An Identity Related to Derivations of Standard Operator Algebras and Semisimple H*-Algebra¹

In this paper we prove the following result. Let X be a real or complex Banach space, let L (X) be the algebra of all bounded linear operators on X, and let <img border=0 width=120 height=19 src="http:/fbpe/img/cubo/v12n1/img13.jpg">be a standard operator algebra. Suppose <img bor...

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Autores principales:	Kosi-Ulbl,Irena, Vukman,Joso
Lenguaje:	English
Publicado:	Universidad de La Frontera. Departamento de Matemática y Estadística. 2010
Materias:	Prime ring semiprime ring, Banach space standard operator algebra H*-algebra derivation Jordan derivation
Acceso en línea:	http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0719-06462010000100009
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id	oai:scielo:S0719-06462010000100009
record_format	dspace
institution	Scielo Chile
collection	Scielo Chile
language	English
topic	Prime ring semiprime ring, Banach space standard operator algebra H*-algebra derivation Jordan derivation
spellingShingle	Prime ring semiprime ring, Banach space standard operator algebra H-algebra derivation Jordan derivation Kosi-Ulbl,Irena Vukman,Joso An Identity Related to Derivations of Standard Operator Algebras and Semisimple H-Algebra¹
description	In this paper we prove the following result. Let X be a real or complex Banach space, let L (X) be the algebra of all bounded linear operators on X, and let <img border=0 width=120 height=19 src="http:/fbpe/img/cubo/v12n1/img13.jpg">be a standard operator algebra. Suppose <img border=0 width=150 height=23 src="http:/fbpe/img/cubo/v12n1/img16.jpg">is a linear mapping satisfying the relation <img border=0 width=300 height=27 src="http:/fbpe/img/cubo/v12n1/img14.jpg">. In this case D is of the form <img border=0 width=230 height=24 src="http:/fbpe/img/cubo/v12n1/img15.jpg">and some <img border=0 width=108 height=19 src="http:/fbpe/img/cubo/v12n1/img17.jpg">, which means that D is a linear derivation. In particular, D is continuous. We apply this result, which generalizes a classical result of Chernoff, to semisimple H- algebras. This research has been motivated by the work of Herstein [4], Chernoff [2] and Molnár [5] and is a continuation of our recent work [8] and [9] .Throughout, R will represent an associative ring. Given an integer <img border=0 width=38 height=14 src="http:/fbpe/img/cubo/v12n1/img18.jpg">, a ring R is said to be n−torsion free, if for <img border=0 width=92 height=17 src="http:/fbpe/img/cubo/v12n1/img19.jpg">implies x = 0. Recall that a ring R is prime if for a, b <img border=0 width=13 height=14 src="http:/fbpe/img/cubo/v12n1/img20.jpg">R, aRb = (0) implies that either a = 0 or b = 0, and is semiprime in case aRa = (0) implies a = 0. Let A be an algebra over the real or complex field and let B be a subalgebra of A. A linear mapping D : B <img border=0 width=15 height=15 src="http:/fbpe/img/cubo/v12n1/img21.jpg">A is called a linear derivation in case <img border=0 width=142 height=13 src="http:/fbpe/img/cubo/v12n1/img23.jpg">holds for all pairs x, y <img border=0 width=13 height=14 src="http:/fbpe/img/cubo/v12n1/img20.jpg">R. In case we have a ring R an additive mapping D : R<img border=0 width=15 height=15 src="http:/fbpe/img/cubo/v12n1/img21.jpg"> R is called a derivation if <img border=0 width=142 height=13 src="http:/fbpe/img/cubo/v12n1/img23.jpg">holds for all pairs x, y <img border=0 width=13 height=14 src="http:/fbpe/img/cubo/v12n1/img20.jpg">R and is called a Jordan derivation in case <img border=0 width=138 height=17 