arXiv subjects

Guangyu An

Publications and source records attributed to Guangyu An.

7 recordsLinked to original sources

arXiv2022

Derivations, local and 2-local derivations of standard operator algebras

Let X be a Banach space over field F (R or C). Denote by B(X) the set of all bounded linear operators on X and by F(X) the set of all finite rank operators on X. A subalgebra A of B(X) is called a standard operator algebra if A contain F(X). We give a brief proof of a well-known result that every derivation from A into B(X) is inner. There is another classical result that every local derivation on B(X) is a derivation. We extend the result by proving that every local derivation from A into B(X) is a derivation. Based on these two results, we prove that every 2-local derivation from A into B(X) is a derivation.

math.FA↗

arXiv2020

Characterizing linear mappings through zero products or zero Jordan products

Let $\mathcal{A}$ be a $*$-algebra and $\mathcal{M}$ be a $*$-$\mathcal A$-bimodule, we study the local properties of $*$-derivations and $*$-Jordan derivations from $\mathcal{A}$ into $\mathcal{M}$ under the following orthogonality conditions on elements in $\mathcal A$: $ab^*=0$, $ab^*+b^*a=0$ and $ab^*=b^*a=0$. We characterize the mappings on zero product determined algebras and zero Jordan product determined algebras. Moreover, we give some applications on $C^*$-algebras, group algebra, matrix algebras, algebras of locally measurable operators and von Neumann algebras.

math.OA↗

arXiv2018

Characterizations of centralizable mappings on algebras of locally measurable operators

A linear mapping $ϕ$ from an algebra $\mathcal{A}$ into its bimodule $\mathcal M$ is called a centralizable mapping at $G\in\mathcal{A}$ if $ϕ(AB)=ϕ(A)B=Aϕ(B)$ for each $A$ and $B$ in $\mathcal{A}$ with $AB=G$. In this paper, we prove that if $\mathcal M$ is a von Neumann algebra without direct summands of type $\mathrm{I}_1$ and type $\mathrm{II}$, $\mathcal A$ is a $*$-subalgebra with $\mathcal M\subseteq\mathcal A\subseteq LS(\mathcal{M})$ and $G$ is a fixed element in $\mathcal A$, then every continuous (with respect to the local measure topology $t(\mathcal M)$) centralizable mapping at $G$ from $\mathcal A$ into $\mathcal M$ is a centralizer.

math.OA↗

arXiv2018

Local Lie derivations on von Neumann algebras and algebras of locally measurable operators

Let $\mathcal{A}$ be a unital associative algebra and $\mathcal{M}$ be an $\mathcal{A}$-bimodule. A linear mapping $φ$ from $\mathcal{A}$ into an $\mathcal{A}$-bimodule $\mathcal{M}$ is called a Lie derivation if $φ[A,B]=[φ(A),B]+[A,φ(B)]$ for each $A,B$ in $\mathcal{A}$, and $φ$ is called a \emph{local Lie derivation} if for every $A$ in $\mathcal{A}$, there exists a Lie derivation $φ_{A}$ (depending on $A$) from $\mathcal{A}$ into $\mathcal{M}$ such that $φ(A)=φ_{A}(A)$. In this paper, we prove that every local Lie derivation on von Neumann algebras is a Lie derivation; and we show that if $\mathcal M$ is a type I von Neumann algebra with atomic lattice of projections, then every local Lie derivation on $LS(\mathcal M)$ is a Lie derivation.

math.OA↗

arXiv2018

Characterizations of $(m,n)$-Jordan derivations on some algebras

Let $\mathcal R$ be a ring, $\mathcal{M}$ be a $\mathcal R$-bimodule and $m,n$ be two fixed nonnegative integers with $m+n\neq0$. An additive mapping $δ$ from $\mathcal R$ into $\mathcal{M}$ is called an \emph{$(m,n)$-Jordan derivation} if $(m+n)δ(A^{2})=2mAδ(A)+2nδ(A)A$ for every $A$ in $\mathcal R$. In this paper, we prove that every $(m,n)$-Jordan derivation from a $C^{*}$-algebra into its Banach bimodule is zero. An additive mapping $δ$ from $\mathcal R$ into $\mathcal{M}$ is called a $(m,n)$-Jordan derivable mapping at $W$ in $\mathcal R$ if $(m+n)δ(AB+BA)=2mδ(A)B+2mδ(B)A+2nAδ(B)+2nBδ(A)$ for each $A$ and $B$ in $\mathcal R$ with $AB=BA=W$. We prove that if $\mathcal{M}$ is a unital $\mathcal A$-bimodule with a left (right) separating set generated algebraically by all idempotents in $\mathcal A$, then every $(m,n)$-Jordan derivable mapping at zero from $\mathcal A$ into $\mathcal{M}$ is identical with zero. We also show that if $\mathcal{A}$ and $\mathcal{B}$ are two unital algebras, $\mathcal{M}$ is a faithful unital $(\mathcal{A},\mathcal{B})$-bimodule and $\mathcal{U}={\left[\begin{array}{cc}\mathcal{A} &\mathcal{M} \\\mathcal{N} & \mathcal{B} \\\end{array}\right]}$ is a generalized matrix algebra, then every $(m,n)$-Jordan derivable mapping at zero from $\mathcal{U}$ into itself is equal to zero.

math.OA↗

arXiv2018

Characterizations of Jordan derivations on algebras of locally measurable operators

We prove that if $\mathcal M$ is a properly infinite von Neumann algebra and $LS(\mathcal M)$ is the local measurable operator algebra affiliated with $\mathcal M$, then every Jordan derivation from $LS(\mathcal M)$ into itself is continuous with respect to the local measure topology $t(\mathcal M)$. We construct an extension of a Jordan derivation from $\mathcal M$ into $LS(\mathcal M)$ up to a Jordan derivation from $LS(\mathcal M)$ into itself. Moreover, we prove that if $\mathcal M$ is a properly von Neumann algebra and $\mathcal A$ is a subalgebra of $LS(\mathcal M)$ such that $\mathcal M\subset\mathcal A$, then every Jordan derivation from $\mathcal A$ into $LS(\mathcal M)$ is continuous with respect to the local measure topology $t(\mathcal M)$.

math.OA↗

arXiv2016

Characterizations of 2-local derivations and local Lie derivations on some algebras

We prove that every 2-local derivation from the algebra $M_n(\mathcal{A})(n>2)$ into its bimodule $M_n(\mathcal{M})$ is a derivation, where $\mathcal{A}$ is a unital Banach algebra and $\mathcal{M}$ is a unital $\mathcal{A}$-bimodule such that each Jordan derivation from $\mathcal{A}$ into $\mathcal{M}$ is an inner derivation, and that every 2-local derivation on a C*-algebra with a faithful traceable representation is a derivation. We also characterize local and 2-local Lie derivations on some algebras such as von Neumann algebras, nest algebras, Jiang-Su algebra and UHF algebras.

math.OA↗