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.

Jordan *-Derivations of Finite-Dimensional Semiprime Algebras

In this paper, we characterize Jordan *-derivations of a 2-torsion free, finite-dimensional semiprime algebra R with involution *. To be precise, we prove the following. Let δ : R → R be a Jordan *-derivation. Then there exists a *-algebra decomposition R = U ⊕ V such that both U and V are invariant under δ. Moreover, * is the identity map of U and δ|U is a derivation, and the Jordan *-derivation δ|V is inner. We also prove the following. Let R be a noncommutative, centrally closed prime algebra with involution *, char R ≠ 2, and let δ be a nonzero Jordan *-derivation of R. If δ is an elementary operator of R, then dimCR < ∞ and δ is inner.

Centralizers and Vanishing Derivations on Multi-linear Polynomials in Prime Rings

Let R be a prime algebra over a commutative ring K with characteristic not equal to 2. Let d and δ be non-zero derivations of R, f(x1,…, xn) a multi-linear polynomial over K with n non-commuting variables, and m ≥ 1 a fixed integer. We prove that if δ (d(f(r1,…, rn))m) = 0 for any r1,…, rn ∈ R, then either f(x1,…, xn) is central valued on R or R satisfies the standard identity s4.

Generalized Derivations with Power-Central Values on Multilinear Polynomials

Let R be a prime algebra over a commutative ring K, Z and C the center and extended centroid of R, respectively, g a generalized derivation of R, and f (X1, …,Xt) a multilinear polynomial over K. If g(f (X1, …,Xt))n ∈ Z for all x1, …, xt ∈ R, then either there exists an element λ ∈ C such that g(x)= λx for all x ∈ R or f(x1, …,xt) is central-valued on R except when R satisfies s4, the standard identity in four variables.

On surjective linear maps preserving commutativity

We describe surjective linear maps preserving commutativity from (symmetric elements of) any algebra (with involution) onto (symmetric elements of) a prime algebra (with involution) not satisfying polynomial identities of low degree. Bijective commutativity preservers on skew elements of centrally closed prime algebras with involution of the first kind are also investigated.

A Result on Derivations with Algebraic Values

Let R be a prime algebra over a field F and let d be a non-zero derivation in R such that for every x ∊ R, d(x) is algebraic over F of bounded degree. Then R is a primitive ring with a minimal right ideal eR, where e2 = e and eRe is a finite dimensional central division algebra.