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In the present paper we prove that every 2-local inner derivation on the Lie ring of skew-adjoint matrices over a commutative $*$-ring is an inner derivation. We also apply our technique to various Lie algebras of infinite-dimensional skew-adjoint matrix-valued maps on a set and prove that every 2-local spatial derivation on such algebras is a spatial derivation. We also show that every local spatial derivation on the above Lie algebras is a derivation., Comment: 21 pages. arXiv admin note: text overlap with arXiv:2108.03993, arXiv:1803.06281

The present paper is devoted to the description of local and 2-local automorphisms on Cayley algebras over an arbitrary field $\mathbb{F}$. Given a Cayley algebra $\mathcal{C}$ with norm $n$, let $O(\mathcal{C},n)$ be the corresponding orthogonal group. We prove that the group of all local automorphisms of $\mathcal{C}$ coincides with the group $\{\varphi\in O(\mathcal{C},n)\mid \varphi(1)=1\}.$ Further we prove that the behavior of 2-local automorphisms depends on the Cayley algebra being split or division. Every 2-local automorphism on the split Cayley algebra is an automorphism, i.e. they form the exceptional Lie group $G_2(\mathbb{F})$ if $\textrm{char}\mathbb{F}\neq 2,3$. On the other hand, on division Cayley algebras over a field $\mathbb{F}$, the groups of 2-local automorphisms and local automorphisms coincide, and they are isomorphic to the group $\{\varphi\in O(\mathcal{C},n)\mid \varphi(1)=1\}.$, Comment: 9 pages. arXiv admin note: text overlap with arXiv:2105.08423

The present paper is devoted to the description of local and $2$-local derivations on Cayley algebras over an arbitrary field $\mathbb{F}$. Given a Cayley algebra $\mathcal{C}$ with norm $\mathfrak{n}$, let $\mathcal{C}_0$ be its subspace of trace $0$ elements. We prove that the space of all local derivations of $\mathcal{C}$ coincides with the Lie algebra $\{d\in (\mathcal{C},\mathfrak{n}) | d(1)=0\}$ which is isomorphic to the orthogonal Lie algebra $(\mathcal{C}_0,\mathfrak{n})$. Further we prove that, surprisingly, the behavior of $2$-local derivations depends on the Cayley algebra being split or division. Every $2$-local derivation on the split Cayley algebra is a derivation, i.e. they form the exceptional Lie algebra $\mathfrak{g}_2(\mathbb{F})$ if $\textrm{char}\mathbb{F}\neq 2,3$. On the other hand, on division Cayley algebras over a field $\mathbb{F}$, the sets of $2$-local derivations and local derivations coincide, and they are isomorphic to the Lie algebra $(\mathcal{C}_0,\mathfrak{n})$. As a corollary we obtain descriptions of local and $2$-local derivations of the seven dimensional simple non-Lie Malcev algebras over fields of characteristic $\neq 2,3$.

The present paper is devoted to study of ring isomorphisms of $\ast$-subalgebras of Murray--von Neumann factors. Let $\cM,$ $\cN$ be von Neumann factors of type II$_1,$ and let $S(\cM),$ $S(\cN)$ be the $\ast$-algebras of all measurable operators affiliated with $\cM$ and $ \cN,$ respectively. Suppose that $\cA\subset S(\cM),$ $\cB\subset S(\cN)$ are their $\ast$-subalgebras such that $\cM\subset \cA,$ $\cN\subset \cB.$ We prove that for every ring isomorphism $\Phi: \cA \to \cB$ there exist a positive invertible element $a \in \cB$ with $a^{-1}\in \cB$ and a real $\ast$-isomorphism $\Psi: \cM \to \cN$ (which extends to a real $\ast$-isomorphism from $\cA$ onto $\cB$) such that $\Phi(x) = a\Psi(x)a^{-1}$ for all $x \in \cA.$ In particular, $\Phi$ is real-linear and continuous in the measure topology. In particular, noncommutative Arens algebras and noncommutative $\cL_{log}$-algebras associated with von Neumann factors of type II$_1$ satisfy the above conditions and the main Theorem implies the automatic continuity of their ring isomorphisms in the corresponding metrics. We also present an example of a $\ast$-subalgebra in $S(\cM),$ which shows that the condition $\cM\subset \cA$ is essential in the above mentioned result., Comment: 12 pages. arXiv admin note: text overlap with arXiv:2010.01627

We give a complete description of ring isomorphisms between algebras of measurable operators affiliated with von Neumann algebras of type II$_1.$, Comment: 21 pages

In the present paper we study local and 2-local derivations of locally finite split simple Lie algebras. Namely, we show that every local and 2-local derivation on such Lie algebra is a derivation.