src="http:/fbpe/img/cubo/v12n1/img24.jpg">is fulfilled for all x R. A derivation D is inner in case there exists a <img border=0 width=13 height=14 src="http:/fbpe/img/cubo/v12n1/img20.jpg">R, such that <img border=0 width=98 height=13 src="http:/fbpe/img/cubo/v12n1/img25.jpg">holds for all x <img border=0 width=13 height=14 src="http:/fbpe/img/cubo/v12n1/img20.jpg">R. Every derivation is a Jordan derivation. The converse is in general not true. A classical result of Herstein [4] asserts that any Jordan derivation on a prime ring of characteristic different from two is a derivation. Cusack [3] generalized Herstein’s result to 2 -torsion free semiprime rings. Let us recall that a semisimple H-algebra is a semisimple Banach * -algebra whose norm is a Hilbert space norm such that <img border=0 width=154 height=17 src="http:/fbpe/img/cubo/v12n1/img26.jpg">is fulfilled for all x, y, z <img border=0 width=13 height=14 src="http:/fbpe/img/cubo/v12n1/img20.jpg">A (see [1]). Let X be a real or complex Banach space and let L(X) and F(X) denote the algebra of all bounded linear operators on X and the ideal of all finite rank operators in L(X), respectively. An algebra A(X) <img border=0 width=12 height=14 src="http:/fbpe/img/cubo/v12n1/img22.jpg">L(X) is said to be standard in case F(X) <img border=0 width=12 height=14 src="http:/fbpe/img/cubo/v12n1/img22.jpg">A(X). Let us point out that any standard algebra is prime, which is a consequence of Hahn-Banach theorem.
author	Kosi-Ulbl,Irena Vukman,Joso
author_facet	Kosi-Ulbl,Irena Vukman,Joso
author_sort	Kosi-Ulbl,Irena
title	An Identity Related to Derivations of Standard Operator Algebras and Semisimple H*-Algebra¹
title_short	An Identity Related to Derivations of Standard Operator Algebras and Semisimple H*-Algebra¹
title_full	An Identity Related to Derivations of Standard Operator Algebras and Semisimple H*-Algebra¹
title_fullStr	An Identity Related to Derivations of Standard Operator Algebras and Semisimple H*-Algebra¹
title_full_unstemmed	An Identity Related to Derivations of Standard Operator Algebras and Semisimple H*-Algebra¹
title_sort	identity related to derivations of standard operator algebras and semisimple h*-algebra¹
publisher	Universidad de La Frontera. Departamento de Matemática y Estadística.
publishDate	2010
url	http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0719-06462010000100009
work_keys_str_mv	AT kosiulblirena anidentityrelatedtoderivationsofstandardoperatoralgebrasandsemisimplehalgebra1 AT vukmanjoso anidentityrelatedtoderivationsofstandardoperatoralgebrasandsemisimplehalgebra1 AT kosiulblirena identityrelatedtoderivationsofstandardoperatoralgebrasandsemisimplehalgebra1 AT vukmanjoso identityrelatedtoderivationsofstandardoperatoralgebrasandsemisimplehalgebra1
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spelling	oai:scielo:S0719-064620100001000092018-10-08An Identity Related to Derivations of Standard Operator Algebras and Semisimple H-Algebra¹Kosi-Ulbl,IrenaVukman,Joso Prime ring semiprime ring, Banach space standard operator algebra H-algebra derivation Jordan derivation In this paper we prove the following result. Let X be a real or complex Banach space, let L (X) be the algebra of all bounded linear operators on X, and let <img border=0 width=120 height=19 src="http:/fbpe/img/cubo/v12n1/img13.jpg">be a standard operator algebra. Suppose <img border=0 width=150 height=23 src="http:/fbpe/img/cubo/v12n1/img16.jpg">is a linear mapping satisfying the relation <img border=0 width=300 height=27 src="http:/fbpe/img/cubo/v12n1/img14.jpg">. In this case D is of the form <img border=0 width=230 height=24 