arXiv subjects
Publications and source records attributed to Jiankui Li.
Let $D$ be a noncommutative division ring and let $R = M_{m}(D)$ with $m > 1$. We characterize additive mappings $f,$ $g:R\rightarrow R$ satisfying the identity $f(X) = X^{n}g(X^{-1})$ for every invertible element $X$ in $R$, where $n$ is a nonnegative integer. We show that the solutions are precisely given by $f = g$ and $f(X) = Xf(I)$ for all $X$ in $R$. Moreover, if $n \neq2$, both $f$ and $g$ are identically zero.
Let $A$ be a Banach algebra admitting a bounded approximate unit and satisfying property $\mathbb{B}$. Suppose $T: A \rightarrow X$ is a continuous linear map, where $X$ is an essential Banach $A$-bimodule. We prove that the following statements are equivalent: $(i)$ $T$ is anti-derivable at zero (i.e., $a b =0$ in $A$ $\Rightarrow T(b)\cdot a + b\cdot T(a) =0$); $(ii)$ There exist an element $\xi \in X^{**}$ and a linear map (actually a bounded Jordan derivation) $d: A\to X$ satisfying $\xi \cdot a = a \cdot \xi \in X$, $T(a) = d(a) +\xi \cdot a$, and $d(b)\cdot a + b\cdot d(a)= - 2 \xi \cdot (b a),$ for all $a,b\in A$ with $a b =0$. Assuming that $A$ is a C$^*$-algebra we show that a bounded linear mapping $T: A\to X$ is anti-derivable at zero if, and only if, there exist an element $\eta \in X^{**}$ and an anti-derivation $d: A \rightarrow X$ satisfying $\eta \cdot a = a \cdot \eta \in X$, $\eta \cdot [a,b] = 0$ {\rm(}i.e., $L_{\eta}: A \to A$, $L_{\eta} (a) = \eta \cdot a$ vanishes on commutators{\rm)}, and $T(a) = d(a) +\eta \cdot a$, for all $a,b \in A$. The results are also applied for some special operator algebras.
The purpose of this paper is to characterize several classes of functional identities involving inverses with related mappings from a unital Banach algebra $\mathcal{A}$ over the complex field into a unital $\mathcal{A}$-bimodule $\mathcal{M}$. Let $N$ be a fixed invertible element in $\mathcal{A}$, $M$ be a fixed element in $\mathcal{M}$, and $n$ be a positive integer. We investigate the forms of additive mappings $f$, $g$ from $\mathcal{A}$ into $\mathcal{M}$ satisfying one of the following identities: \begin{equation*} \begin{aligned} &f(A)A- Ag(A) = 0\\ &f(A)+ g(B)\star A= M\\ &f(A)+A^{n}g(A^{-1})=0\\ &f(A)+A^{n}g(B)=M \end{aligned} \qquad \begin{aligned} &\text{for each invertible element}~A\in\mathcal{A}; \\ &\text{whenever}~ A,B\in\mathcal{A}~\text{with}~AB=N;\\ &\text{for each invertible element}~A\in\mathcal{A}; \\ &\text{whenever}~ A,B\in\mathcal{A}~\text{with}~AB=N, \end{aligned} \end{equation*} where $\star$ is either the Jordan product $A\star B = AB+BA$ or the Lie product $A\star B = AB-BA$.
Let $\mathfrak{M}(\mathbb{D}, m, n, P)$ be the ring of all $m \times n$ matrices over a division ring $\mathbb{D}$, with the product given by $A \bullet B=A P B$, where $P$ is a fixed $n \times m$ matrix over $\mathbb{D}$. When $2\leq m, n <\infty$ and $\operatorname{rank} P \geq 2$, we demonstrate that every element in $\mathcal{A}=\mathfrak{M}(\mathbb{D}, m, n, P)$ is a sum of finite products of pairs of commutators. We also estimate the minimal number $N$ such that $\mathcal{A}= \sum^N [\mathcal{A}, \mathcal{A}][\mathcal{A}, \mathcal{A}]$. Furthermore, if $\operatorname{char}\mathbb{D}\neq 2$, we prove that $\mathfrak{M}(\mathbb{D}, m, n, P)$ is additively spanned by Jordan products of idempotents. For a field $\mathbb{F}$ with $\operatorname{char}\mathbb{F}\neq 2, 3$, we show that the Munn algebra $\mathfrak{M}(\mathbb{F}, m, n, P)$ is zero product determined and zero Jordan product determined.