In the present paper we introduce and investigate the notion of 2-local linear map on vector spaces. A sufficient condition is obtained for linearity of a 2-local linear map on finite dimensional vector spaces. Based on this result we prove that every 2-local derivation on a finite dimensional formally real Jordan algebra is a derivation. Also we show that every 2-local 1-automorphism (i.e. implemented by single symmetries) of mentioned Jordan algebras is an automorphism., Comment: 17 pages

The paper is devoted to 2-local automorphisms on $AW^\ast$-algebras. Using the technique of matrix algebras over a unital Banach algebra we prove that any 2-local automorphism on an arbitrary $AW^\ast$-algebra without finite type~I direct summands is a global automorphism., Comment: 10 pages. arXiv admin note: substantial text overlap with arXiv:1702.07629

The present paper is devoted to study 2-local derivations on infinite-dimensional Lie algebras over a field of characteristic zero. We prove that all 2-local derivations on the Witt algebra as well as on the positive Witt algebra are (global)derivations, and give an example of infinite-dimensional Lie algebra with a 2-local derivation which is not a derivation., Comment: 9 pages

In the present paper we prove that every 2-local inner derivation on the Lie ring of skew-symmetric matrices over a commutative ring is an inner derivation. We also apply our technique to various Lie algebras of infinite dimensional skew-adjoint matrix-valued maps on a set and prove that every 2-local spatial derivation on such algebras is a spatial derivation., Comment: 12 pages

We prove that a linear mapping on the algebra \(\mathfrak{sl}_n\) of all trace zero complex matrices is a local automorphism if and only if it is an automorphism or an anti-automorphism. We also show that a linear mapping on a simple Leibniz algebra of the form \(\mathfrak{sl}_n\dot +\mathcal{I}\) is a local automorphism if and only if it is an automorphism. We give examples of finite-dimensional nilpotent Lie algebras \(\mathcal{L}\) with \(\dim \mathcal{L} \geq 3\) which admit local automorphisms which are not automorphisms., Comment: 11 pages. arXiv admin note: text overlap with arXiv:1602.05187, arXiv:1703.10506

The present paper is devoted to the description of finite-dimensional semisimple Leibniz algebras over complex numbers, their derivations and automorphisms., Comment: 13 pages

In the present paper we prove that every 2-local inner derivation on the matrix ring over a commutative ring is an inner derivation and every derivation on an associative ring has an extension to a derivation on the matrix ring over this associative ring. We also develop a Jordan analog of the above method and prove that every 2-local inner derivation on the Jordan matrix ring over a commutative ring is a derivation., Comment: 19 pages, Linear Algebra and its Applications 2017

The present paper is devoted to local and 2-local derivations and automorphism of complex finite-dimensional simple Leibniz algebras. We prove that all local derivations and 2-local derivations on a finite-dimensional complex simple Leibniz algebra are automatically derivations. We show 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. We prove that every 2-local automorphism on a complex finite-dimensional simple Leibniz algebra is an automorphism and show 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., Comment: 28 pages

Let \(\mathcal{A}\) be a unital Banach algebra such that any Jordan derivation from \(\mathcal{A}\) into any \(\mathcal{A}\)-bimodule \(\mathcal{M}\) is a derivation. We prove that any 2-local derivation from the algebra $M_n(\mathcal{A})$ into $M_n(\mathcal{M})$ $(n\geq 3)$ is a derivation. We apply this result to show that any 2-local derivation on the algebra of locally measurable operators affiliated with a von Neumann algebra without direct abelian summands is a derivation., Comment: 12 pages

We prove that a von Neumann algebra $M$ is abelian if and only if the square of every derivation on the algebra $S(M)$ of measurable operators, affiliated with $M$, is a local derivation. We also show that for general associative unital algebras this is not true., Comment: 8 pages. arXiv admin note: text overlap with arXiv:1602.04959

There are Jordan analogues of annihilators in Jordan algebras which are called Jordan annihilators. The present paper is devoted to investigation of those Jordan algebras every Jordan annihilator of which is generated by an idempotent as an inner ideal. We prove that a finite dimensional unital Jordan algebra satisfies this condition if and only if it has no nilpotent elements, and in this case it is a direct sum of simple Jordan algebras., Comment: 15 pages

We introduce and investigate new classes of Jordan algebras which are close to but wider than Rickart and Baer Jordan algebras considered in our previous paper. Such Jordan algebras are called RJ- and BJ-algebras respectively. Criterions are given for a Jordan algebra to be a BJ-algebra. Also, it is proved that every finite dimensional Jordan algebra without nilpotent elements, which have square roots, is a BJ-algebra., Comment: 10 pages

We prove that every $2$-local automorphism on a finite-dimensional semi-simple Lie algebra $\mathcal{L}$ over an algebraically closed field of characteristic zero is an automorphism. We also show that each finite-dimensional nilpotent Lie algebra $\mathcal{L}$ with $\dim \mathcal{L}\geq 2$ admits a $2$-local automorphism which is not an automorphism., Comment: 8 pages

The present paper presents a survey of some recent results devoted to derivations, local derivations and 2-local derivations on various algebras of measurable operators affiliated with von Neumann algebras. We give a complete description of derivation on these algebras, except the case where the von Neumann algebra is of type II$_1$. In the latter case the result is obtained under an extra condition of measure continuity of derivations. Local and 2-local derivations on the above algebras are also considered. We give sufficient conditions on a von Neumann algebra $M$, under which every local or 2-local derivation on the algebra of measurable operators affiliated with $M$ is automatically becomes a derivation. We also give examples of commutative algebras of measurable operators admitting local and 2-local derivations which are not derivations., Comment: accepted to Contem. Math. AMS, 21 pages. arXiv admin note: text overlap with arXiv:0901.2983, arXiv:0808.0149

