On left Jordan derivations of rings and Banach algebras

Abstract

It is well known that there are no nonzero linear derivations on complex commutative semisimple Banach algebras. In this paper we prove the following extension of this result. Let ▫$A$▫ be a complex semisimple Banach algebra and let ▫$D: A \to A$▫ be a linear mapping satisfying the relation ▫$D(x^2) = 2xD(x)$▫ for all ▫$x \in R$▫. In this case ▫$D = 0$▫. Throughout, ▫$R$▫ will represent an associative ring with center Z(R). A ring ▫$R$▫ is ▫$n$▫-torsion free, where ▫$n > 1$▫ is an integer, if ▫$nx = 0$▫, ▫$x \in R$▫ implies ▫$x = 0$▫. As usual the commutator ▫$xy - yx$▫ will be denoted by ▫$[x,y]$▫. We shall use the commutator identities ▫$[xy,z] = [x,z]y + x[y,z]$▫ and ▫$[x,yz] = [x,y]z + y[x,z]$▫ for all ▫$x,y,z \in R$▫. Recall that a ring ▫$R$▫ is prime if for ▫$a,b \in R$▫, ▫$aRb = (0)$▫ implies that either ▫$a=0$▫ or ▫$b=0$▫, and is semiprime in case ▫$aRa = (0)$▫ implies that ▫$a=0$▫. An additive mapping ▫$D$▫ is called a derivation if ▫$D(xy) = D(x)y + xD(y)$▫ holds for all pairs ▫$x,y \in R$▫, and is called a Jordan derivation in case ▫$D(x^2) = D(x)x + xD(x)$▫ is fulfilled for all ▫$x \in R▫$. Obviously, any derivation is a Jordan derivation. The converse is in general not true. Herstein has proved that any Jordan derivation on a 2-torsion free prime ring is a derivation. Cusack has generalized Herstein's result to 2-torsion free semiprime rings. An additive mapping ▫$D: R \to R$▫ is called a left derivation if ▫$D(xy) = yD(x) + xD(y)$▫ holds for all pairs ▫$x,y \in R$▫ and is called a left Jordan derivation (or Jordan left derivation) in case ▫$D(x^2) = 2xD(x)$▫ is fulfilled for all ▫$x \in R$▫. In this paper by a Banach algebra we mean a Banach algebra over the complex field.

Keywords

matematika;prakolobar;polprakolobar;Banachova algebra;odvajanje;jordansko odvajanje;levo odvajanje;levo jordansko odvajanje;komutirajoče preslikave;centralizirajoče preslikave;mathematics;prime ring;semiprime ring;Banach algebra;derivation;Jordan derivation;left derivation;left Jordan derivation;commuting mappings;centralizing mapping;

Data

Language:	English
Year of publishing:	2008
Typology:	1.01 - Original Scientific Article
Organization:	UM FNM - Faculty of Natural Sciences and Mathematics
UDC:	512.552.34
COBISS:	14792537
ISSN:	0001-9054
Views:	91
Downloads:	19
Average score:	0 (0 votes)
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Other data

Secondary language:	English
Secondary keywords:	matematika;prakolobar;polprakolobar;Banachova algebra;odvajanje;jordansko odvajanje;levo odvajanje;levo jordansko odvajanje;komutirajoče preslikave;centralizirajoče preslikave;
URN:	URN:SI:UM:
Type (COBISS):	Not categorized
Pages:	str. 260-266
Volume:	ǂVol. ǂ75
Issue:	ǂno. ǂ3
Chronology:	2008
ID:	8724006