Let $\mathcal{L}$ be a completely distributive commutative subspace lattice or a subspace lattice with two atoms, we use a unified approach to study the derivations, homomorphisms on $\mathrm{Alg} \mathcal{L}$. We verify that the multiplier algebra of $\mathrm{Alg} \mathcal{L}\cap \mathcal{K}(\mathcal{H})$ is isomorphic to $\mathrm{Alg} \mathcal{L}$ and $\mathrm{Alg} \mathcal{L}$ is zero product determined. For $T$ in $M_{n}(\mathbb{C})$, $n\geq 2$, we show that $\mathcal{A}_{T}$ is zero product determined if and only if every local derivation from $\mathcal{A}_{T}$ into any Banach $\mathcal{A}_{T}$-bimodule is a derivation. In addition, we establish some equivalent conditions for an algebra to be zero product determined. For countable dimensional locally matrix algebras and triangular UHF algebras, we also show that they are zero Lie product determined.
Let $\mathcal{A}$ be a unital Banach $*$-algebra and $\mathcal{M}$ be a unital $*$-$\mathcal{A}$-bimodule. If $W$ is a left separating point of $\mathcal{M}$, we show that every $*$-derivable mapping at $W$ is a Jordan derivation, and every $*$-left derivable mapping at $W$ is a Jordan left derivation under the condition $W \mathcal{A}=\mathcal{A}W$. Moreover we give a complete description of linear mappings $δ$ and $τ$ from $\mathcal{A}$ into $\mathcal{M}$ satisfying $δ(A)B^*+Aτ(B)^*=0$ for any $A, B\in \mathcal{A}$ with $AB^*=0$ or $δ(A)\circ B^*+A\circτ(B)^*=0$ for any $A, B\in \mathcal{A}$ with $A\circ B^*=0$, where $A\circ B=AB+BA$ is the Jordan product.
We prove that each local Lie $n$-derivation is a Lie $n$-derivation under mild assumptions on the unital algebras with a nontrivial idempotent. As applications, we obtain descriptions of local Lie $n$-derivations on generalized matrix algebras, triangular algebras, nest algebras, von Neumann algebras, and the algebras of locally measurable operators affiliated with a von Neumann algebra.
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.
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.
We characterize derivations and 2-local derivations from $M_{n}(\mathcal{A})$ into $M_{n}(\mathcal{M})$, $n \ge 2$, where $\mathcal{A}$ is a unital algebra over $\mathbb{C}$ and $\mathcal{M}$ is a unital $\mathcal{A}$-bimodule. We show that every derivation $D: M_{n}(\mathcal{A}) \to M_{n}(\mathcal{M})$, $n \ge 2,$ is the sum of an inner derivation and a derivation induced by a derivation from $\mathcal{A}$ to $\mathcal{M}$. We say that $\mathcal{A}$ commutes with $\mathcal{M}$ if $am=ma$ for every $a\in\mathcal{A}$ and $m\in\mathcal{M}$. If $\mathcal{A}$ commutes with $\mathcal{M}$ we prove that every inner 2-local derivation $D: M_{n}(\mathcal{A}) \to M_{n}(\mathcal{M})$, $n \ge 2$, is an inner derivation. In addition, if $\mathcal{A}$ is commutative and commutes with $\mathcal{M}$, then every 2-local derivation $D: M_{n}(\mathcal{A}) \to M_{n}(\mathcal{M})$, $n \ge 2$, is a derivation.
