Derivations with a hereditary domain, II

A. Villena

Studia Mathematica (1998)

  • Volume: 130, Issue: 3, page 275-291
  • ISSN: 0039-3223

Abstract

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The nilpotency of the separating subspace of an everywhere defined derivation on a Banach algebra is an intriguing question which remains still unsolved, even for commutative Banach algebras. On the other hand, closability of partially defined derivations on Banach algebras is a fundamental problem motivated by the study of time evolution of quantum systems. We show that the separating subspace S(D) of a Jordan derivation defined on a subalgebra B of a complex Banach algebra A satisfies B [ B ∩ S ( D ) ] B ⊂ R a d B ( A ) provided that BAB ⊂ A and d i m ( R a d J ( A ) ∩ ⋂ n = 1 ∞ B n ) < ∞ , where R a d J ( A ) and R a d B ( A ) denote the Jacobson and the Baer radicals of A respectively. From this we deduce the closability of partially defined derivations on complex semiprime Banach algebras with appropriate domains. The result applies to several relevant classes of algebras.

How to cite

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Villena, A.. "Derivations with a hereditary domain, II." Studia Mathematica 130.3 (1998): 275-291. <http://eudml.org/doc/216558>.

@article{Villena1998,
abstract = {The nilpotency of the separating subspace of an everywhere defined derivation on a Banach algebra is an intriguing question which remains still unsolved, even for commutative Banach algebras. On the other hand, closability of partially defined derivations on Banach algebras is a fundamental problem motivated by the study of time evolution of quantum systems. We show that the separating subspace S(D) of a Jordan derivation defined on a subalgebra B of a complex Banach algebra A satisfies $B[B ∩ S(D)]B ⊂ Rad_B(A)$ provided that BAB ⊂ A and $dim(Rad_J(A) ∩ ⋂_\{n=1\}^∞ B^n) < ∞$, where $Rad_J(A)$ and $Rad_B(A)$ denote the Jacobson and the Baer radicals of A respectively. From this we deduce the closability of partially defined derivations on complex semiprime Banach algebras with appropriate domains. The result applies to several relevant classes of algebras.},
author = {Villena, A.},
journal = {Studia Mathematica},
keywords = {Jordan derivation; Banach algebra; Jacobson radical; Baer radical; separating subspace},
language = {eng},
number = {3},
pages = {275-291},
title = {Derivations with a hereditary domain, II},
url = {http://eudml.org/doc/216558},
volume = {130},
year = {1998},
}

TY - JOUR
AU - Villena, A.
TI - Derivations with a hereditary domain, II
JO - Studia Mathematica
PY - 1998
VL - 130
IS - 3
SP - 275
EP - 291
AB - The nilpotency of the separating subspace of an everywhere defined derivation on a Banach algebra is an intriguing question which remains still unsolved, even for commutative Banach algebras. On the other hand, closability of partially defined derivations on Banach algebras is a fundamental problem motivated by the study of time evolution of quantum systems. We show that the separating subspace S(D) of a Jordan derivation defined on a subalgebra B of a complex Banach algebra A satisfies $B[B ∩ S(D)]B ⊂ Rad_B(A)$ provided that BAB ⊂ A and $dim(Rad_J(A) ∩ ⋂_{n=1}^∞ B^n) < ∞$, where $Rad_J(A)$ and $Rad_B(A)$ denote the Jacobson and the Baer radicals of A respectively. From this we deduce the closability of partially defined derivations on complex semiprime Banach algebras with appropriate domains. The result applies to several relevant classes of algebras.
LA - eng
KW - Jordan derivation; Banach algebra; Jacobson radical; Baer radical; separating subspace
UR - http://eudml.org/doc/216558
ER -

References

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