scholarly journals Characterization of Multiplicative Lie Triple Derivations on Rings

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Xiaofei Qi
Keyword(s):  
Von Neumann Algebras ◽  
Idempotent Element ◽  
Prime Rings ◽  
Von Neumann ◽  
Mild Conditions ◽  
Lie Triple Derivation

LetRbe a ring having unit 1. Denote byZRthe center ofR. Assume that the characteristic ofRis not 2 and there is an idempotent elemente∈Rsuch thataRe=0⇒a=0  and  aR1-e=0⇒a=0. It is shown that, under some mild conditions, a mapL:R→Ris a multiplicative Lie triple derivation if and only ifLx=δx+hxfor allx∈R, whereδ:R→Ris an additive derivation andh:R→ZRis a map satisfyingha,b,c=0for alla,b,c∈R. As applications, all Lie (triple) derivations on prime rings and von Neumann algebras are characterized, which generalize some known results.

scholarly journals Dual Banach algebras: Connes-amenability, normal, virtual diagonals, and injectivity of the predual bimodule

2004 ◽  
Vol 95 (1) ◽  
pp. 124 ◽  
Author(s):  
Volker Runde
Keyword(s):  
Abelian Subgroup ◽  
Von Neumann Algebras ◽  
Banach Algebras ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Von Neumann ◽  
Connes Amenability ◽  
Definition Of ◽  
Dual Banach Algebra

Let $\mathcal A$ be a dual Banach algebra with predual $\mathcal A_*$ and consider the following assertions: (A) $\mathcal A$ is Connes-amenable; (B) $\mathcal A$ has a normal, virtual diagonal; (C) $\mathcal A_*$ is an injective $\mathcal A$-bimodule. For general $\mathcal A$, all that is known is that (B) implies (A) whereas, for von Neumann algebras, (A), (B), and (C) are equivalent. We show that (C) always implies (B) whereas the converse is false for $\mathcal A = M(G)$ where $G$ is an infinite, locally compact group. Furthermore, we present partial solutions towards a characterization of (A) and (B) for $\mathcal A = B(G)$ in terms of $G$: For amenable, discrete $G$ as well as for certain compact $G$, they are equivalent to $G$ having an abelian subgroup of finite index. The question of whether or not (A) and (B) are always equivalent remains open. However, we introduce a modified definition of a normal, virtual diagonal and, using this modified definition, characterize the Connes-amenable, dual Banach algebras through the existence of an appropriate notion of virtual diagonal.

A characterization of von Neumann algebras

2010 ◽  
Vol 76 (3-4) ◽  
pp. 561-580
Author(s):  
János Kristóf
Keyword(s):  
Von Neumann Algebras ◽  
Von Neumann
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