Isomorphisms and Derivations in Lie C ∗-Algebras

Ulam [1] gave a talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of unsolved problems, containing the stability problem of homomorphisms. Hyers [2] proved the stability problem of additive mappings in Banach spaces. Rassias [3] provided a generalization of Hyers’ theorem which allows the Cauchy difference to be unbounded: Let f : E→ E′ be a mapping from a normed vector space E into a Banach space E′ subject to the inequality


Introduction and preliminaries
Ulam [1] gave a talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of unsolved problems, containing the stability problem of homomorphisms.Hyers [2] proved the stability problem of additive mappings in Banach spaces.Rassias [3] provided a generalization of Hyers' theorem which allows the Cauchy difference to be unbounded: Let f : E → E be a mapping from a normed vector space E into a Banach space E subject to the inequality for all x, y ∈ E, where and p are constants with > 0 and p < 1.The inequality (1.1) that was introduced by Rassias [3] provided a lot of influence in the development of a generalization of the Hyers-Ulam stability concept.This new concept is known as Hyers-Ulam-Rassias stability of functional equations.Gȃvruta [4] provided a further generalization of Th.M. Rassias' theorem.Several mathematicians have contributed works on these subjects (see [4][5][6][7][8][9][10][11][12][13][14]).Rassias [15] provided an alternative generalization of Hyers' stability theorem which allows the Cauchy difference to be unbounded, as follows.
Theorem 1.1.Let f : E → E be a mapping from a normed vector space E into a Banach space E subject to the inequality for all x, y ∈ E, where and p are constants with > 0 and 0 ≤ p < 1/2.Then the limit exists for all x ∈ E and L : E → E is the unique additive mapping which satisfies for all x ∈ E. If p < 0, then inequality (1.2) holds for x, y = 0, and (1.4) for x = 0.If p > 1/2, then inequality (1.2) holds for all x, y ∈ E, and the limit exists for all x ∈ E and A : E → E is the unique additive mapping which satisfies for all x ∈ E.
This paper is organized as follows.In Section 2, we investigate isomorphisms and derivations in C * -algebras associated with the Cauchy-Jensen functional equation.In Section 3, we investigate isomorphisms and derivations in Lie C * -algebras associated with the Cauchy-Jensen functional equation.In Section 4, we investigate isomorphisms and derivations in JC * -algebras associated with the Cauchy-Jensen functional equation.

Isomorphisms and derivations in C * -algebras
Throughout this section, assume that A is a C * -algebra with norm • A , and that B is a In this section, we investigate C * -algebra isomorphisms between C * -algebras and linear derivations on C * -algebras associated with the Cauchy-Jensen functional equation.
Theorem 2.2.Let r > 1 and θ be nonnegative real numbers, and let f : A → B be a bijective mapping such that ) 4 Abstract and Applied Analysis Letting y = −μx and z = 0, we get for all x ∈ A and all μ ∈ T 1 .Hence f (μx) = μ f (x) for all x ∈ A and all μ ∈ T 1 .By the same reasoning as in the proof of [8, Theorem 2.1], the mapping f : for all x, y ∈ A. Thus for all x ∈ A. Thus

4). Then the mapping
Proof.The proof is similar to the proof of Theorem 2.2.
Theorem 2.4.Let r > 1 and θ be nonnegative real numbers, and let f : A → A be a mapping satisfying (2.2) such that for all x, y ∈ A.Then the mapping f : A → A is a linear derivation.
Proof.By the same reasoning as in the proof of Theorem 2.2, the mapping f : It follows from (2.11) that for all x, y ∈ A. Thus the mapping f : A → A is a linear derivation.
Theorem 2.5.Let r < 1 and θ be nonnegative real numbers, and let f : A → A be a mapping satisfying (2.2) and (2.11).Then the mapping f : A → A is a linear derivation.
Proof.The proof is similar to the proofs of Theorems 2.2 and 2.4.
Theorem 2.6.Let r > 1 and θ be nonnegative real numbers, and let f : A → B be a bijective mapping satisfying (2.2) such that for all μ ∈ T and all x, y ∈ A. Then the mapping f : Proof.By the same reasoning as in the proof of Theorem 2.2, the mapping f : It follows from (2.14) that 6 Abstract and Applied Analysis for all x, y ∈ A. Thus for all x, y ∈ A.
It follows from (2.15) that for all x ∈ A. Thus for all x ∈ A. Hence the bijective mapping f : Theorem 2.7.Let r < 1 and θ be nonnegative real numbers, and let f : A → B be a bijective mapping satisfying (2.2), (2.14), and (2.15).Then the mapping f : Proof.The proof is similar to the proofs of Theorems 2.2 and 2.6.
Theorem 2.8.Let r > 1 and θ be nonnegative real numbers, and let f : A → A be a mapping satisfying (2.2) such that for all x, y ∈ A.Then the mapping f : A → A is a linear derivation.
Proof.By the same reasoning as in the proof of Theorem 2.2, the mapping f : It follows from (2.20) that for all x, y ∈ A. So for all x, y ∈ A. Thus the mapping f : A → A is a linear derivation.
Theorem 2.9.Let r < 1 and θ be nonnegative real numbers, and let f : A → A be a mapping satisfying (2.2) and (2.20).Then the mapping f : A → A is a linear derivation.
Proof.The proof is similar to the proofs of Theorems 2.2 and 2.8.

