Let $\mathcal{A}$ be a unital algebra and $\mathcal{M}$ be a unital $\mathcal{A}$-bimodule. A characterization of generalized derivations and generalized Jordan derivations from $\mathcal{A}$ into $\mathcal{M}$, through zero products or zero Jordan products, is given. Suppose that $\mathcal{M}$ is a unital left $\mathcal{A}$-module. It is investigated when a linear mapping from $\mathcal{A}$ into $\mathcal{M}$ is a Jordan left derivation under certain conditions. It is also studied whether an algebra with a nontrivial idempotent is zero Jordan product determined, and Jordan homomorphisms, Lie homomorphisms and Lie derivations on zero Jordan product determined algebras are characterized.

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Characterizations of linear mappings through zero products or zero jordan products

A linear mapping 
$$\delta $$

 from a 
$${*}$$

-algebra 
$$\mathcal {A}$$

 into a 
$${*}$$

-
$$\mathcal {A}$$

-bimodule 
$$\mathcal {M}$$

 is a 
$${*}$$

-derivable mapping at 
$$G\in \mathcal {A}$$

 if 
$$A\delta (B)^{*}+\delta (A)B=\delta (G)$$

 for each A, B in 
$$\mathcal {A}$$

 with 
$$AB^{*}=G$$

. We prove that every (continuous) 
$${*}$$

-derivable mapping at G from a (unital 
$$C^{*}$$

-algebra) factor von Neumann algebra into its Banach 
$${*}$$

-bimodule is a 
$${*}$$

-derivation if and only if G is a left separating point. A linear mapping 
$$\delta $$

 from a 
$${*}$$

-algebra 
$$\mathcal {A}$$

 into a 
$${*}$$

-left 
$$\mathcal {A}$$

-module 
$$\mathcal {M}$$

 is a 
$${*}$$

-left derivable mapping at 
$$G\in \mathcal {A}$$

 if 
$$A\delta (B)^{*}+B\delta (A)=\delta (G)$$

 for each A, B in 
$$\mathcal {A}$$

 with 
$$AB^{*}=G$$

. We prove that every continuous 
$${*}$$

-left derivable mapping at a left separating point from a unital 
$$C^{*}$$

-algebra or von Neumann algebra into its Banach 
$${*}$$

-left 
$$\mathcal {A}$$

-module is identical with zero under certain conditions.

Characterizations of $${*}$$∗ and $${*}$$∗ -left derivable mappings on some algebras

A linear mapping ϕ from an algebra 
$${\cal A}$$

 into its bimodule 
$${\cal M}$$

 is called a centralizable mapping at G ∈ 
$${\cal A}$$

 if ϕ(AB) = ϕ(A)B = Aϕ(B) for each A and B in 
$${\cal A}$$

 with AB = G. In this paper, we prove that if 
$${\cal M}$$

 is a von Neumann algebra without direct summands of type I1 and type II, 
$${\cal A}$$

 is a *-subalgebra with 
$${\cal M}$$

 ⊆ 
$${\cal A}$$

 ⊆ LS (
$${\cal M}$$

) and G is a fixed element in 
$${\cal A}$$

, then every continuous (with respect to the local measure topology t(
$${\cal M}$$

)) centralizable mapping at G from 
$${\cal A}$$

 into 
$${\cal M}$$

 is a centralizer.

Characterizations of Centralizable Mappings on Algebras of Locally Measurable Operators

Generalized derivable mappings and centralizable mappings on B(X)

Abstract The main goal of this paper is to characterize Jordan two-sided centralizers, Jordan centralizers and related maps on triangular rings without identity. As an application of our main theorem, we characterize Jordan generalized derivations on triangular rings. Precisely, we prove that every Jordan generalized derivation on a triangular ring is a two-sided generalized derivation. As consequences, and apart from proving the other results, many known theorems can be either generalized or deduced. 

Characterization of Jordan two-sided centralizers and related maps on triangular rings

A linear mapping $\phi$ on an algebra $\mathcal{A}$ is called a centralizable mapping at $G\in\mathcal{A}$ if $\phi(AB)=\phi(A)B=A\phi(B)$ for each $A$ and $B$ in $\mathcal{A}$ with $AB=G$, and $\phi$ is called a derivable mapping at $G\in\mathcal{A}$ if $\phi(AB)=\phi(A)B+A\phi(B)$ 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.

Wenhua Qian

Papers

Characterizations of centralizers and derivations on some algebras

Characterizations of centralizers and derivations on some algebras