For a commutative C*-algebra $\mathcal A$ with unit $e$ and a Hilbert~$\mathcal A$-module $\mathcal M$, denote by End$_{\mathcal A}(\mathcal M)$ the algebra of all bounded $\mathcal A$-linear mappings on $\mathcal M$, and by End$^*_{\mathcal A}(\mathcal M)$ the algebra of all adjointable mappings on $\mathcal M$. We prove that if $\mathcal M$ is full, then each derivation on End$_{\mathcal A}(\mathcal M)$ is $\mathcal A$-linear, continuous, and inner, and each 2-local derivation on End$_{\mathcal A}(\mathcal M)$ or End$^{*}_{\mathcal A}(\mathcal M)$ is a derivation. If there exist $x_0$ in $\mathcal M$ and $f_0$ in $\mathcal M^{'}$, such that $f_0(x_0)=e$, where $\mathcal M^{'}$ denotes the set of all bounded $\mathcal A$-linear mappings from $\mathcal M$ to $\mathcal A$, then each $\mathcal A$-linear local derivation on End$_{\mathcal A}(\mathcal M)$ is a derivation.
Let $\mathcal{U}=\left[ \begin{array}{cc} \mathcal{A} & \mathcal{M} \mathcal{N}& \mathcal{B} \end{array} \right]$ be a generalized matrix ring, where $\mathcal{A}$ and $\mathcal{B}$ are 2-torsion free. We prove that if $ϕ:\mathcal{U}\rightarrow \mathcal{U}$ is an additive mapping such that $ϕ(U)\circ V+U\circ ϕ(V)=0$ whenever $UV=VU=0,$ then $ϕ=δ+η$, where $δ$ is a Jordan derivation and $η$ is a multiplier. As its applications, we prove that the similar conclusion remains valid on full matrix algebras, unital prime rings with a nontrivial idempotent, unital standard operator algebras, CDCSL algebras and von Neumann 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.
A linear mapping $ϕ$ on an algebra $\mathcal{A}$ 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 $AB=G$, and $ϕ$ is called a derivable mapping at $G\in\mathcal{A}$ if $ϕ(AB)=ϕ(A)B+Aϕ(B)$ for each $A$ and $B$ in $\mathcal{A}$ with $AB=G$. A point $G$ in $\mathcal{A}$ is called a full-centralizable point (resp. full-derivable point) if every centralizable (resp. derivable) mapping at $G$ is a centralizer (resp. derivation). We prove that every point in a von Neumann algebra or a triangular algebra is a full-centralizable point. We also prove that a point in a von Neumann algebra is a full-derivable point if and only if its central carrier is the unit.
Let d be a linear mapping from a unital Banach algebra A into a unital left A-module M, and w in Z(A) be a left separating point of M. We show that the following three conditions are equivalent: (i) d is a Jordan left derivation; (ii) d is left derivable at w; (iii) d is Jordan left derivable at w. Let A be a Banach algebra with the property (B), and M be a Banach left A-module. We consider the relations between generalized (Jordan) left derivations and (Jordan) left derivable mappings at zero from A into M.
In the paper, we consider the question whether a unital full amalgamated free product of quasidiagonal C*-algebras is quasidiagonal again. We give a sufficient condition such that a unital full amalgamated free product of quasidiagonal C*-algebras with amalgamation over a finite dimensional C*- algebra is quasidiagonal. Applying this result, we conclude that a unital full free product of two AF algebras with amalgamation over a finite-dimensional C*-algebra is AF if there are faithful tracial states on each of these two AF algebras such that the restrictions on the common subalgebra agree.
We prove that if M is a von Neumann algebra whose abelian summand is discrete, then every local derivation on the algebra of all measurable operators affilated with M is a derivation. This answers a question of Richard Kadison.
In this paper, we characterize Jordan derivable mappings in terms of Peirce decomposition and determine Jordan all-derivable points for some general bimodules. Then we generalize the results to the case of Jordan higher derivable mappings. An immediate application of our main results shows that for a nest $\mathcal{N}$ on a Banach $X$ with the associated nest algebra $alg\mathcal{N}$, if there exists a non-trivial element in $\mathcal{N}$ which is complemented in $X$, then every $C\in alg\mathcal{N}$ is a Jordan all-derivable point of $L(alg\mathcal{N}, B(X))$ and a Jordan higher all-derivable point of $L(alg\mathcal{N})$.