Multiplicative Jordan triple (θ,ϕ) -derivations of rings and standard operator algebras

Ring Derivations on Standard Operator Algebras

Some remarks on derivations in semiprime rings and standard operator algebras

In this paper functional equations related to derivations on semiprime rings and standard operator algebras are investigated.We prove, for example, the following result, which is related to a classical result of Chernoff.Let X be a real or complex Banach space, let L(X) be the algebra of all bounded linear operators of X into itself and let A(X) ⊂ L(X) be a standard operator algebra.Suppose there exist linear mappings D, G : A(X) → L(X) satisfying the relationsThroughout, R will represent an associative ring with center Z(R).As usual we write [x, y] for xyyx.Given an integer n ≥ 2, a ring R is said to be n-torsion free, if for x ∈ R, nx = 0 implies x = 0. Recall that a ring R is prime, if for a, b ∈ R, aRb = (0) implies a = 0 or b = 0, and is semiprime in case aRa = (0) implies a = 0. We denote by Q s the symmetric Martindale ring of quotients.For the explanation of Q s we refer the reader to [2].Let A be an algebra over the real or complex field and let B be a subalgebra of A. A linear mapping D : B → A is called a linear derivation in case D(xy) = D(x)y + xD(y) holds for all pairs x, y ∈ R. In case we have a ring R, an additive mapping D : R → R is called a derivation, if D(xy) = D(x)y + xD(y) holds for all pairs x, y ∈ R and is called a Jordan derivation in case D(x 2 ) = D(x)x + xD(x) is fulfilled for all x ∈ R. A derivation D is inner in case there exists a ∈ R such that D(x) = [x, a] holds

The purpose of this paper is to establish some results concerning generalized left derivations in rings and Banach algebras. In fact, we prove the following results: Let R be a 2-torsion free semiprime ring, and let \({G: R \longrightarrow R}\) be a generalized Jordan left derivation with associated Jordan left derivation \({\delta: R \longrightarrow R}\). Then every generalized Jordan left derivation is a generalized left derivation on R. This result gives an affirmative answer to the question posed as a remark in Ashraf and Ali (Bull. Korean Math. Soc. 45:253–261, 2008). Also, the study of generalized left derivation has been made which acts as a homomorphism or as an anti-homomorphism on some appropriate subset of the ring R. Further, we introduce the notion of generalized left bi-derivation and prove that if a prime ring R admits a generalized left bi-derivation G with associated left bi-derivation B then either R is commutative or G is a right bi-centralizer (or bi-multiplier) on R. Finally, it is shown that every generalized Jordan left derivation on a semisimple Banach algebra is continuous.

In this paper, we shall show the following two results: (1) Let A be a standard operator algebra with I, if (D is a linear mapping on A which satisfies that 4>(T) maps ker T into ranT for all T E A, then (D is of the form P(T) = TA + BT for some A, B in B(X). (2) Let X be a Hilbert space, if b is a norm-continuous linear mapping on B(X) which satisfies that 4>(P) maps ker P into ran P for all self-adjoint projection P in then (D is of the form P(T) = TA + BT for some A, B in B(X). In what follows X stands for a Banach space (or Hilbert space) and X* for its norm dual. We denote by (x, f) the duality pairing between elements E X* and E X, and we use the symbols B(X), L(X), F(X), I and x 0 f to denote the set of all linear bounded operators on X, the set of all linear mappings on X, the set of all finite rank operators on X, the identity operator and the rank one operator (*, f)x on X, respectively. If A is a Banach algebra, and A1 is a Banach subalgebra of A, we say that a linear mapping 4D: A1 -A is a derivation if (ab) = (a)b + a4(b) for any a and b in A1. The derivation 1D is called inner if there exists an element a in A such that 4(b) = ba ab for any b in A1. We say that a linear mapping 4D: A1 -A is a local derivation if for every a in A1, there exists a derivation &a: A1 -A, depending on a, such that 4(a) = ba(a). A linear mapping 4i is called a Jordan derivation if (a 2) = a4(a) + a4(a) for every a in A1. We give the notion of bilocal derivation as follows: Definition 1. If A is a Banach subalgebra of a linear mapping pp: A B(X) is called a bilocal derivation if for every T in A and u in X, there exists a derivation 6T,u A -B(X), depending on T and u, such that 4D(T)u = 5T,U(T)U. Definition 2. Let X be a Banach space, a Banach subalgebra A of B(X) is called a standard operator algebra if A contains F(X). D. R. Larson and A. R. Sourour [5] have proved that every local derivation on B(X) is a derivation. R. Kadison [4] and M. Bresar [1] have discussed normcontinuous local derivations on von Neumann algebras. It is obvious that every Received by the editors June 14, 1995 and, in revised form, November 8, 1995. 1991 Mathematics Subject Classification. Primary 47D30, 47D25, 47B47.

