JORDAN AUTOMORPHISMS, JORDAN DERIVATIONS OF GENERALIZED TRIANGULAR MATRIX ALGEBRA

We investigate Jordan automorphisms and Jordan derivations of a class of algebras called generalized triangular matrix algebras. We prove that any Jordan automorphism on such an algebra is either an automorphism or an antiautomorphism and any Jordan derivation on such an algebra is a derivation.


Introduction
Throughout this paper, let R be a 2-torsion-free commutative ring with identity 1.Consider an associative algebra A over R, then A can be viewed as a Jordan algebra with the usual product x • y = (1/2)(xy + yx).An R-linear map δ : A → A is called a derivation (resp., Jordan derivation) of A if An R-linear map θ : A → A is said to be a Jordan homomorphism of A if or, equivalently, θ a 2 = θ(a) 2 , ∀a ∈ A. (1.2) Derivations, Jordan derivations, as well as automorphisms and Jordan automorphisms of the algebra of triangular matrices and some class of their subalgebras have been the object of active research for a long time [1,2,5,6,9,10].
A well-know result of Herstein [11] states that every Jordan isomorphism on a prime ring of characteristic different from 2 is either an isomorphism or an anti-isomorphism.We remark that the situation where the rings are semiprime rings does not hold.In the same time, he showed that every Jordan derivation on a prime ring of characteristic different from 2 is a derivation [12].A brief proof of this result can be found in [4].This result is extended by [3,8] to the semiprime case.
Let now ᐁ be the algebra of the form where A and B are unital R-algebras and M is an (A,B)-bimodule.This algebra ᐁ, endowed with the usual formal matrix addition and multiplication, will be called a generalized triangular matrix algebra.Many widely studied algebras, including upper-triangular matrix algebras, block-triangular matrix algebras, nest algebras, semi-nest algebras, and triangular Banach algebras, may be viewed as triangular algebras.Khazal et al. [13] discuss the automorphism group of ᐁ so that A and B have only trivial idempotents.Cheung [7] gives sufficient conditions under which every Lie derivation is a sum of derivation on ᐁ and a mapping from ᐁ to its center.
In this paper, we consider linear operators on a class of algebras of the form ᐁ; specifically, Jordan derivations and Jordan automorphisms.M is assumed to be faithful as a left A-module as well as a right B-module.We will prove that if both A and B have only trivial idempotents, any Jordan automorphism of the ring ᐁ is either an automorphism or an antiautomorphism, and we will prove that any Jordan derivation of such an algebra ᐁ is a derivation of ᐁ.

The Jordan automorphism of generalized triangular matrix algebra
In this section, we suppose that ᐁ is the algebra of the form where A and B are unital R-algebras and M is an (A,B)-bimodule, both A and B have only trivial idempotents.This section is devoted to prove the following result.
Theorem 2.1.If M is faithful as a left A-module as well as a right B-module and if both A and B have only trivial idempotents, then any Jordan automorphism θ of ᐁ is either an automorphism or an antiautomorphism.
We now introduce the notations E x = ( 1 x 0 ) , F x = ( 0 x 1 ), and X = ( 0 x 0 ) , for some Then it is easy to check the following relations: On the other hand, if θ is a Jordan automorphism of ᐁ, then either θ(E 0 ) = E u or θ(E 0 ) = F u for some u ∈ M, since E 0 is an idempotent.
The proof of Theorem 2.1 is an immediate consequence of the following two lemmas.
By applying θ 1 to aE 0 = E 0 • (aE 0 ) and bF 0 = F 0 • (bF 0 ) for a ∈ A and b ∈ B, we get that θ 1 (aE 0 ) = ϕ A (a)E 0 and θ 1 (bF 0 ) = ϕ B (b)F 0 , where ϕ A : A → A and ϕ B : B → B are additive and bijective maps. 2 , that is ϕ A and ϕ B are Jordan automorphisms of A and B, respectively. Applying Applying again θ 1 to ( 0 ax 0 ) = 2(aE 0 ) • X and ( where ᐁ ∈ M, and ϕ A : As a consequence, the following two identities are valid for all a 1 ,a 2 ∈ A and x ∈ M: Since M is faithful, we have ϕ A (a 1 a 2 ) = ϕ A (a 1 )ϕ A (a 2 ), which means that ϕ A is an automorphism of A. Similarly, ϕ B is an automorphism of B. Finally, in view of these arguments, one can easily check that θ is an automorphism of ᐁ, which concludes the proof of the lemma.Lemma 2.4.Assume that θ(E 0 ) = F u for some u ∈ M, then θ is an antiautomorphism of ᐁ.

