# Derivations with a hereditary domain, II

Studia Mathematica (1998)

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

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topVillena, 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 -

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