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Shavkat Ayupov

Publications and source records attributed to Shavkat Ayupov.

At least 19 recordsLinked to original sources
arXiv2026

Local and 2-local $\frac{1}2$-derivations of infinite-dimensional Lie algebras

In this work, we describe local and 2-local $\frac12$-derivations of infinite-dimensional Lie algebras. We prove that all local and 2-local $\frac12$-derivations of the Witt algebra as well as of the positive Witt algebra and the classical one-sided Witt algebra are $\frac12$-derivations. We also give an example of an infinite-dimensional Lie algebra with a local (2-local) $\frac12$-derivation which is not a $\frac12$-derivation. Further we prove that all local (2-local) $\frac12$-derivations on the $\mathcal{W}(a,b)$ algebra are $\frac12$-derivations.

math.RA↗
arXiv2022

Local and 2-local derivations on Lie matrix rings over commutative involutive rings

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.

math.RA↗
arXiv2021

Local and $2$-local automorphisms of Cayley algebras

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 $\{φ\in O(\mathcal{C},n)\mid φ(1)=1\}.$ 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 $\{φ\in O(\mathcal{C},n)\mid φ(1)=1\}.$

math.RA↗
arXiv2021

Local and $2$-local derivations of Cayley algebras

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 $\neq 2,3$.

math.RA↗
arXiv2020

Ring isomorphisms of $\ast$-subalgebras of Murray-von Neumann factors

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 $Φ: \cA \to \cB$ there exist a positive invertible element $a \in \cB$ with $a^{-1}\in \cB$ and a real $\ast$-isomorphism $Ψ: \cM \to \cN$ (which extends to a real $\ast$-isomorphism from $\cA$ onto $\cB$) such that $Φ(x) = aΨ(x)a^{-1}$ for all $x \in \cA.$ In particular, $Φ$ 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.

math.OA↗
arXiv2020

Description of 2-local derivations and automorphisms on finite dimensional Jordan algebras

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.

math.RA↗
arXiv2019

2-Local automorphisms on $AW^\ast$-algebras

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.

math.OA↗
arXiv2019

2-local derivations on infinite-dimensional Lie algebras

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.

math.RA↗
arXiv2018

Description of 2-local derivations on some Lie rings of skew-adjoint matrices

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.

math.RA↗
arXiv2018

Local automorphisms on finite-dimensional Lie and Leibniz algebras

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.

math.RA↗
arXiv2017

Local and 2-local derivations and automorphisms on simple Leibniz algebras

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.

math.RA↗
arXiv2017

Two-Local derivations on associative and Jordan matrix rings over commutative rings

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.

math.RA↗
arXiv2017

2-Local derivations on matrix algebras and algebras of measurable operators

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.

math.OA↗
arXiv2016

Local derivations on measurable operators and commutativity

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.

math.OA↗
arXiv2016

Jordan counterparts of Rickart and Baer $*$-algebras

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.

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