2-Local derivations of real AW*-algebras are derivation

Abstract2-Local derivations on real matrix algebras over unital semi-prime Banach algebras are considered. Using the real analogue of the result that any 2-local derivation on the algebra $$M_{2^n}(A)$$ M 2 n ( A ) ($$n\ge 2$$ n ≥ 2 ) is a derivation, it is shown that any 2-local derivation on real AW$$^*$$ ∗ -algebra for which the enveloping algebra is (complex) AW*-algebra, is a derivation, where A is a unital semi-prime Banach algebra with the inner derivation property.

Derivations of matrix semirings

It is well known that if [Formula: see text] is a derivation in semiring [Formula: see text], then in the semiring [Formula: see text] of [Formula: see text] matrices over [Formula: see text], the map [Formula: see text] such that [Formula: see text] for any matrix [Formula: see text] is a derivation. These derivations are used in matrix calculus, differential equations, statistics, physics and engineering and are called hereditary derivations. On the other hand (in sense of [Basic Algebra II (W. H. Freeman & Company, 1989)]) [Formula: see text]-derivation in matrix semiring [Formula: see text] is a [Formula: see text]-linear map [Formula: see text] such that [Formula: see text], where [Formula: see text]. We prove that if [Formula: see text] is a commutative additively idempotent semiring any [Formula: see text]-derivation is a hereditary derivation. Moreover, for an arbitrary derivation [Formula: see text] the derivation [Formula: see text] in [Formula: see text] is of a special type, called inner derivation (in additively, idempotent semiring). In the last section of the paper for a noncommutative semiring [Formula: see text] a concept of left (right) Ore elements in [Formula: see text] is introduced. Then we extend the center [Formula: see text] to the semiring LO[Formula: see text] of left Ore elements or to the semiring RO[Formula: see text] of right Ore elements in [Formula: see text]. We construct left (right) derivations in these semirings and generalize the result from the commutative case.

Generalised Inner Derivations in Semi Prime Rings

Let A be any ring and f(xy) = f(x)y+xha(y), where f be any generalised inner derivation(G.I.D ) a be the fixed element of A. In this paper, it is shown that (i) ha must necessarily be a derivation for semi prime ring A. (ii) ∃ no generalized inner derivations f : A → A such that f(x ◦ y) = x ◦ y or f(x ◦ y) + x ◦ y = 0 ∀ x,y ∈ A, We have proved Havala [2] def. p.1147, Herstein [3] Lemma 3.1 p. 1106 as corollaries, along with other results.

Derivations of polynomial semirings

The aim of this paper is the investigation of derivations in semiring of polynomials over idempotent semiring. For semiring [Formula: see text], where [Formula: see text] is a commutative idempotent semiring we construct derivations corresponding to the polynomials from the principal ideal [Formula: see text] and prove that the set of these derivations is a non-commutative idempotent semiring closed under the Jordan product of derivations — Theorem 3.3. We introduce generalized inner derivations defined as derivations acting only over the coefficients of the polynomial and consider [Formula: see text]-derivations in classical sense of Jacobson. In the main result, Theorem 5.3, we show that any derivation in [Formula: see text] can be represented as a sum of a generalized inner derivation and an [Formula: see text]-derivation.

Multiplicative δ-derivation in alternative algebras

Every multiplicative [Formula: see text]-derivation of an alternative algebra [Formula: see text] is additive if there exists an idempotent [Formula: see text] in [Formula: see text] satisfying the following conditions: (i) [Formula: see text] implies [Formula: see text]; (ii) [Formula: see text] implies [Formula: see text]; (iii) [Formula: see text] implies [Formula: see text] for [Formula: see text]. In particular, every [Formula: see text]-derivation of a prime alternative algebra with a nontrivial idempotent is additive. This generalizes the known result obtained by Rodrigues, Guzzo and Ferreira for [Formula: see text]-derivations. As an application, we apply multiplicative [Formula: see text]-derivation to an alternative complex algebra [Formula: see text] of all [Formula: see text] complex matrices to see how it decomposes into a sum of [Formula: see text]-inner derivation and a [Formula: see text]-derivation on [Formula: see text] given by an additive derivation [Formula: see text] on [Formula: see text].

Double Derivations of n-Lie Superalgebras

Let 𝔤 be an n-Lie superalgebra. We study the double derivation algebra [Formula: see text] and describe the relation between [Formula: see text] and the usual derivation Lie superalgebra Der(𝔤). We show that the set [Formula: see text] of all double derivations is a subalgebra of the general linear Lie superalgebra gl(𝔤) and the inner derivation algebra ad(𝔤) is an ideal of [Formula: see text]. We also show that if 𝔤 is a perfect n-Lie superalgebra with certain constraints on the base field, then the centralizer of ad(𝔤) in [Formula: see text] is trivial. Finally, we give that for every perfect n-Lie superalgebra 𝔤, the triple derivations of the derivation algebra Der(𝔤) are exactly the derivations of Der(𝔤).

Local and 2-local derivations on noncommutative Arens algebras

The paper is devoted to so-called local and 2-local derivations on the noncommutative Arens algebra L ω(M,τ) associated with a von Neumann algebra M and a faithful normal semi-finite trace τ. We prove that every 2-local derivation on L ω(M,τ) is a spatial derivation, and if M is a finite von Neumann algebra, then each local derivation on L ω(M,τ) is also a spatial derivation and every 2-local derivation on M is in fact an inner derivation.

On the eigenvector algebra of the product of elements with commutator one in the first Weyl algebra

Let A1 = K〈X, Y|[Y, X]=1〉 be the (first) Weyl algebra over a field K of characteristic zero. It is known that the set of eigenvalues of the inner derivation ad(YX) of A1 is ℤ. Let A1 → A1, X ↦ x, Y ↦ y, be a K-algebra homomorphism, i.e. [y, x] = 1. It is proved that the set of eigenvalues of the inner derivation ad(yx) of the Weyl algebra A1 is ℤ and the eigenvector algebra of ad(yx) is K〈x, y〉 (this would be an easy corollary of the Problem/Conjecture of Dixmier of 1968 [still open]: is an algebra endomorphism of A1 an automorphism?).

(σ, τ)-amenability of C*-algebras

Suppose that is an algebra, σ, τ : → are two linear mappings such that both σ() and τ() are subalgebras of and 𝒳 is a (τ(), σ())-bimodule. A linear mapping D : → 𝒳 is called a (σ, τ)-derivation if D(ab) = D(a) · σ(b) + τ(a) · D(b) (a, b ∈ ). A (σ, τ)-derivation D is called a (σ, τ)-inner derivation if there exists an x ∈ 𝒳 such that D is of the form either or . A Banach algebra is called (σ, τ)-amenable if every (σ, τ)-derivation from into a dual Banach (τ(), σ())-bimodule is (σ, τ)-inner. Studying some general algebraic aspects of (σ, τ)-derivations, we investigate the relation between the amenability and the (σ, τ)-amenability of Banach algebras in the case where σ, τ are homomorphisms. We prove that if 𝔄 is a C*-algebra and σ, τ are *-homomorphisms with ker(σ) = ker(τ), then 𝔄 is (σ, τ)-amenable if and only if σ(𝔄) is amenable.