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This paper is devoted to study of local and 2-local derivations on octonion algebras. We shall give a general form of local derivations on the real octonion algebra ℝ . This description implies that the space of all local derivations on ℝ when equipped with Lie bracket is isomorphic to the Lie algebra 7 (ℝ) of all real skew-symmetric 7 × 7 -matrices. We also consider 2 -local derivations on an octonion algebra over an algebraically closed field of characteristic zero and prove that every 2 -local derivation on is a derivation. Further, we apply these results to similar problems for the simple seven-dimensional Malcev algebra. As a corollary, we obtain that the real octonion algebra ℝ and Malcev algebra M 7 (ℝ) are simple non-associative algebras which admit pure local derivations, that is, local derivations which are not derivation. [ABSTRACT FROM AUTHOR]

The present paper is devoted to the study of the dimension of the space of all derivations on homogeneous commutative regular algebras. We shall show that if Ω , Σ , μ is a Maharam homogeneous measure space with a finite countable-additive measure μ and A is a homogeneous regular unital subalgebra in S (Ω) with the homogeneous Boolean algebra of idempotents ∇ (A) , then dim Der (A) = τ (∇ (A)) trdeg (A) , where τ (∇ (A)) is the weight of the Boolean algebra ∇ (A) and trdeg (A) is the transcendence degree of A . [ABSTRACT FROM AUTHOR]

The present paper is devoted to study of band preserving isomorphisms of commutative unital regular algebras. Let A be a commutative unital regular algebra over an algebraically closed field F of characteristic zero and let ∇ = ∇ (A) be the Boolean algebra of all idempotents in A. Assume that μ is a finite strictly positive countable-additive measure on ∇ and let A be complete with respect to the metric ρ (x , y) = μ (s (x - y)) , x , y ∈ A. We prove that if B is a subalgebra of A such that B ⊃ ∇ , then for any band preserving monomorphism Φ : B → B there exists a band preserving monomorphism Ψ : A → A such that Ψ | B = Φ. Further we introduce a notion of transcendence degree of a commutative unital regular algebra and prove that two homogeneous unital regular subalgebras of S (Ω) – the algebra of all classes of measurable complex-valued functions on a Maharam homogeneous measure space (Ω , Σ , μ) , are isomorphic if and only if their Boolean algebras of idempotents are isomorphic and their transcendence degrees coincide. As an application we obtain that the regular algebra S(0; 1) – of all classes of measurable complex-valued functions and the algebra AD(0; 1) – of all classes of approximately differentiable functions on [0; 1] are isomorphic. [ABSTRACT FROM AUTHOR]

In the present paper, we study 2-local derivations on the so-called infinite dimensional generalized Witt algebras. Firstly, we prove that every 2-local derivation on the generalized Witt algebra W n (F) over the vector space F n is a derivation, where F is a field of characteristic zero. Further we consider generalized Witt algebras of the form W = W (G , I) over the field F , where I is an infinite index set and G = ⨁ i ∈ I Z = (a i) i ∈ I : a i = 0 e x c e p t f o r a f i n i t e n u m b e r o f i ∈ I , and prove that all 2-local derivations on W (G , I) are also derivations. Finally, we show that every 2-local derivation on B (Z n , I) , the Borel subalgebra of W n (F) , is a derivation. [ABSTRACT FROM AUTHOR]

The present paper is devoted to the description of the structures of finite-dimensional semisimple Leibniz algebras over complex numbers, their derivations and automorphisms. [ABSTRACT FROM AUTHOR]

We show that any local derivation on the solvable Leibniz algebras with model or abelian nilradicals, whose dimension of complementary space is maximal is a derivation. We show that solvable Leibniz algebras with abelian nilradicals, which have 1 dimension complementary space, admit local derivations which are not derivations. Moreover, similar problem concerning 2 -local derivations of such algebras is investigated and an example of solvable Leibniz algebra is given such that any 2 -local derivation on it is a derivation, but which admits local derivations which are not derivations. [ABSTRACT FROM AUTHOR]