On Lie-Type Derivations of Rings and Standard Operator Algebras by Local Actions

On commutators and derivations in rings

Let X be a complex Hilbert space, let L(X) be the algebra of all bounded linear operators on X, and let A(X) ⊂ L(X) be a standard operator algebra, which is closed under the adjoint operation. Suppose there exists a linear mapping D: A(X) → L(X) satisfying the relation D(AA*A) = D(A) A*A + AD(A*)A + AA*D(A), for all A ∈ A(X). In this case D is of the form D(A) = AB-BA, for all A∈ A(X) and some B ∈ L(X), which means that D is a derivation. We apply this result to semisimple H*-algebras.

The purpose of this paper is to prove the following result. Let X be a complex Hilbert space, let L(X) be an algebra of all bounded linear operators on X and let A(X) ⊂ L(X) be a standard operator algebra, which is closed under the adjoint operation. Suppose there exists a linear mapping D : A(X) → L(X) satisfying the relation 2D(AA*A) = D(AA*)A + AA*D(A) + D(A)A*A + AD(A*A) for all A ∈ A(X). In this case, D is of the form D(A) = [A,B] for all A ∈ A(X) and some fixed B ∈ L(X), which means that D is a derivation.

We present a new simple proof of the structural result for elementary operators on standard operator algebras. Using the idea of this short proof we can characterize surjective maps between standard operator algebras having a certain multiplicativity-like property appearing in the abstract definition of elementary operators of length one. In particular, we show that such maps are automatically additive.

Let be a standard operator algebra on an infinite dimensional complex Hilbert space containing identity operator I. In this paper it is shown that if is closed under the adjoint operation, then every multiplicative ∗-Lie triple derivation is a linear ∗-derivation. Moreover, if there exists an operator S ∈ such that S + S∗ = 0 then d(U) = U S − SU for all U ∈ , that is, d is inner. Furthermore, it is also shown that any multiplicative ∗-Lie triple higher derivation D = {δn}n∈ℕ of is automatically a linear inner higher derivation on with d(U)∗ = d(U∗).

In this paper we prove the following result. LetXbe a real or complex Banach space, letL(X) be the algebra of all bounded linear operators onX, and letA(X) ⊂L(X) be a standard operator algebra. Suppose we have a linear mappingD:A(X) →L(X) satisfying the relationD(A3) =D(A)A2+AD(A)A+A2D(A), for allA∈A(X). In this caseDis of the formD(A) =AB−BA, for allA∈A(X) and someB∈L(X). We apply this result, which generalizes a classical result of Chernoff, to semisimpleH*-algebras.

Let be a standard operator algebra on an infinite dimensional complex Hilbert space containing identity operator I. Let be the polynomial defined by n indeterminates and their multiple *-Lie products and be the set of non-negative integers. In this paper, it is shown that if is closed under the adjoint operation and is the family of mappings such that the identity map on satisfying for all and for each , then is an additive *-higher derivation. Moreover, is inner.

Let ℋ be an infinite dimensional complex Hilbert space and 𝒜 be a standard operator algebra on ℋ which is closed under the adjoint operation. We prove that every nonlinear *-Lie derivation δ of 𝒜 is automatically linear. Moreover, δ is an inner *-derivation.

Let H H be a real or complex Hilbert space, dim ⁡ H > 1 \dim H > 1 , and B ( H ) \mathcal {B}(H) the algebra of all bounded linear operators on H H . Assume that A \mathcal {A} is a standard operator algebra on H H . Then every additive Jordan ∗ {\ast } -derivation J : A → B ( H ) J:\mathcal {A} \to \mathcal {B}(H) is of the form J ( A ) = A T − T A ∗ J(A) = AT - T{A^{\ast }} for some T ∈ B ( H ) T \in \mathcal {B}(H) .