arXiv subjects

Meiyun Liu

Publications and source records attributed to Meiyun Liu.

2 recordsLinked to original sources

arXiv2016

Strong $k$-commutativity preserving maps on 2$\times$2 matrices

Let ${\mathcal M}_2(\mathbb F)$ be the algebra of 2$\times$2 matrices over the real or complex field $\mathbb F$. For a given positive integer $k\geq 1$, the $k$-commutator of $A$ and $B$ is defined by $[A,B]_k=[[A,B]_{k-1},B]$ with $[A,B]_0=A$ and $[A,B]_1=[A,B]=AB-BA$. The main result is shown that a map $Φ: {\mathcal M}_2(\mathbb F)\to {\mathcal M}_2(\mathbb F)$ with range containing all rank one matrices satisfies that $[Φ(A),Φ(B)]_k = [A,B]_k $ for all $A, B\in{\mathcal M}_2(\mathbb F)$ if and only if there exist a functional $h :{\mathcal M}_2(\mathbb F) \rightarrow {\mathbb F}$ and a scalar $λ\in{\mathbb F}$ with $λ^{k+1} = 1$ such that $Φ(A) = λA + h(A)I$ for all $A \in{\mathcal M}_2(\mathbb F)$.

math.RA↗

arXiv2016

Strong $3$-Commutativity Preserving Maps on Standard Operator Algebras

Let $X$ be a Banach space of dimension $\geq 2$ over the real or complex field ${\mathbb F}$ and ${\mathcal A}$ a standard operator algebra in ${\mathcal B}(X)$. A map $Φ:{\mathcal A} \rightarrow {\mathcal A}$ is said to be strong $3$-commutativity preserving if $[Φ(A),Φ(B)]_3 = [A,B]_3$ for all $A, B\in{\mathcal A}$, where $[A,B]_3$ is the 3-commutator of $A,B$ defined by $[A,B]_3=[[[A,B],B],B]$. The main result in this paper is shown that, if $Φ$ is a surjective map on ${\mathcal A}$, then $Φ$ is strong $3$-commutativity preserving if and only if there exist a functional $h :{\mathcal A} \rightarrow {\mathbb F}$ and a scalar $λ\in{\mathbb F}$ with $λ^4 = 1$ such that $Φ(A) = λA + h(A)I$ for all $A \in{\mathcal A}$.

math.FA↗