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Sh. A. Ayupov

Publications and source records attributed to Sh. A. Ayupov.

At least 19 recordsLinked to original sources
arXiv2026

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

In the present paper we prove that every 2-local inner derivation on the Jordan ring of self-adjoint matrices over a commutative involutive ring is a derivation. We also apply our technique to various Jordan algebras of infinite dimensional self-adjoint matrix-valued maps on a set and prove that every 2-local spatial derivation on such algebras is a spatial derivation. It is also proved that every local spatial derivation on the same Jordan algebras is a derivation.

math.RA↗
arXiv2023

Local and 2-local derivations on filiform associative algebras

This paper is devoted to the study of local and 2-local derivations of nullfiliform, filiform and naturally graded quasi-filiform associative algebras. We prove that these algebras as a rule admit local derivations which are not derivations. We show that filiform and naturally graded quasi-filiform associative algebras admit 2-local derivations which are not derivations and any 2-local derivation of null-filiform associative algebras is a derivation.

math.RA↗
arXiv2019

Local and 2-local derivations of solvable Leibniz algebras

We show that any local derivation on the solvable Leibniz algebras with model or abelian nilradicals, whose the 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 1-local derivations of such algebras are investigated and an example of solvable Leibniz algebra given such that any 2-local derivation on it is a derivation, but which admit local derivations which are not derivations.

math.RA↗
arXiv2018

Local derivations on Solvable Lie algebras

We show that in the class of solvable Lie algebras there exist algebras which admit local derivations which are not ordinary derivation and also algebras for which every local derivation is a derivation. We found necessary and sufficient conditions under which any local derivation of solvable Lie algebras with abelian nilradical and one-dimensional complementary space is a derivation. Moreover, we prove that every local derivation on a finite-dimensional solvable Lie algebra with model nilradical and maximal dimension of complementary space is a derivation.

math.RA↗
arXiv2014

Leibniz algebras associated with representations of filiform Lie algebras

In this paper we investigate Leibniz algebras whose quotient Lie algebra is a naturally graded filiform Lie algebra $n_{n,1}.$ We introduce a Fock module for the algebra $n_{n,1}$ and provide classification of Leibniz algebras $L$ whose corresponding Lie algebra $L/I$ is the algebra $n_{n,1}$ with condition that the ideal $I$ is a Fock $n_{n,1}$-module, where $I$ is the ideal generated by squares of elements from $L$.

math.RA↗
arXiv2013

2-Local derivations on matrix algebras over commutative regular algebras

The paper is devoted to 2-local derivations on matrix algebras over commutative regular algebras. We give necessary and sufficient conditions on a commutative regular algebra to admit 2-local derivations which are not derivations. We prove that every 2-local derivation on a matrix algebra over a commutative regular algebra is a derivation. We apply these results to 2-local derivations on algebras of measurable and locally measurable operators affiliated with type I von Neumann algebras.

math.OA↗
arXiv2011

2-Local derivations and automorphisms on $B(H)$

The paper is devoted to 2-local derivations and 2-local automorphisms on the algebra $B(H)$ of all bounded linear operators on a Hilbert space $H.$ We prove that every 2-local derivation on $B(H)$ is a derivation. A similar result is obtained for automorphisms.

math.OA↗
arXiv2011

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.

math.OA↗
arXiv2011

Orlicz Spaces associated with a Semi-Finite Von Neumann Algebra

In the present paper we introduce a certain class of non commutative Orlicz spaces, associated with arbitrary faithful normal locally-finite weights on a semi-finite von Neumann algebra $M.$ We describe the dual spaces for such Orlicz spaces and, in the case of regular weights, we show that they can be realized as linear subspaces of the algebra of $LS(M)$ of locally measurable operators affiliated with $M.$

math.OA↗
arXiv2011

Topologies on Central Extensions of Von Neumann Algebras

Given a von Neumann algebra $M$ we consider the central extension $E(M)$ of $M.$ We introduce the topology $t_c(M)$ on $E(M)$ generated by a center-valued norm and prove that it coincides with the topology of convergence locally in measure on $E(M)$ if and only if $M$ does not have direct summands of type II. We also show that $t_c(M)$ restricted on the set $E(M)_h$ of self-adjoint elements of $E(M)$ coincides with the order topology on $E(M)_h$ if and only if $M$ is a $σ$-finite type I$_{fin}$ von Neumann algebra.

math.OA↗
arXiv2011

Index Theory for Real Factors

The notion of index for arbitrary real factors is introduced and investigated. The main tool in our approach is the reduction of real factors to involutive *-anti-automorphisms of their complex enveloping von Neumann algebras. Similar to the complex case the values of the index for real factors are calculated.

math.OA↗
arXiv2011

Automorphisms of central extensions of type I von Neumann algebras

Given a von Neumann algebra $M$ we consider the central extension $E(M)$ of $M.$ For type I von Neumann algebras $E(M)$ coincides with the algebra $LS(M)$ of all locally measurable operators affiliated with $M.$ In this case we show that an arbitrary automorphism $T$ of $E(M)$ can be decomposed as $T=T_a\circ T_ϕ,$ where $T_a(x)=axa^{-1}$ is an inner automorphism implemented by an element $a\in E(M),$ and $T_ϕ$ is a special automorphism generated by an automorphism $ϕ$ of the center of $E(M).$ In particular if $M$ is of type I$_\infty$ then every band preserving automorphism of $E(M)$ is inner.

math.OA↗
arXiv2010

Additive derivations on generalized Arens algebras

Given a von Neumann algebra $M$ with a faithful normal finite trace $τ$ denote by $L^Λ(M, τ)$ the generalized Arens algebra with respect to $M.$ We give a complete description of all additive derivations on the algebra $L^Λ(M, τ).$ In particular each additive derivation on the algebra $L^Λ(M, τ),$ where $M$ is a type II von Neumann algebra, is inner.

math.OA↗
arXiv2009

On a certain class of operator algebras and their derivations

Given a von Neumann algebra $M$ with a faithful normal finite trace, we introduce the so called finite tracial algebra $M_f$ as the intersection of $L_p$-spaces $L_p(M, μ)$ over all $p \geq 1$ and over all faithful normal finite traces $μ$ on $M.$ Basic algebraic and topological properties of finite tracial algebras are studied. We prove that all derivations on these algebras are inner.

math.OA↗