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ISSN: 1736-4337
Journal of Generalized Lie Theory and Applications
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On The Stability of Conditional Homomorphisms in Lie C*-algebras

Eshaghi M1,2, Abbaszadeh S1* and Manuel De la Sen3

1Department of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, Semnan 35195-363, Iran

2Center of Excellence in Nonlinear Analysis and Applications, Semnan University, Semnan, Iran

3Department of Electricity and Electronics, University of the Basque Country, Spain

Corresponding Author:
Abbaszadeh S
Department of Mathematics
Faculty of Mathematics, Statistics and Computer Sciences
Semnan University, Iran
Tel: 982333366296
E-mail: [email protected]

Received date March 24, 2015; Accepted date June 27, 2015; Published date June 30, 2015

Citation: Eshaghi M, Abbaszadeh S, Sen MD (2015) On The Stability of Conditional Homomorphisms in Lie C*-algebras. J Generalized Lie Theory Appl 9: 220. doi:10.4172/1736-4337.1000220

Copyright: © 2015 Eshaghi M, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

 

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Abstract

During the last years several papers studying conditional functional equations have appeared. They mostly deal with equations satisfied on some restricted domain and many among them concern equations postulated for orthogonal vectors. In this paper, we define the conditional homomorphisms with the predecessor defined by γ (x)=γ (y) with an even mapping γ. Then, using a fixed point theorem, we investigate the stability of the conditional homomorphisms in Lie C* -algebras.

Keywords

Conditional homomorphism; Contractively sub homogeneous; Expansively super homogeneous; Fixed point theorem; Stability

Introduction

The problem of solving equations on spheres is an important part of computational aspects, ordinary differential equations and partial differential equations [1-3]. The development of computational methods for solving partial differential equations on the sphere is complicated by problems that result from the spherical coordinate system itself. However, some of the numerical methods for solving vector differential equations are applicable to any vector differential equation on the sphere.

The functional equation (ξ) is stable if any function f satisfying the equation (ξ) approximately is near to the true solution of (ξ). An interested reader can find more information on such problems with the emphasis on functional equations in [4-8].

Let X and Y are two Banach spaces. Consider f : X →Y to be a mapping such that f (tx) is continuous in imagefor each fixed x ∈ X. Assume that there exist constants ε ≥ 0 and p ∈ [0, 1) such that

image

For all x, y ∈ X, Rassias [6] showed that there exists a unique image -linear mapping T : X →Y such that

image

For all x ∈ X, A generalization of the theorem of Rassias was obtained by Gavruta [9] by replacing the unbounded Cauchy difference by a general control function image in the spirit of Rassias approach.

In 1994, Alsina and Garcia-Roig [10] solved the conditional equation

f (x + y)=f (x) + f (y) (1.1)

whenever ∥x∥=∥y∥ for continuous mapping f from a real inner product space (X, (.|.)) with dim X ≥ 2 into a real topological linear space Y. They recognized the connection between this equation and the orthogonally additive functional equation. They also obtained the linearity of such a function f in the case where Y=Rn.

Sikorska [11] studied a generalized stability of Cauchy and Jensen functional equations, where the respective Cauchy or Jensen differences are approximated by arbitrary functions. Moreover, [12] Sikorska solved the conditional Pexider functional equation on prescribed sets being generalizations of spheres. We refer the reader to [13-17] for some interesting results on the stability of conditional functional equations.

In this paper, we apply a fixed point theorem to prove the stability of conditional homomorphisms using contractively subhomogeneous and expansively superhomogeneous functions.

The paper is organized as follows. In Section 2, some necessary preliminaries and summarization of some previous known results are presented. The concepts of conditional homomorphism and conditional Jordan homomorphism in Lie C*-algebras are introduced. In Section 3, we deal with the stability of conditional homomorphisms and conditional Jordan homomorphisms in Lie C*-algebras. Finally, a conclusion is given in Section 4.

Preliminaries

Ger and Sikorska [18] solved the conditional functional equation (1.1) with the norm replaced by an abstract function fulfilling suitable conditions.

Theorem 2.1: [18] Let X be a real linear space with dim X ≥ 2, (Y, +) be an abelian group, Z be a given nonempty set and let γ : X →Z be an even mapping γ(−x) = γ(x),x∈X such that

(c1) for any two linearly independent vectors x, y ∈ X there exist linearly independent vectors u, v ∈ Lin{x, y} such that γ (u+v)=γ (u−v),

(c2) if x, y ∈ X, γ (x + y)=γ (x − y), then γ (αx + y)=γ (αx−y) for all image,

(c3) for all x ∈ X and image there exists a y ∈ X such that γ (x+y)=γ (x−y) and γ ((λ+1) x)=γ ((λ−1) x−2y).