This paper is devoted to study local and 2-local derivations and automorphism of complex finite-dimensional simple Leibniz algebras. We show that all local derivations and 2-local derivations on a finite-dimensional complex simple Leibniz algebra are automatically derivations. We prove that nilpotent Leibniz algebras as a rule admit local derivations and 2-local derivations which are not derivations. Further, we consider automorphisms of simple Leibniz algebras. It is established that every 2-local automorphism on a complex finite-dimensional simple Leibniz algebra is an automorphism and that nilpotent Leibniz algebras admit 2-local automorphisms which are not automorphisms. A similar problem concerning local automorphism on simple Leibniz algebras is reduced to the case of simple Lie algebras. [ABSTRACT FROM AUTHOR]

In the present paper, we study 2-local derivations of infinite-dimensional Lie algebras over a field of characteristic zero. We prove that all 2-local derivations of the Witt algebra as well as of the positive Witt algebra and the classical one-sided Witt algebra are (global) derivations. We also give an example of an infinite-dimensional Lie algebra with a 2-local derivation which is not a derivation. [ABSTRACT FROM AUTHOR]

Let E2 be the 2-dimensional Euclidean space, and let O(2,R) be the group of all orthogonal transformations of E2. Put MO(2,R) = {F : E2 → E2 | Fx = gx + b, g ∊ O(2,R), b ∊ E2}, SO(2,R) = {g ∊ O(2,R)|detg = 1} and MSO(2,R) = {F ∊ M(2,R)|detg = 1}.The present paper is devoted to solutions of problems of G-equivalence of m-tuples in E2 for groups G = O(2,R), SO(2,R), MO(2,R), MSO(2,R). Complete systems of Ginvariants of m-tuples in E2 for these groups are obtained. Complete systems of relations between elements of the obtained complete systems of G-invariants are given for these groups. [ABSTRACT FROM AUTHOR]

The paper is devoted to the description of 2-local derivations on von Neumann algebras. Earlier it was proved that every 2-local derivation on a semi-finite von Neumann algebra is a derivation. In this paper, using the analogue of Gleason Theorem for signed measures, we extend this result to type $$III$$ von Neumann algebras. This implies that on arbitrary von Neumann algebra each 2-local derivation is a derivation. [ABSTRACT FROM AUTHOR]

This paper is devoted to derivations on the algebra S0( M, τ) of all τ-compact operators affiliated with a von Neumann algebra M and a faithful normal semi-finite trace τ. The main result asserts that every tτ-continuous derivation $${D : S_0(M, \tau) \rightarrow S_{0}(M, \tau)}$$ D : S 0 ( M , τ ) → S 0 ( M , τ ) is spatial and implemented by a τ-measurable operator affiliated with M, where tτ denotes the measure topology on S0( M, τ). We also show the automatic tτ-continuity of all derivations on S0( M, τ) for properly infinite von Neumann algebras M. Thus in the properly infinite case the condition of tτ-continuity of the derivation is redundant for its spatiality. [ABSTRACT FROM AUTHOR]

In the given article, enveloping C*-algebras of AJW-algebras are considered. Conditions are given, when the enveloping C*-algebra of an AJW-algebra is an AW*-algebra, and corresponding theorems are proved. In particular, we proved that if $\mathcal{A}$ is a real AW*-algebra, $\mathcal{A}_{sa}$ is the JC-algebra of all self-adjoint elements of $\mathcal{A}$, $\mathcal{A}+i\mathcal{A}$ is an AW*-algebra and $\mathcal{A}\cap i\mathcal{A} = \{0\}$ then the enveloping C*-algebra $C^*(\mathcal{A}_{sa})$ of the JC-algebra $\mathcal{A}_{sa}$ is an AW*-algebra. Moreover, if $\mathcal{A}+i\mathcal{A}$ does not have nonzero direct summands of type I, then $C^*(\mathcal{A}_{sa})$ coincides with the algebra $\mathcal{A}+i\mathcal{A}$, i.e. $C^*(\mathcal{A}_{sa})= \mathcal{A}+i\mathcal{A}$. [ABSTRACT FROM AUTHOR]

Given a von Neumann algebra M, we consider the central extension E( M) of M. We introduce the topology t( M) on E( M) generated by a center-valued norm and prove that it coincides with the topology of local convergence in measure on E( M) if and only if M does not have direct summands of type II. We also show that t( M) restricted to the set E( M) of self-adjoint elements of E( M) coincides with the order topology on E( M) if and only if M is a σ-finite type I von Neumann algebra. [ABSTRACT FROM AUTHOR]