- Research Article
- Open access
- Published:
Approximately
-Jordan Homomorphisms on Banach Algebras
Journal of Inequalities and Applications volume 2009, Article number: 870843 (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.
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F870843/MediaObjects/13660_2008_Article_2023_Equ1_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F870843/MediaObjects/13660_2008_Article_2023_Equ2_HTML.gif)
for all Then, there exists a unique n-Jordan homomorphism
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F870843/MediaObjects/13660_2008_Article_2023_Equ3_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F870843/MediaObjects/13660_2008_Article_2023_Equ4_HTML.gif)
Hence, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F870843/MediaObjects/13660_2008_Article_2023_Equ5_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F870843/MediaObjects/13660_2008_Article_2023_Equ6_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F870843/MediaObjects/13660_2008_Article_2023_Equ7_HTML.gif)
Hence, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F870843/MediaObjects/13660_2008_Article_2023_Equ8_HTML.gif)
On the other hand, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F870843/MediaObjects/13660_2008_Article_2023_Equ9_HTML.gif)
It follows from (2.8) and (2.9) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F870843/MediaObjects/13660_2008_Article_2023_Equ10_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F870843/MediaObjects/13660_2008_Article_2023_Equ11_HTML.gif)
for all Then, there exists a unique n-ring homomorphism
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F870843/MediaObjects/13660_2008_Article_2023_Equ12_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F870843/MediaObjects/13660_2008_Article_2023_Equ13_HTML.gif)
Now, replacing by
in (2.13), we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F870843/MediaObjects/13660_2008_Article_2023_Equ14_HTML.gif)
By (2.13) and (2.14), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F870843/MediaObjects/13660_2008_Article_2023_Equ15_HTML.gif)
By (2.15) it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F870843/MediaObjects/13660_2008_Article_2023_Equ16_HTML.gif)
Replacing by
in (2.16), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F870843/MediaObjects/13660_2008_Article_2023_Equ17_HTML.gif)
Replacing by
in (2.17), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F870843/MediaObjects/13660_2008_Article_2023_Equ18_HTML.gif)
Hence, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F870843/MediaObjects/13660_2008_Article_2023_Equ19_HTML.gif)
By (2.19) it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F870843/MediaObjects/13660_2008_Article_2023_Equ20_HTML.gif)
Replacing by
in (2.20), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F870843/MediaObjects/13660_2008_Article_2023_Equ21_HTML.gif)
Replacing by
in (2.21), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F870843/MediaObjects/13660_2008_Article_2023_Equ22_HTML.gif)
Replacing by
in above equality to get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F870843/MediaObjects/13660_2008_Article_2023_Equ23_HTML.gif)
Replacing by
in (2.23), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F870843/MediaObjects/13660_2008_Article_2023_Equ24_HTML.gif)
Combining (2.23) by (2.24), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F870843/MediaObjects/13660_2008_Article_2023_Equ25_HTML.gif)
Replacing by
in (2.25) to obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F870843/MediaObjects/13660_2008_Article_2023_Equ26_HTML.gif)
replacing by
in (2.26), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F870843/MediaObjects/13660_2008_Article_2023_Equ27_HTML.gif)
Now, replace by
in (2.27), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F870843/MediaObjects/13660_2008_Article_2023_Equ28_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F870843/MediaObjects/13660_2008_Article_2023_Equ29_HTML.gif)
for all Then, there exists a unique n-ring homomorphism
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F870843/MediaObjects/13660_2008_Article_2023_Equ30_HTML.gif)
for all
Proof.
It follows from Theorems 2.1, 2.2, and 2.5.
References
Hejazian S, Mirzavaziri M, Moslehian MS:
-homomorphisms. Bulletin of the Iranian Mathematical Society 2005,31(1):13–23.
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.
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.
Eshaghi Gordji M: -Jordan homomorphisms. preprint
Ulam SM: Problems in Modern Mathematics. John Wiley & Sons, New York, NY, USA; 1940.
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
Czerwik S: Functional Equations and Inequalities in Several Variables. World Scientific, River Edge, NJ, USA; 2002:x+410.
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
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.
Jung S-M: Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor, Fla, USA; 2001:ix+256.
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
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
Gajda Z: On stability of additive mappings. International Journal of Mathematics and Mathematical Sciences 1991,14(3):431–434. 10.1155/S016117129100056X
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
Badora R: On approximate derivations. Mathematical Inequalities & Applications 2006,9(1):167–173.
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
Eshaghi Gordji M: On approximate -ring homomorphisms and -ring derivations. preprint
Author information
Authors and Affiliations
Corresponding author
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.
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
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2009/870843