If f : X →Y satisfies the condition

γ (x)=γ (y) implies f (x + y)=f (x) + f (y), x, y ∈ X ,

Then f is additive.

They dealt also with the Hyers-Ulam stability problem cf. [4,5] for such a more general version of (1.1). In order to make the above assumptions more readable, they gave the following example.

Example 2.2: Let (X, (.|.)) be a real inner product space with dim X ≥ 2, image and γ (x)=∥x∥, x ∈ X. Then function γ satisfies (c1) − (c3).

A C*-algebra A endowed with the Lie product [x, y]=xy −yx on A is called a Lie C*-algebra. The stability problems of functional equations between C*-algebras have been investigated by a number of authors [19-22].

Definition 2.3: A image -linear mapping H of a Lie C*-algebra A to a Lie C*-algebra B is called a conditional homomorphism if

H ([x, y])=[H (x), H (y)] (2.1)

Holds for all x, y ∈ A with γ (x)=γ (y).

Definition 2.4: A image -linear mapping H : A→ B is said to be a conditional Jordan homomorphism if

H ([x, y] + [y, x])=[H (x), H (y)] + [H (y), H (x)] (2.2)

For all x, y ∈ A with γ (x)=γ (y).

For explicitly later use, we state the following theorem.

Theorem 2.5: (Banach) Let (X, d) be a complete metric space and consider a mapping Λ : X → X as a strictly contractive mapping, that is

d (Λx, Λy) ≤ Ld(x, y)

For all x, y ∈ X and for some (Lipschitz constant) 0<L<1. Then there exists a unique a ∈ X such that Λa=a. Moreover, for each x ∈ X,

image

and in fact for each x ∈ X,

image

Let A, B be real vector spaces. We recall that if there exists a constant L with 0<L<1 such that a n−times mapping image satisfies

image

For all x, xj ∈ A (1≤ j ≠ i ≤ n ) and all positive integers λ, then we say that ρ is n-contractively sub homogeneous if ℓ=-1, and ρ is n-expansively superhomogeneous if l=−1. It follows by the above inequality that ρ satisfies the following properties:

image image

For all x, xj ∈ A (1≤ j ≠ i ≤ n ) and all positive integers λ.

Remark 2.6: If ρ is n-contractively sub-additive then ρ is contractively sub homogeneous of degree n and if ρ is n-expansively super-additive, then ρ is and expansively super homogeneous of degree n.

The Main Results

Throughout this section, let A and B be Lie C*-algebras, dim A ≥ 2 and γ be an even mapping from A to a nonempty set Z, satisfying the conditions (c1) − (c3) and for all x, y ∈ A, γ (x)=γ (y) implies that image where image. Let image

Let image be either 3-expansively super homogeneous mappings for ℓ=−1 or 3-contractively sub homogeneous mappings for ℓ=1, with constant image. Let imagebe either 2- expansively super homogeneous mappings for ℓ=−1 or 2 -contractively sub homogeneous mappings for ℓ=1, with constant 0<L=L (ℓ)<1.

Lemma 3.1: Let image be an additive function such that f (tx)=t f (x) for all image and x ∈ A. Then the function f is image -linear [23].

Theorem 3.2: Suppose that image (i=1, 2, 3) are mappings fulfilling

image (3.1) image (3.2)

For all image and x, y, z ∈ A with γ (x)=γ (y). If f3 is an odd mapping and fi (0)=0 (i=1, 2, 3), then there exists a unique conditional homomorphism image such that

image image(3.3)

 

image

Where the mapping imageis defined by

image(3.4)

 

For all x ∈ A and some y0 ∈ A for which x and y0 satisfy the condition (c3) with λ=1.

Proof: We define

image

Define d: ε × ε −→ [0, ∞] by

image

It is easy to see that (ε, d) is a complete metric space. Let us consider the linear mapping Λ : ε →ε defined by

image

For all x ∈ A. Let f , g ∈ε and let C ∈ [0, ∞) be an arbitrary constant with d (f, g)<C. From the definition of d, we have

image

For all x, y ∈ A with γ (x)=γ (y). By the assumption and the last inequality, we get

image

For all x, y ∈ A with γ (x)=γ (y) and therefore

d(Λf ,Λg) ≤ Ld( f , g).

So, Λ is a strictly contractive mapping with the Lipschitz constant L.

From (3.1) we have

image (3.5) image (3.6)

For all x ∈ A

Let x ∈ A. By (c3) for λ=1, there exists y0 ∈ A such that γ (x + y0)=γ (x − y0) and γ (2x)=γ (2y0).