src="http:/fbpe/img/cubo/v12n1/img15.jpg">and some <img border=0 width=108 height=19 src="http:/fbpe/img/cubo/v12n1/img17.jpg">, which means that D is a linear derivation. In particular, D is continuous. We apply this result, which generalizes a classical result of Chernoff, to semisimple H- algebras. This research has been motivated by the work of Herstein [4], Chernoff [2] and Molnár [5] and is a continuation of our recent work [8] and [9] .Throughout, R will represent an associative ring. Given an integer <img border=0 width=38 height=14 src="http:/fbpe/img/cubo/v12n1/img18.jpg">, a ring R is said to be n−torsion free, if for <img border=0 width=92 height=17 src="http:/fbpe/img/cubo/v12n1/img19.jpg">implies x = 0. Recall that a ring R is prime if for a, b <img border=0 width=13 height=14 src="http:/fbpe/img/cubo/v12n1/img20.jpg">R, aRb = (0) implies that either a = 0 or b = 0, and is semiprime in case aRa = (0) implies a = 0. Let A be an algebra over the real or complex field and let B be a subalgebra of A. A linear mapping D : B <img border=0 width=15 height=15 src="http:/fbpe/img/cubo/v12n1/img21.jpg">A is called a linear derivation in case <img border=0 width=142 height=13 src="http:/fbpe/img/cubo/v12n1/img23.jpg">holds for all pairs x, y <img border=0 width=13 height=14 src="http:/fbpe/img/cubo/v12n1/img20.jpg">R. In case we have a ring R an additive mapping D : R<img border=0 width=15 height=15 src="http:/fbpe/img/cubo/v12n1/img21.jpg"> R is called a derivation if <img border=0 width=142 height=13 src="http:/fbpe/img/cubo/v12n1/img23.jpg">holds for all pairs x, y <img border=0 width=13 height=14 src="http:/fbpe/img/cubo/v12n1/img20.jpg">R and is called a Jordan derivation in case <img border=0 width=138 height=17 src="http:/fbpe/img/cubo/v12n1/img24.jpg">is fulfilled for all x R. A derivation D is inner in case there exists a <img border=0 width=13 height=14 src="http:/fbpe/img/cubo/v12n1/img20.jpg">R, such that <img border=0 width=98 height=13 src="http:/fbpe/img/cubo/v12n1/img25.jpg">holds for all x <img border=0 width=13 height=14 src="http:/fbpe/img/cubo/v12n1/img20.jpg">R. Every derivation is a Jordan derivation. The converse is in general not true. A classical result of Herstein [4] asserts that any Jordan derivation on a prime ring of characteristic different from two is a derivation. Cusack [3] generalized Herstein’s result to 2 -torsion free semiprime rings. Let us recall that a semisimple H-algebra is a semisimple Banach * -algebra whose norm is a Hilbert space norm such that <img border=0 width=154 height=17 src="http:/fbpe/img/cubo/v12n1/img26.jpg">is fulfilled for all x, y, z <img border=0 width=13 height=14 src="http:/fbpe/img/cubo/v12n1/img20.jpg">A (see [1]). Let X be a real or complex Banach space and let L(X) and F(X) denote the algebra of all bounded linear operators on X and the ideal of all finite rank operators in L(X), respectively. An algebra A(X) <img border=0 width=12 height=14 src="http:/fbpe/img/cubo/v12n1/img22.jpg">L(X) is said to be standard in case F(X) <img border=0 width=12 height=14 src="http:/fbpe/img/cubo/v12n1/img22.jpg">A(X). Let us point out that any standard algebra is prime, which is a consequence of Hahn-Banach theorem.info:eu-repo/semantics/openAccessUniversidad de La Frontera. Departamento de Matemática y Estadística.Cubo (Temuco) v.12 n.1 20102010-01-01text/htmlhttp://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0719-06462010000100009en10.4067/S0719-06462010000100009

An Identity Related to Derivations of Standard Operator Algebras and Semisimple H*-Algebra¹

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