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.

Characterization of linear mappings on (Banach) ⋆-algebras by similar properties to derivations

Let 𝓐 be a ⋆-algebra, δ : 𝓐 → 𝓐 be a linear map, and z ∈ 𝓐 be fixed. We consider the condition that δ satisfies xδ(y)⋆ + δ(x)y⋆ = δ(z) (x⋆δ(y) + δ(x)⋆y = δ(z)) whenever xy⋆ = z (x⋆y = z), and under several conditions on 𝓐, δ and z we characterize the structure of δ. In particular, we prove that if 𝓐 is a Banach ⋆-algebra, δ is a continuous linear map, and z is a left (right) separating point of 𝓐, then δ is a Jordan derivation. Our proof is based on complex variable techniques. Also, we describe a linear map δ satisfying the above conditions with z = 0 on two classes of ⋆-algebras: zero product determined algebras and standard operator algebras.

Maps Preserving k-Jordan Products on Operator Algebras

For any positive integer k, the k-Jordan product of a , b in a ring R is defined by { a , b } k = { { a , b } k − 1 , b } 1 , where { a , b } 0 = a and { a , b } 1 = a b + b a . A map f on R is k-Jordan zero-product preserving if { f ( a ) , f ( b ) } k = 0 whenever { a , b } k = 0 for a , b ∈ R ; it is strong k-Jordan product preserving if { f ( a ) , f ( b ) } k = { a , b } k for all a , b ∈ R . In this paper, strong k-Jordan product preserving nonlinear maps on general rings and k-Jordan zero-product preserving additive maps on standard operator algebras are characterized, generalizing some known results.

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)}} .

The structure of triple homomorphisms onto prime algebras

Triple homomorphisms on C*-algebras and JB*-triples have been studied in the literature. From the viewpoint of associative algebras, we characterise the structure of triple homomorphisms from an arbitrary ⋆-algebra onto a prime *-algebra. As an application, we prove that every triple homomorphism from a Banach ⋆-algebra onto a prime semisimple idempotent Banach *-algebra is continuous. The analogous results for prime C*-algebras and standard operator *-algebras on Hilbert spaces are also described.