Multiplicative Lie triple derivations on standard operator algebras

Let χ be a Banach space of dimension n > 1 and 𝒰 ⊂ ℬ(χ) be a standard operator algebra. In the present paper it is shown that if a mapping d : 𝒰 → 𝒰 (not necessarily linear) satisfies d ( [ [ U , V ] , W ] ) = [ [ d ( U ) , V ] , W ] + [ [ U , d ( V ) , W ] ] + [ [ U , V ] , d ( W ) ] d\left( {\left[ {\left[ {U,V} \right],W} \right]} \right) = \left[ {\left[ {d\left( U \right),V} \right],W} \right] + \left[ {\left[ {U,d\left( V \right),W} \right]} \right] + \left[ {\left[ {U,V} \right],d\left( W \right)} \right] for all U, V, W ∈ 𝒰, then d = ψ + τ, where ψ is an additive derivation of 𝒰 and τ : 𝒰 → 𝔽I vanishes at second commutator [[U, V ], W ] for all U, V, W ∈ 𝒰. Moreover, if d is linear and satisfies the above relation, then there exists an operator S ∈ 𝒰 and a linear mapping τ from 𝒰 into 𝔽I satisfying τ ([[U, V ], W ]) = 0 for all U, V, W ∈ 𝒰, such that d(U) = SU − US + τ (U) for all U ∈ 𝒰.

On certain equations in semiprime rings and standard operator algebras

The purpose of this paper is to prove the following result which is related to a classical result of Chernoff. Let X be a real or complex Banach space, let {\mathcal{L}(X)} be the algebra of all bounded linear operators of X into itself and let {\mathcal{A}(X)\subset\mathcal{L}(X)} be a standard operator algebra. Suppose there exist linear mappings {\mathcal{H},\mathcal{G}\colon\mathcal{A(X)}\to\mathcal{L(X)}} satisfying the relations \displaystyle\mathcal{H}(\mathcal{A}^{m+n})=\mathcal{H}(\mathcal{A}^{m})% \mathcal{A}^{n}+\mathcal{A}^{m}\mathcal{G}(\mathcal{A}^{n}), \displaystyle\mathcal{G}(\mathcal{A}^{m+n})=\mathcal{G}(\mathcal{A}^{m})% \mathcal{A}^{n}+\mathcal{A}^{m}\mathcal{H}(\mathcal{A}^{n}) for all {\mathcal{A}\in\mathcal{A(X)}} and some fixed integers {m,n\geq 1} . Then there exists {\mathcal{B}\in\mathcal{L(X)}} , such that {\mathcal{H(A)}=\mathcal{AB}-\mathcal{BA}} for all {\mathcal{A}\in\mathcal{F(X)}} , where {\mathcal{F(X)}} denotes the ideal of all finite rank operators in {\mathcal{L}(X)} , and {\mathcal{H}(\mathcal{A}^{m})=\mathcal{A}^{m}\mathcal{B}-\mathcal{B}\mathcal{A% }^{m}} for all {\mathcal{A}\in\mathcal{A(X)}} .

Nonlinear *-Lie Higher Derivations of Standard Operator Algebras

Let ℌ be an in finite-dimensional complex Hilbert space and A be a standard operator algebra on ℌ which is closed under the adjoint operation. It is shown that every nonlinear *-Lie higher derivation D = {δn}gn∈N of A is automatically an additive higher derivation on A. Moreover, D = {δn}gn∈N is an inner *-higher derivation.

On Derivations of Operator Algebras with Involution

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.

On derivations of standard operator algebras and semisimple H *-algebras

In this paper we prove the following result. Let X be a real or complex Banach 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. Suppose we have a linear mapping D : A ( X ) → L ( X ) satisfying the relation D ( A3 ) = D ( A ) A2 + AD ( A ) A + A2D ( 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 ). We apply this result, which generalizes a classical result of Chernoff, to semisimple H *-algebras.