arXiv subjects

Jiankui Li

Publications and source records attributed to Jiankui Li.

23 records · Page 2Linked to original sources

arXiv2012

On (m, n)-derivations of Some Algebras

Let $\mathcal{A}$ be a unital algebra, $δ$ be a linear mapping from $\mathcal{A}$ into itself and $m$, $n$ be fixed integers. We call $δ$ an (\textit{m, n})-derivable mapping at $Z$, if $mδ(AB)+nδ(BA)=mδ(A)B+mAδ(B)+nδ(B)A+nBδ(A)$ for all $A, B\in \mathcal{A}$ with $AB=Z$. In this paper, (\textit{m, n})-derivable mappings at 0 (resp. $I_\mathcal{A}\oplus0$, $I$) on generalized matrix algebras are characterized. We also study (\textit{m, n})-derivable mappings at 0 on CSL algebras. We reveal the relationship between this kind of mappings with Lie derivations, Jordan derivations and derivations.

math.OA↗

arXiv2011

Mappings on some reflexive algebras characterized by action on zero products or Jordan zero products

Let $\mathcal{L}$ be a subspace lattice on a Banach space $X$ and let $δ:\mathrm{Alg}\mathcal{L}\rightarrow B(X)$ be a linear mapping. If $\vee\{L\in \mathcal{L}: L_-\nsupseteq L\}=X$ or $\wedge\{L_-:L\in \mathcal{L}, L_-\nsupseteq L\}=(0)$, we show that the following three conditions are equivalent: (1) $δ(AB)=δ(A)B+Aδ(B)$ whenever $AB=0$; (2) $δ(AB+BA)=δ(A)B+Aδ(B)+δ(B)A+Bδ(A)$ whenever $AB+BA=0$; (3) $δ$ is a generalized derivation and $δ(I)\in (\mathrm{Alg}\mathcal{L})^\prime$. If $\vee\{L\in \mathcal{L}: L_-\nsupseteq L\}=X$ or $\wedge\{L_-:L\in \mathcal{L}, L_-\nsupseteq L\}=(0)$ and $δ$ satisfies $δ(AB+BA)=δ(A)B+Aδ(B)+δ(B)A+Bδ(A)$ whenever $AB=0$, we obtain that $δ$ is a generalized derivation and $δ(I)A\in(\mathrm{Alg}\mathcal{L})^\prime$ for every $A\in \mathrm{Alg}\mathcal{L}$. We also prove that if $\vee\{L\in \mathcal{L}: L_-\nsupseteq L\}=X$ and $\wedge\{L_-:L\in \mathcal{L}, L_-\nsupseteq L\}=(0)$, then $δ$ is a local generalized derivation if and only if $δ$ is a generalized derivation.

math.FA↗

arXiv2011

On (m,n,l)-Jordan Centralizers of Some Algebras

Let $\mathcal{A}$ be a unital algebra over the complex field $\mathbb{C}$. A linear mapping $δ$ from $\mathcal{A}$ into itself is called a weak (\textit{m,n,l})-Jordan centralizer if $(m+n+l)δ(A^2)-mδ(A)A-nAδ(A)-lAδ(I)A\in \mathbb{C}I$ for every $A\in \mathcal{A}$, where $m\geq0, n\geq0, l\geq0$ are fixed integers with $m+n+l\neq 0$. In this paper, we study weak (\textit{m,n,l})-Jordan centralizer on generalized matrix algebras and some reflexive algebras alg$\mathcal{L}$, where $\mathcal{L}$ is CSL or satisfies $\vee\{L: L\in \mathcal{J}(\mathcal{L})\}=X$ or $\wedge\{L_-: L\in \mathcal{J}(\mathcal{L})\}=(0)$, and prove that each weak (\textit{m,n,l})-Jordan centralizer of these algebras is a centralizer when $m+l\geq1$ and $n+l\geq1$.

math.OA↗

arXiv2008

Reduced free products of unital AH algebras and Blackadar and Kirchberg's MF algebras

In the paper, we prove that reduced free products of unital AH algebras with respect to given faithful tracial states, in the sense of Voiculescu, are Blackadar and Kirhcberg's MF algebras. We also show that the reduced free products of unital AH algebras with respect to given faithful tracial states, under mild conditions, are not quasidiagonal. Therefore we conclude, for a large class of AH algebras, the Brown-Douglas-Fillmore extension semigroups of the reduced free products of these AH algebras with respect to given faithful tracial states are not groups. Our result is based on Haagerup and Thorbjørsen's work on the reduced C$^*$-algebras of free groups.

math.OA↗

arXiv2004

Absolutely Continuous Representations and a Kaplansky Density Theorem for Free Semigroup Algebras

We introduce notions of absolutely continuous functionals and representations on the non-commutative disk algebra $A_n$. Absolutely continuous functionals are used to help identify the type L part of the free semigroup algebra associated to a $*$-extendible representation $σ$. A $*$-extendible representation of $A_n$ is ``regular'' if the absolutely continuous part coincides with the type L part. All known examples are regular. Absolutely continuous functionals are intimately related to maps which intertwine a given $*$-extendible representation with the left regular representation. A simple application of these ideas extends reflexivity and hyper-reflexivity results. Moreover the use of absolute continuity is a crucial device for establishing a density theorem which states that the unit ball of $σ(A_n)$ is weak-$*$ dense in the unit ball of the associated free semigroup algebra if and only if $σ$ is regular. We provide some explicit constructions related to the density theorem for specific representations. A notion of singular functionals is also defined, and every functional decomposes in a canonical way into the sum of its absolutely continuous and singular parts.

math.OA↗