An identity with generalized derivations on Lie ideals, right ideals and Banach algebras

Vincenzo de Filippis; Giovanni Scudo; Mohammad S. Tammam El-Sayiad

Czechoslovak Mathematical Journal (2012)

  • Volume: 62, Issue: 2, page 453-468
  • ISSN: 0011-4642

Abstract

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Let R be a prime ring of characteristic different from 2 , U the Utumi quotient ring of R , C = Z ( U ) the extended centroid of R , L a non-central Lie ideal of R , F a non-zero generalized derivation of R . Suppose that [ F ( u ) , u ] F ( u ) = 0 for all u ∈ L , then one of the following holds: (1) there exists α ∈ C such that F ( x ) = α x for all x ∈ R ; (2) R satisfies the standard identity s 4 and there exist a ∈ U and α ∈ C such that F ( x ) = a x + x a + α x for all x ∈ R . We also extend the result to the one-sided case. Finally, as an application we obtain some range inclusion results of continuous or spectrally bounded generalized derivations on Banach algebras.

How to cite

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de Filippis, Vincenzo, Scudo, Giovanni, and Tammam El-Sayiad, Mohammad S.. "An identity with generalized derivations on Lie ideals, right ideals and Banach algebras." Czechoslovak Mathematical Journal 62.2 (2012): 453-468. <http://eudml.org/doc/246544>.

@article{deFilippis2012,
abstract = {Let $R$ be a prime ring of characteristic different from $2$, $U$ the Utumi quotient ring of $R$, $C=Z(U)$ the extended centroid of $R$, $L$ a non-central Lie ideal of $R$, $F$ a non-zero generalized derivation of $R$. Suppose that $[F(u),u]F(u)=0$ for all $u\in L$, then one of the following holds: (1) there exists $\alpha \in C$ such that $F(x)=\alpha x$ for all $x\in R$; (2) $R$ satisfies the standard identity $s_4$ and there exist $a\in U$ and $\alpha \in C$ such that $F(x)=ax+xa+\alpha x$ for all $x\in R$. We also extend the result to the one-sided case. Finally, as an application we obtain some range inclusion results of continuous or spectrally bounded generalized derivations on Banach algebras.},
author = {de Filippis, Vincenzo, Scudo, Giovanni, Tammam El-Sayiad, Mohammad S.},
journal = {Czechoslovak Mathematical Journal},
keywords = {prime rings; differential identities; generalized derivations; Banach algebra; prime rings; differential identities; generalized derivations; Banach algebras; Lie ideals; additive maps},
language = {eng},
number = {2},
pages = {453-468},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {An identity with generalized derivations on Lie ideals, right ideals and Banach algebras},
url = {http://eudml.org/doc/246544},
volume = {62},
year = {2012},
}

TY - JOUR
AU - de Filippis, Vincenzo
AU - Scudo, Giovanni
AU - Tammam El-Sayiad, Mohammad S.
TI - An identity with generalized derivations on Lie ideals, right ideals and Banach algebras
JO - Czechoslovak Mathematical Journal
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 2
SP - 453
EP - 468
AB - Let $R$ be a prime ring of characteristic different from $2$, $U$ the Utumi quotient ring of $R$, $C=Z(U)$ the extended centroid of $R$, $L$ a non-central Lie ideal of $R$, $F$ a non-zero generalized derivation of $R$. Suppose that $[F(u),u]F(u)=0$ for all $u\in L$, then one of the following holds: (1) there exists $\alpha \in C$ such that $F(x)=\alpha x$ for all $x\in R$; (2) $R$ satisfies the standard identity $s_4$ and there exist $a\in U$ and $\alpha \in C$ such that $F(x)=ax+xa+\alpha x$ for all $x\in R$. We also extend the result to the one-sided case. Finally, as an application we obtain some range inclusion results of continuous or spectrally bounded generalized derivations on Banach algebras.
LA - eng
KW - prime rings; differential identities; generalized derivations; Banach algebra; prime rings; differential identities; generalized derivations; Banach algebras; Lie ideals; additive maps
UR - http://eudml.org/doc/246544
ER -

References

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