Proof. By hypothesis, we have θ(F
Applying θ to aE 0 = E 0 • (aE 0 ) and bF 0 = F 0 • (bF 0 ) for a ∈ A and b ∈ B, we have where ϕ 1 : A → B and ϕ 2 : B → A are obviously additive and bijective maps.In addition, by application of θ to (a 2 E 0 ) = (aE 0 ) 2 and (b 2 F 0 ) = (bF 0 ) 2 for a ∈ A and b ∈ B, we can show by simple calculus that ϕ 1 and ϕ 2 are Jordan isomorphisms.
Applying now θ to It follows that where u ∈ M and ϕ 2 : . Hence, we have the following two identities: which shows that ϕ 1 is an anti-isomorphism from A onto B, since M is faithful.It is proved analogously that ϕ 2 is an anti-isomorphism from B onto A. Finally, the preceding arguments allows us to get by simple calculus that θ is an antiautomorphism of ᐁ.This completes the proof of the lemma.

The Jordan derivations of generalized triangular matrix algebra
In this section, we suppose that ᐁ is the algebra of the form where A and B are unital R-algebras and M is an (A,B)-bimodule.The first principal result of this paper is the following.
Theorem 3.1.If M is faithful as a left A-module as well as a right B-module, then any Jordan derivation of ᐁ is an ordinary derivation.
Before proving this theorem, we need to describe all Jordan derivations of ᐁ.
where u ∈ M and g A : where g A : , and k B : M → B are clearly linear maps.Take X = ( 1 0 0 ) and Y = ( 0 x 0 ) in the equation Hence, k A (x) = 0 and k B (x) = 0, which allow us to have Putting now X = ( a 0 0 ) and Y = ( a 0 0 ), we obtain , that is, g A is a Jordan derivation of A, and h A (a • a ) = 0. Replacing a by 1 in the last relation yields h A (a) = 0. Therefore,

∂ a x b
showing that ).We continue with the same method by taking X = ( a 0 b ) and Now, if we take X = ( a 0 0 ) and Y = ( 1 x 0 ), we find that deducing the identity f (ax where u ∈ M, g A : A → A, g B : B → B, and f : M → M are linear maps satisfying that (i) g A is a Jordan derivation of A, f (ax Hence, we have the following two identities:

Mathematical Problems in Engineering
Special Issue on Time-Dependent Billiards

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.

. 9 )
which means that g B is a Jordan derivation of B, and h B (bb + b b) = 0. Substituting now b = 1 in the latter identity implies h B (b) = 0. Consequently, ∂( a x b ) = ( gA(a) fA(a)+fB(b)+ f (x) gB(b) .13) A. H. A. Driss and B. Y. l'Moufadal 2131 Hence, f (xb) = xg B (b) + f (x)b, which ends the proof of the lemma.Now we are ready to establish our first principal theorem.Proof of Theorem 3.1.Let ∂ be a Jordan derivation of ᐁ, we have ∂ a x b = g A (a) au − ub + f (x) g B (b) , (3.14) .15)As a consequence, we get thatg A (aa )x = g A (a)a x + ag A (a )x.(3.16)Since M is faithful, g A (aa ) = g A (a)a + ag A (a ) and g A is a derivation of A. A similar argument shows that g B is a derivation of B.Finally, one can now easily check that ∂ is a derivation of ᐁ.Indeed, let X = ( a x b ) and Y = ( a x b ) be arbitrary elements in ᐁ.By straightforward computations, we have∂(XY) = g A (aa ) aa u − ubb + f (ax + xb ) g B (bb ) , ∂(X)Y + X∂(Y )= g A (a)a +ag A (a ) aa u − ubb +g A (a)x + a f (x )+xg B (b )+ f (x)b g B (b)b +bg B (b ).(3.17)This shows that ∂(XY) = ∂(X)Y + X∂(Y ) and concludes the proof of the theorem.