Hence

image (3.7) image (3.8) image (3.9)

Also, by the evenness of γ we have

image (3.10)

Substituting both x, y in (3.1) by x + y0 and x − y0, we have

image (3.11) image (3.12)

It follows from the inequalities (3.5) − (3.12) and the triangle inequality that

image image image image image image image(3.13) image image image image image image

If we replace x and y0 in (3.13) with image and image

respectively, and divide by 2 the resulting inequality, then we have

image(3.14)

If we replace x and y0 in (3.13) with image and image, respectively, and using 3-expansivity of φ, then

image(3.15)

We can reduce (3.14) and (3.15) to

image

and then

image

That is, image By Theorem 2.5, there exists a unique mapping ∈ which is the fixed point of

Λ and satisfies

image

Also

image

This yield

image(3.16)

Let x ∈ A and y0 ∈ A be the element for which x and y0 satisfy the condition (c3) with λ=1. Utilizing the triangle inequality, the oddness of f3 and the inequalities (3.5) and (3.6), we infer that

image image(3.17) image image image(3.18) image

Combining (3.17) and (3.18) with (3.16), we get

image image

Using the above inequalities and the fact that H (2x)=2H (x), we obtain

image(3.19) image(3.20)

From the inequalities (3.17) and (3.18), we get

image image image image

For all image , whence

image(3.21)

Let x, y ∈ A with γ (x)=γ (y). By the assumption, image and then image for all n ∈ image . By 3-contractivity of φ we infer that

image

And by 3-expansivity of φ we conclude that

image

Since image, from n → ∞ in the above inequalities, one proves by (3.21) that H (x + y) − H (x) − H (y)=0. Hence H is conditionally additive.

In addition, it is clear from (3.1) that the following inequality

image image image

Holds for all x ∈ A and image. By Lemma 3.1, H is image -linear.

We claim that the mapping H satisfies the functional equation (2.1). Define image by r (x, y) =f1 ([x, y]) − [f2 (x), f3 (y)] for all x, y ∈ A with γ (x)=γ (y). From (3.2) it follows that

image(3.22)

Making use of (3.21) and (3.22), we get

image(3.23)

For all x, y ∈ A with γ (x)=γ (y).

Now, letting x, y ∈ A with γ (x)=γ (y) and image , by (3.23) and conditional additivity of H one obtains

image

This yield

image(3.24)

The comparison of the above equality with (3.23) shows that

image

 

For all x, y ∈ A with γ (x)=γ (y) and image . Taking the limit as n → ∞, we conclude that H ([x, y])=[H (x), H (y)]. This completes the proof of Theorem 3.2.

In particular, given image and image for ε, θ ≥ 0 and some real numbers p in the main theorem, one gets the following corollary.

Corollary 3.3: Let : → i f image be mappings satisfying

image image

For all image and x, y, z ∈ A with ∥x∥=∥y∥, ε, θ ≥ 0 and real numbers p such that p<1 for ℓ=1 and p>1 for ℓ=−1. If f3 is an odd mapping and fi (0)=0 (i=1, 2, 3), then there exists a unique conditional homomorphism image such that

image

For all x ∈ A and some y0 ∈ A for which x and y0 satisfy the condition (c3) with λ=1.

Proof: Let γ:=∥.∥ (cf. Example 2.2). Define φ and ϕ as above and apply Theorem 3.2 with L=2ℓ(p−1).

In the next theorem, we prove the Hyers-Ulam stability problem for conditional Jordan homomorphisms.

Theorem 3.4: Suppose that : → i f image are mappings satisfying the functional inequalities

image image

For all image and x, y, z ∈ A with γ (x)=γ (y). If f3 is an odd mapping and fi (0)=0 (i=1, 2, 3), then there exists a unique conditional Jordan homomorphism imagesatisfying (3.3).

Proof: Applying a similar argument to the corresponding part of Theorem 3.2, we can conclude that there exists a unique conditional image -linear mapping image satisfying (3.3). Moreover,

image(3.27)

For all x ∈ A. We will show that H satisfies the functional equation (2.2). Define image by image for all x, y ∈ A with γ(x)=γ(y). It follows from (3.26) that

image(3.28)

By (3.27) and (3.28), we obtain

image(3.29)

For all x, y ∈ A with γ (x)=γ (y).

Now let x, y ∈ A with γ (x)=γ (y) and image be fixed. By (3.29) and conditional additivity of H, it can be shown that

image

and then

image

Sending n to infinity, we obtain

image

This completes the proof of the theorem.

Corollary 3.5: Let : → i f image be mappings satisfying

image image

For all image and x, y, z ∈ A with ∥x∥=∥y∥, ε, θ ≥ 0 and real numbers p such that p<1 for ℓ=1 and p > 1 for ℓ=−1. If f3 is an odd mapping and fi (0)=0 (i=1, 2, 3), then there exists a unique conditional Jordan homomorphismimagesatisfying (3.25).

Proof: Define image, image and image apply Theorem 3.4 with L=2ℓ(p−1).

Acknowledgement

The authors are very grateful to the Spanish Government for its support of this research through Grant DPI2012-30651.

References

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