In the present paper 2-local derivations on various algebras of infinite dimensional matrix-valued functions on a compact are considered. It is proved that every 2-local derivation on such algebra is a derivation. Also we explain that the method developed in the given paper can be applied to associative, Jordan and Lie algebras of infinite dimensional matrix-valued functions on a compact., Comment: 8 pages. arXiv admin note: substantial text overlap with arXiv:1112.6236

We prove that every local derivation on a finite-dimensional semisimple Lie algebra over an algebraically closed field of characteristic zero is a derivation. We also give examples of finite-dimensional nilpotent Lie algebras $\mathcal{L}$ with $\dim\mathcal{L}\geq 3$ which admit local derivations which are not derivations., Comment: 14 pages

In this article it is proved that for every special AJW-algebra $A$ there exist central projections $e$, $f$, $g\in A$, $e+f+g=1$ such that (1) $eA$ is reversible and there exists a norm-closed two sided ideal $I$ of $C^*(eA)$ such that $eA={{}^\perp}(^\perp(I_{sa})_+)_+$; (2) $fA$ is reversible and $R^*(fA)\cap iR^*(fA)=\{0\}$; (3) $gA$ is a totally nonreversible AJW-algebra., Comment: 7 pages

We survey the results on local and 2-local derivations on C*-algebras, von Neumann algebras and JB*-triples., Comment: 60 pages

The paper is devoted to 2-local derivations on matrix algebras over unital semi-prime Banach algebras. For a unital semi-prime Banach algebra $\mathcal{A}$ with the inner derivation property we prove that any 2-local derivation on the algebra $M_{2^n}(\mathcal{A}),$ $n\geq 2,$ is a derivation. We apply this result to $AW^\ast$-algebras and show that any 2-local derivation on an arbitrary $AW^\ast$-algebra is a derivation., Comment: 12 pages

It is proved that every 2-local derivation on an AW$^*$-algebra of type I is a derivation. Also an analog of Gleason theorem for signed measures on projections of homogenous AW$^*$-algebras except the cases of an AW$^*$-algebra of type I$_2$ and a factor of type I$_m$, $2

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., Comment: 7 pages

This paper is devoted to derivations on the algebra $S_0(M, \tau)$ of all $\tau$-compact operators affiliated with a von Neumann algebra $M$ and a faithful normal semi-finite trace $\tau.$ The main result asserts that every $t_\tau$-continuous derivation $D:S_0(M, \tau)\rightarrow S_0(M, \tau)$ is spatial and implemented by a $\tau$-measurable operator affiliated with $M$, where $t_\tau$ denotes the measure topology on $S_0(M, \tau)$. We also show the automatic $t_\tau$-continuity of all derivations on $S_0(M, \tau)$ for properly infinite von Neumann algebras $M$. Thus in the properly infinite case the condition of $t_\tau$-continuity of the derivation is redundant for its spatiality., Comment: 15 pages. arXiv admin note: substantial text overlap with arXiv:1306.0251

This paper is devoted to derivations on the algebra $S(M)$ of all measurable operators affiliated with a finite von Neumann algebra $M.$ We prove that if $M$ is a finite von Neumann algebra with a faithful normal semi-finite trace $\tau$, equipped with the locally measure topology $t,$ then every $t$-continuous derivation $D:S(M)\rightarrow S(M)$ is inner. A similar result is valid for derivation on the algebra $S(M,\tau)$ of $\tau$-measurable operators equipped with the measure topology $t_{\tau}$., Comment: Accepted for publication in the Journal of Mathematical Analysis and Applications

In the present paper it is proved that every inner 2-local derivation on the matrix ring $M_n(\Re)$ of $n\times n$ matrices over a commutative associative ring $\Re$ is an inner derivation. Also, it is proved that, every derivation on an associative ring $\Re$ has an extension to a derivation on the matrix ring $M_n(\Re)$ of $n\times n$ matrices over $\Re$., Comment: 9 pages

In the present paper we prove that every 2-local derivation on a semi-finite von Neumann algebra is a derivation., Comment: 3 pages

In the present paper we prove that every 2-local derivation on a von Neumann algebra of type I is a derivation., Comment: 8 pages

Given a von Neumann algebra $M$ we introduce so called central extension $mix(M)$ of $M$. We show that $mix(M)$ is a *-subalgebra in the algebra $LS(M)$ of all locally measurable operators with respect to $M,$ and this algebra coincides with $LS(M)$ if and only if $M $ does not admit type II direct summands. We prove that if $M$ is a properly infinite von Neumann algebra then every additive derivation on the algebra $mix(M)$ is inner. This implies that on the algebra $LS(M)$, where $M$ is a type I$_\infty$ or a type III von Neumann algebra, all additive derivations are inner derivations.