematical Inequalities & Applications 2006,9(1):167–173.

    MathSciNet  Article  MATH  Google Scholar 

  17. 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.2005.435

    MathSciNet  Article  MATH  Google Scholar 

  18. Eshaghi Gordji M: On approximate -ring homomorphisms and -ring derivations. preprint

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to M. Eshaghi Gordji.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Gordji, M.E., Karimi, T. & Kaboli Gharetapeh, S. Approximately -Jordan Homomorphisms on Banach Algebras. J Inequal Appl 2009, 870843 (2009). https://doi.org/10.1155/2009/870843

Download citation

  • Received:

  • Accepted:

  • Published:

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

Keywords

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