Isomorphisms and derivations in Lie C * -algebras
Throughout this section, assume that A is a Lie C * -algebra with norm for all x, y ∈ A.
In this section, we investigate Lie C * -algebra isomorphisms between Lie C * -algebras and Lie derivations on Lie C * -algebras associated with the Cauchy-Jensen functional equation.
Theorem 3.3.Let r > 1 and θ be nonnegative real numbers, and let f : A → B be a bijective mapping satisfying (2.2) such that for all x, y ∈ A. Then the mapping f : A → B is a Lie C * -algebra isomorphism.
Proof.By the same reasoning as in the proof of Theorem 2.2, the mapping f : for all x, y ∈ A. Hence the bijective mapping f : A → B is a Lie C * -algebra isomorphism, as desired.
Theorem 3.4.Let r < 1 and θ be nonnegative real numbers, and let f : A → B be a bijective mapping satisfying (2.2) and (3.3).Then the mapping f : Proof.The proof is similar to the proofs of Theorems 2.2 and 3.3.
Theorem 3.5.Let r > 1 and θ be nonnegative real numbers, and let f : for all x, y ∈ A. Then the mapping f : A → A is a Lie derivation.
Proof.By the same reasoning as in the proof of Theorem 2.2, the mapping f : It follows from (3.6) that for all x, y ∈ A. Thus the mapping f : A → A is a Lie derivation.
Theorem 3.6.Let r < 1 and θ be nonnegative real numbers, and let f : A → A be a mapping satisfying (2.2) and (3.6).Then the mapping f : Proof.The proof is similar to the proofs of Theorems 2.2 and 3.5.
Theorem 3.7.Let r > 1 and θ be nonnegative real numbers, and let f : A → B be a bijective mapping satisfying (2.2) such that for all x, y ∈ A. Then the mapping f : Proof.By the same reasoning as in the proof of Theorem 2.2, the mapping f : It follows from (3.9) that for all x, y ∈ A. Hence the bijective mapping f : A → B is a Lie C * -algebra isomorphism, as desired.
Choonkil Park et al. 9 Theorem 3.8.Let r < 1 and θ be nonnegative real numbers, and let f : A → B be a bijective mapping satisfying (2.2) and (3.9).Then the mapping f : Proof.The proof is similar to the proofs of Theorems 2.2, 2.6, and 3.7.
Theorem 3.9.Let r > 1 and θ be nonnegative real numbers, and let f : for all x, y ∈ A. Then the mapping f : Proof.By the same reasoning as in the proof of Theorem 2.2, the mapping f : It follows from (3.12) that for all x, y ∈ A. Thus the mapping f : A → A is a Lie derivation.
Theorem 3.10.Let r < 1 and θ be nonnegative real numbers, and let f : A → A be a mapping satisfying (2.2)and (3.12).Then the mapping f : Proof.The proof is similar to the proofs of Theorems 2.2, 2.8, and 3.9.

Isomorphisms and derivations in JC * -algebras
Throughout this section, assume that A is a JC * -algebra with norm • A , and that B is a JC * -algebra with norm • B .
In this section, we investigate JC * -algebra isomorphisms between JC * -algebras and Jordan derivations on JC * -algebras associated with the Cauchy-Jensen functional equation.
Theorem 4.3.Let r > 1 and θ be nonnegative real numbers, and let f : A → B be a bijective mapping satisfying (2.2) such that for all x, y ∈ A. Then the mapping f : A → B is a JC * -algebra isomorphism.
Proof.By the same reasoning as in the proof of Theorem 2.2, the mapping f : It follows from (4.3) that for all x, y ∈ A. Thus for all x, y ∈ A. Hence the bijective mapping f : A → B is a JC * -algebra isomorphism, as desired.
Theorem 4.4.Let r < 1 and θ be nonnegative real numbers, and let f : A → B be a bijective mapping satisfying (2.2) and (4.3).Then the mapping f : Proof.The proof is similar to the proofs of Theorems 2.2 and 4.3.
Theorem 4.5.Let r > 1 and θ be nonnegative real numbers, and let f : A → A be a mapping satisfying (2.2) such that for all x, y ∈ A. Then the mapping f : A → A is a Jordan derivation.
Proof.By the same reasoning as in the proof of Theorem 2.2, the mapping f : It follows from (4.6) that Choonkil Park et al. 11 for all x, y ∈ A. So for all x, y ∈ A. Thus the mapping f : A → A is a Jordan derivation.
Theorem 4.6.Let r < 1 and θ be positive real numbers, and let f : A → A be a mapping satisfying (2.2) and (4.6).Then the mapping f : A → A is a Jordan derivation.
Proof.The proof is similar to the proofs of Theorems 2.2 and 4.5.
Theorem 4.7.Let r > 1 and θ be nonnegative real numbers, and let f : A → B be a bijective mapping satisfying (2.2) such that for all x, y ∈ A. Then the mapping f : A → B is a JC * -algebra isomorphism.
Proof.By the same reasoning as in the proof of Theorem 2.2, the mapping f : It follows from (4.9) that for all x, y ∈ A. Thus for all x, y ∈ A. Hence the bijective mapping f : A → B is a JC * -algebra isomorphism, as desired.
Theorem 4.8.Let r < 1 and θ be nonnegative real numbers, and let f : A → B be a bijective mapping satisfying (2.2) and (4.9).Then the mapping f : Proof.The proof is similar to the proofs of Theorems 2.2, 2.6, and 4.7.
Theorem 4.9.Let r > 1 and θ be nonnegative real numbers, and let f : A → A be a mapping satisfying (2.2) such that for all x, y ∈ A.Then the mapping f : A → A is a Jordan derivation.
Proof.By the same reasoning as in the proof of Theorem 2.2, the mapping f :

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: • A , and that B is a Lie C * -algebra with norm • B .
C-linear mapping H : A → B is called a Lie C * -algebra isomorphism if H : A → B satisfies